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Essays on long-term mortality and interest rate risk

de Kort, J.P.

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Essays on long-term mortality and interest rate risk

Jan de Kort

Essays on long-term m

ortality and interest rate risk

Jan de Kort

UITNODIGING

j.p. de kort

E S S AY S O N L O N G - T E R M M O RTA L I T Y A N D I N T E R E S TR AT E R I S K

ISBN 978-94-6332-193-8

J.P. de Kort: Essays on long-term mortality and interest rate risk, Academisch proef-schrift, February 15, 2017

This work has been partially funded by Netspar (Center for Studies on Pen-sions, Ageing and Retirement).

E S S AY S O N L O N G - T E R M M O RTA L I T Y A N DI N T E R E S T R AT E R I S K

academisch proefschrift

ter verkrijging van de graad van doctoraan de Universiteit van Amsterdamop gezag van de Rector Magnificus

prof. dr. ir. K.I.J. Maexten overstaan van een door het College voor Promoties ingestelde

commissie, in het openbaar te verdedigen in de Agnietenkapelop vrijdag 7 juli 2017, te 10.00 uur

door

Johan Pieter de Kort

geboren te Stein.

P R O M O T I E C O M M I S S I E

promotor: Prof. dr. ir. M.H. Vellekoop, Universiteit van Amsterdam

copromotor: Prof. dr. R.J.A. Laeven, Universiteit van Amsterdam

overige leden: Prof. dr. H.P. Boswijk, Universiteit van Amsterdam

Prof. dr. F.R. Kleibergen, Universiteit van Amsterdam

Prof. dr. A.A.J. Pelsser, Universiteit Maastricht

Prof. dr. P.J.C. Spreij, Universiteit van Amsterdam

Prof. dr. M. Stadje, Universität Ulm

Faculteit Economie en Bedrijfskunde

A C K N O W L E D G M E N T S

This thesis is the result of five-and-a-half years of part-time research at theQuantitative Economics department of the University of Amsterdam. Prepar-ing this dissertation has been an enjoyable experience. For this I am indebtedto a number of people.

First, I would like thank my supervisor, Michel Vellekoop, for his encourage-ment and support. I am grateful for his generous advice, and for his carefulreading of many drafts.

Netspar is thanked for financial support. My gratitude also goes to PeterBoswijk, Frank Kleibergen, Roger Laeven, Antoon Pelsser, Peter Spreij andMitja Stadje for volunteering to serve on my thesis committee.

I also thank Frank van Berkum, Zhenzhen Fan and Jitze Hooijsma for sharingan office in the J/K building and for the pleasurable atmosphere.

Special thanks are due to my colleagues at SNS Reaal and VIVAT Verzekerin-gen. In particular, to Sandra Muijs for taking me onboard the Modelling team,and to Erwin Charlier, Aart Konijnenberg, Robby Koreman, Rianne Legerstee,Maurice Lutterot, Patrick Quodbach, Patrick Stolk and Gijsbert de Wolff. Myappreciation also goes to Koos Gubbels, Michel van Meer and Ramon vanden Akker for the discussions in our reading group, and to André Ploegh forsimilar conversations in Mijas and on many other occasions.

Lastly, and most importantly, I thank my family, Annelies, Daan and my latefather Hans, to whom I dedicate this work.

v

C O N T E N T S

1 introduction 1

1.1 On Part I 1

1.2 On Part II 3

i longevity risk 7

2 optimal consumption strategies in markets with longevity

risk 9

2.1 Introduction 9

2.2 The model for the economy 12

2.3 The optimal investment and consumption problem 17

2.4 The Laplace transform of a CIR process 19

2.5 The optimal consumption and investment strategy 29

2.6 Economic analysis of the model 40

2.7 Conclusion 43

2.8 Proofs 44

ii long-term interest rate risk 53

3 convergence of the long-term interest rate in hjm mod-els 55

3.1 The HJM framework 55

3.2 Almost sure convergence on a finite horizon 59

3.3 Convergence on an infinite horizon 61

3.3.1 Ucp convergence of long-term rates 61

3.3.2 Almost sure convergence of long-term rates 65

3.3.3 S1 convergence of long-term rates 67

3.4 Proofs 76

4 on the long rate in factor models of the term struc-ture 83

4.1 Introduction 83

4.2 The long rate in factor models 85

4.3 The two-dimensional case 90

4.4 The higher-dimensional cases 95

5 term structure extrapolation and asymptotic forward

rates 101

5.1 Introduction 101

vii

viii contents

5.2 Term structure interpolation 104

5.3 A class of interpolating functions 106

5.4 The constrained optimization problem 109

5.5 The unconstrained optimization problem 111

5.6 An alternative method 112

5.7 Estimates of asymptotic forward rates 118

5.8 Conclusions 121

5.9 Proofs 122

samenvatting (summary in dutch) 133

bibliography 135

L I S T O F F I G U R E S

Figure 5.6.1 Scaled functions −W(t, u)/(αu) and W(t, u) for α =α = 1/10 and −W(−t, u)/(αu) and W(−t, u) for α =α = 1/2. 114

Figure 5.7.1 Inter- and extrapolation of Datastream swap data up tomaturity 20 on March 29, 2013. 117

Figure 5.7.2 Inter- and extrapolation based on Datastream swap dataup to maturity T = 20, 30, 50 when applying our func-tional L. 119

Figure 5.7.3 The same quantities are shown as in Figure 5.7.2 butnow on a daily basis. 120

ix

L I S T O F S Y M B O L S

Part I:

r Short rate

λ Mortality rate

β Money-market account

F Survival probability

τ Time of death of investor

S Stock price

P Zero-coupon bond price

F Price of survival bond

X Wealth process

c Rate of consumption

π Asset allocation expressed in currency units

η Asset allocation as a fraction of wealth XA Set of admissible portfolio and consumption strategies

θ Market price of risk

|| · || `2 (Euclidean) vector norm

|| · ||2 L2 norm of a function

Part II:

f Forward rate

z Continuously compounded (or exponential) rate

p Zero-coupon bond price

f∞ Long forward rate

z∞ Exponential long rate

σ Forward rate volatility

x

symbols xi

α Risk-free drift

ϕ Market price of interest rate risk

Ai Airy function of the first kind

Bi Airy function of the second kind

I Set of liquidly traded bonds

J Set of times at which bond payments may occur

m Observed market bond prices

π Discount factor

SW Smith-Wilson function

W Exponential tension spline

E Set of square-integrable functions on [0, ∞[ with vanishing limit

at infinity

Fa Set of two times differentiable functions g satisfying g(0) = awith g′, g′′ ∈ E

R+ Set of positive real numbers [0, ∞[

1 I N T R O D U C T I O N

This dissertation comprises a study of long-term risks which play a majorrole in actuarial science. In Part I we analyse long-term mortality risk and itsimpact on consumption and investment decisions of economic agents, whilePart II focuses on the mathematical modelling of long-term interest rates. Inthis first chapter, an introduction is provided to the problems treated in thethesis and their mathematical and economic context. Moreover, we describethe motivation and the contribution to the literature of the subsequent chap-ters.

1.1 on part i

In Part I we study long-term mortality risk and its effect on optimal consump-tion and portfolio selection. We extend techniques from classical portfolio the-ory to characterise the consumption and investment decisions of individualinvestors whose lifetime is governed by stochastically evolving survival prob-abilities.

Portfolio theory is a branch of (mathematical) economics that seeks to describethe optimal behaviour of economic agents who wish to maximise the benefitsfrom their portfolio of investments in the face of uncertainty. This uncertaintymay stem from financial risks, such as changes in the price of a stock or abond, but it may also originate from other causes, such as the variability inthe mortality rate of the investor. Typically, the agent also tries to maximiseconsumption, which is financed by his accumulated wealth. To compare theuncertain outcomes of different investment and consumption strategies, pref-erences are defined which, under suitable assumptions, can be represented bymeans of a utility function.

Historically, portfolio theory was pioneered by Markowitz (1959). He formu-lated a one-period portfolio selection problem in which the investor choosesan asset allocation that is optimal (or, efficient) in the sense that a higher ex-pected return cannot be attained, unless one is willing to accept more risk. Inthe setup of Markowitz, risk is measured by the variance of the returns. Tobin(1965) extended Markowitz’ results to a multi-period setting. Samuelson (1969)subsequently derived the optimal consumption and asset allocation strategy

1

2 introduction

for a certain class of utility functions describing preferences with Constant Rel-ative Risk Aversion (CRRA) in discrete time. Hakansson (1969, 1970) studied arelated problem for an immortal investor and for an investor whose lifetime isuncertain. In an influential paper, Merton (1969), working in continuous time,solved a problem similar to that of Samuelson. For a single economic agentand a given amount of initial wealth, Merton’s portfolio problem consists offinding an investment and consumption strategy that maximises expected util-ity from consumption and, in case of a finite horizon, terminal wealth. Mertonproved that this problem has an explicit solution: it is optimal for the CRRAinvestor to consume a constant fraction of his wealth.

One of the reasons why portfolio theory has been, and still is, a very ac-tive research area is its connection with contemporary macroeconomics. Inresponse to the critique in Lucas (1976, 1978), general dynamic equilibriummodels were developed in which agents act rationally. The modelling of ex-pectations changed from ‘adaptive’, meaning that the future is modelled usingeconometric forecasts based on historical data, to ‘rational’ which means thatall available (market) information is taken into account. This led to the adop-tion of techniques from portfolio theory that had been developed around 1959

by Markowitz and later, for continuous time and Von Neumann-Morgensternpreferences, by Merton. It is argued in Merton (1992) that portfolio theory inits early stages was not adopted by mainstream economics due to its relianceon the mean-variance criterion, which is not compatible with Von Neumann-Morgenstern preferences.

In this thesis we will exclusively discuss the case of a single investor, and wefocus in particular on the effects of mortality changes. The inclusion of anuncertain lifetime for the investor in portfolio selection problems apparentlyoriginates with Yaari (1965). Since then, several authors studied consumptionand asset allocation problems on an uncertain time horizon. In Pliska and Ye(2007), the evolution of the investor’s wealth process depends on a determin-istic income and consumption, an investment in the money-market accountand insurance payments. The expected utility from consumption, from a be-quest and from terminal wealth, is controlled by the consumed amount andthe amount of insurance bought, and optimal strategies are derived for both.In Pliska and Ye (2007), investment opportunities are limited to the money-market account, but Blanchet-Scalliet et al. (2008) and Jeanblanc and Yu (2010)consider a problem in which the agent maximizes the expected utility from be-quest upon death and from terminal utility, in a market that contains a set oftradeable assets with prices that follow diffusion processes. Menoncin (2008)

1.2 on part ii 3

and Shen and Wei further develop these results to more complex dynamicsfor the mortality rate.

There are two approaches to solve optimal consumption and investment prob-lems. The first method, which is commonly referred to as the primal approach,exploits a Markovian assumption on the state price process to derive a nec-essary condition for the optimal strategy in the form of a Hamilton-Jacobi-Bellman (HJB) equation. If explicit solutions to this equation can be obtained,and if such solutions are sufficiently smooth, then a verification theorem, i.e.a sufficient condition for optimality, can usually be proven. If no explicit solu-tions are available, existence can in particular cases be established by analysingviscosity solutions of the HJB equation. The second approach to optimal port-folio selection was initiated in a series of papers by Harrison and Pliska (1981),Harrison and Pliska (1983), and Pliska (1986). This dual or ‘martingale’ ap-proach splits the optimisation problem in two stages, where in the first step anoptimal terminal distribution of wealth is constructed and in the second stepa hedging strategy is derived which replicates this optimal terminal wealthdistribution. Later, Cox and Huang (1989) and Karatzas et al. (1987) extendedthis approach to include consumption.

In Chapter 2 we propose a tractable, yet realistic market model with a stochas-tic mortality rate, for which the optimal consumption and investment strategycan be derived in semi-closed form. From an economic perspective, the noveltyin our approach is twofold. First, the mortality rate, while stochastic, is guar-anteed to be nonnegative. Second, the market price of mortality risk is allowedto be stochastic as well. Proving optimality requires a rigorous mathematicaltreatment of the problem. In the prevailing literature, existence of a solutionis either tacitly assumed, or the conditions imposed to guarantee existenceare too strong to allow for stochastic processes that do not possess momentsof all order. We adopt a dual approach to show that optimal strategies existwhen the investor’s mortality rate follows a square root process, provided thatadditional constraints are imposed on the market price of risk.

1.2 on part ii

In Part II we study long-term interest rate risk. The market for interest ratederivatives such as bonds and swaps is very liquid, but this liquidity only per-sists until maturities of twenty to thirty years. Longer-dated contracts are notfrequently traded, and hence no reliable market prices are available for suchcontracts. Still, insurance firms and pension funds must decide how to value

4 introduction

liabilities with maturities stretching far beyond the liquid part of the yieldcurve. This has led to the introduction of various extrapolation practices inthe financial industry, most notably the Smith-Wilson technique prescribed bythe European Insurance and Occupational Pensions Authority (EIOPA). Thelimit of the forward rate when the maturity goes to infinity in such proce-dures is referred to as the ultimate forward rate (UFR) by the financial indus-try, while in the academic literature this limit, or the limit of the continuouslycompounded yield, is called the long rate. In its most general form the longrate can be a stochastic process, but the UFR is usually assumed to be con-stant. Since the seminal paper by Dybvig et al. (1996), there has been muchdiscussion on whether one could, and should, assume that the long rate mayvary stochastically over time, see also El Karoui et al. (1998) and Hubaleket al. (2002). A consultation paper by EIOPA (2016) has sparked a new debateamongst actuarial practitioners on the very same question.

Chapter 3 gives an overview of the literature on convergence of the long rate inthe Heath-Jarrow-Morton (HJM) framework for stochastic interest rate models.In addition to uniform convergence on compacts and almost sure convergenceof interest rates, which have been studied in Deelstra and Delbaen (1995) andBiagini et al. (2016) building on El Karoui et al. (1998) and Yao (1999a), we alsoconsider uniform convergence in mean. This allows us to derive both neces-sary and sufficient conditions on the volatility function to ensure convergenceof the long rate by using a result due to Barlow and Protter (1990). We thusprovide a novel characterisation of those volatility functions in HJM modelswhich admit a long rate.

In Chapter 4 the behaviour for the long-term interest rate in affine factor mod-els of the term structure is studied. This class of models is important due toits tractability and interpretability, but also because it essentially contains allinterest rate models that admit a finite dimensional realisation, as was shownin a series of publications by Björk and Christensen (1999), Björk and Svens-son (2001) and Filipovic and Teichmann (2003). We demonstrate that withinthe class of affine factor models, stochastic behaviour of the long rate is possi-ble. We thus answer a question which was first put forward in El Karoui et al.(1998). This question has previously received attention in Yao (1999b), Deel-stra (1996) and Deelstra et al. (2000). We extend these results by proving thata model with two state variables is not sufficiently flexible to accommodate afactor model for the term structure that converges almost surely to a stochasticlong rate. We also show that constructing such a model is feasible using threestate variables by providing an explicit example.

1.2 on part ii 5

Chapter 5 introduces a new method to estimate the level of the long ratebased on current prices of market instruments. We show that the term struc-ture extrapolation method proposed in Smith and Wilson (2000), which hasbeen adopted in the European insurance regulation EIOPA (2010), can beinterpreted as the solution to a variational optimization problem where thesmoothest continuation of the term structure is chosen after the last liquidmaturity, for an a priori given level of the ultimate forward rate. We extendthis approach by making the level of the ultimate forward rate part of theoptimization problem. Using historical market prices for bonds and swaps,we determine how the level of the ultimate forward rate resulting from thisoptimisation changes over time, and we show that after the financial crisis of2008, estimated values have dropped substantially below the level imposed inEIOPA (2010).

Part I

L O N G E V I T Y R I S K

2 O P T I M A L C O N S U M P T I O N S T R AT E G I E S I N M A R K E T SW I T H L O N G E V I T Y R I S K

This chapter is based on J. de Kort and M.H. Vellekoop, Existence of optimalconsumption strategies in markets with longevity risk. Insurance: Mathematics andEconomics, 72 (2017): 107–121.

Survival bonds are financial instruments with a payoff that depends on humanmortality rates. In markets that contain such bonds, agents who optimize ex-pected utility of consumption and terminal wealth can mitigate their longevityrisk. To examine how this influences optimal portfolio strategies and consump-tion patterns, we define a model in which the death of the agent is representedby a single jump process with Cox-Ingersoll-Ross intensity. This implies thatour stochastic mortality rate is guaranteed to be nonnegative, in contrast toother models in the literature, including Menoncin (2008) and Blanchet-Scallietet al. (2008). We derive explicit conditions for existence of an optimal con-sumption and investment strategy in terms of model parameters by analysingcertain inhomogeneous Riccati equations. We find that constraints must be im-posed on the market price of longevity risk to have a well-posed problem andwe derive the optimal strategies when such constraints are satisfied.

2.1 introduction

This chapter investigates the optimal consumption and asset allocation of aninvestor in a market which contains financial assets and contracts that are sen-sitive to longevity risk. These contracts, with a payoff that depends on therealised mortality rate in a large population, can be thought of as insuranceproducts that can help to mitigate the effects of changes in survival probabili-ties during the lifetime of the agent. There is at the moment no liquid marketfor such products, but by introducing them in an asset allocation optimizationproblem we formulate a consistent and arbitrage-free way of analyzing theinfluence of the market price of longevity risk on investment behavior. Theprecise value of such a market price of risk may be difficult to estimate inpractice, but it is not realistic to put it at zero, especially for the analysis ofretirement provisions.

9

10 optimal consumption strategies in markets with longevity risk

The classical formulation of the optimal consumption and investment prob-lems that we wish to consider here goes back to Merton (1969, 1971). Inthat setup, markets are complete and mortality is not considered so thereis a fixed time period for investment and consumption. These results havebeen extended in many directions, by changing preferences as in Musiela andZariphopoulou (2010) and Kraft et al. (2011), or by specifying different as-sumptions on asset price dynamics. Some authors, including Kraft (2005) andChacko and Viceira (2005), have considered stochastic volatility processes forequity prices and solved the optimization problem for that case. Others haveintroduced more realistic fixed income markets by introducing stochastic in-terest rates. Many papers use Gaussian models for the short rate but Deelstraet al. (2000) and Kraft (2005, 2009) derive the optimal strategy for an agentmaximizing power utility from terminal wealth by investing in a market witha short-rate following a Cox-Ingersoll-Ross (CIR) process, which will thus re-main positive at all times.

If the investment horizon is uncertain, the optimal strategies need to be ad-justed. We can distinguish between problems where the end of the investmentperiod is chosen by the agent himself, such as the optimal stopping problemfor flexible retirement in Dybvig and Liu (2010), and problems where the endof the investment period cannot be chosen by the agent. Models in whichthe death of the agent is included in the model form an obvious example ofthe latter category. In that case agents face a trade-off between obtaining anamount of utility now with certainty versus an amount of utility in the futurewhich is uncertain due to both financial risk and mortality risk. Yaari (1965)seems to be the first to consider such consumption and investment problemsin which the lifetime of the agent is stochastic. Under the assumption that theprobability of death at a given age is known and constant in time, he solves theoptimal investment problem with uncertain lifetime in continuous time usingdynamic programming methods. The solution shows that one can interpretthe deterministic mortality rate in terms of adjusted discount rates. Hakans-son (1969) obtained similar results in a discrete time. Pliska and Ye (2007),building on previous work by Richard (1975), introduced life insurance asan extra asset for investment. They derive closed form solutions for the in-vestment in stocks, bonds and life insurance products under the assumptionthat mortality rates are time-varying but deterministic. Huang and Milevsky(2008) extended these results to hyperbolic absolute risk aversion (HARA) andinclude a stochastic income process; see also Menoncin and Regis (2015). InCharupat and Milevsky (2002) and Milevsky and Young (2007) annuities in-stead of life insurance contracts are added to the asset mix.

2.1 introduction 11

All these models use time-varying but deterministic mortality rates. One of thefirst authors to study optimal consumption strategies when mortality rates arestochastic is Menoncin (2008). He solves the Hamilton-Jacobi-Bellman equa-tion associated with the optimization problem of an investor who is exposedto a stochastic mortality rate while allowing for a general specification of theasset price dynamics. A longevity bond is available in the economy to mitigatethe effects of this risk. Blanchet-Scalliet et al. (2008) allow the conditional dis-tribution function of an agent’s remaining time-horizon to be stochastic andcorrelated to asset returns. To get closed form expressions, the rate which de-termines the random time at which the investor must liquidate the asset port-folio is allowed to become negative with positive probability. See Maurer (2011)for another example of markets where mortality rates are correlated with as-set returns. Huang et al. (2012) compare the optimal consumption strategy foran agent with a deterministic versus a stochastic mortality rate in a marketwhere only the money market account is available for investment. They showthat, compared to the deterministic case, agents should increase or decreasetheir initial consumption in the stochastic case, depending on whether theirconstant risk aversion coefficient makes them more or less risk averse than aninvestor with a logarithmic utility function. In more recent work Shen and Weisolve an optimal consumption and asset allocation problem in a general diffu-sion framework under the premise that an exponential integrability conditionis satisfied. Guambe and Kufakunesu (2015) generalize these results to Lévyprocesses.

As an extension of these results we provide conditions that guarantee exis-tence of an optimal consumption strategy in a market where rates follow CIRdynamics. We will thus model mortality by a stochastic process, which will al-most surely stay positive at all times. Affine mortality rates lead to analyticallytractable survival probabilities which remain smaller than one, as was notedearlier in work by Biffis (2005) and Schrager (2006); see also Dahl (2004). Weinclude both financial assets and survival bonds in the asset mix and allow astochastic market price of mortality risk. The optimal consumption and invest-ment strategies are derived in semi-closed form and we show that the hedgingdemand in our model is bounded. The impact of stochastic mortality on initialconsumption is characterized in terms of the risk aversion coefficient.

The existence of a solution to the problem studied in the present chapter de-pends on the asymptotic behaviour of the utility function for large values ofwealth and the existence of moments of the state-price density. It was noted inKorn and Kraft (2004) and Kraft (2009) that not all moments of the state-pricedensity in a model based on a CIR short rate may be well-defined. Therefore


the present chapter takes a different approach than the one in Shen and Sherris(2014) and Menoncin and Regis (2015), since we want to characterise explicitlyunder which conditions optimal strategies exist. By assuming that the marketprices of risk are proportional to the square root of the short rate and the mor-tality rate (with a time-varying proportionality coefficient) we can derive suchconditions by analyzing the existence and uniqueness of bounded solutionsfor certain inhomogeneous Riccati equations. Notice that our results are nota special case of Shen and Sherris (2014) and Shen and Wei or the extensionof these results in Guambe and Kufakunesu (2015). The analysis in these pa-pers relies on an exponential integrability condition which is not satisfied ingeneral for our model.

The remainder of this chapter is organized as follows. Section 2.2 introducesthe model for the economy. In Section 2.3 the investment problem is formu-lated and the main result of the chapter is presented. Section 2.4 providesa characterization of the Laplace transform of a Cox-Ingersoll-Ross processwhich is needed to prove the main result. The proof of the main result is givenin Section 2.5. The economic implications of the model are discussed in Sec-tion 2.6. Section 2.7 concludes.

2.2 the model for the economy

In this section we will construct a complete financial market in which assetreturns, interest rates and mortality rates are uncertain. Let (Ω,G, P) be aprobability space on which a three-dimensional standard Brownian motionW(t) = (W1(t), W2(t), W3(t))′ and an exponentially distributed random vari-able Θ are defined. We study the optimal investment and consumption strat-egy of an investor during the timespan [0, T], for some 0 < T < ∞, andwe allow for the possibility that the investor does not survive until T. LetF (t), t ∈ [0, T] be the P-augmentation of the filtration generated by theprocess W. The exponential random variable Θ is taken to be independentof F (T).To model the time of death of the investor we introduce the nonnegative, G-measurable random time

τ = inf

t ≥ 0 :∫ t

0λ(u) du ≥ Θ

, (2.2.1)

2.2 the model for the economy 13

in which the so-called mortality rate λ(t) follows a Cox-Ingersoll-Ross process,that is,

λ(t) = λ0 +∫ t

0(µ2(u)− κ2 λ(u)) du +

∫ t

0ξ2

√λ(u)dW2(u) . (2.2.2)

The constants λ0, κ2, ξ2 and µ2 are assumed to be bounded and strictly pos-itive, and µ2 is a deterministic, continuously differentiable function on [0, T].The extended Feller condition1

2 µ2(t) ≥ ξ22 , for all t ∈ [0, T] , (2.2.3)

ensures that the mortality rate is strictly positive almost surely.

Since∫ t

0 λ(u) du is F (t)-measurable and Θ is independent of F (T), the sur-vival probability of the agent satisfies

F(t) := P [τ > t | F (t)] = P[∫ t

0λ(u) du < Θ

∣∣∣∣F (t)]= exp

(−∫ t

0λ(u) du

). (2.2.4)

To finance his/her consumption and long-term wealth objectives, the agentcan invest in a number of assets: a stock, a zero-coupon bond, a longevitybond and the money-market account.

The value β of the money market account, based on the continuously com-pounded stochastic short rate r, satisfies

β(t) = 1 +∫ t

0β(u)r(u)du .

The short rate is assumed to follow a Cox-Ingersoll-Ross process, i.e.

r(t) = r0 +∫ t

0(µ1(u)− κ1 r(u)) du +

∫ t

0ξ1

√r(u)dW1(u) , (2.2.5)

in which the constants r0, κ1, ξ1 and µ1 are bounded and strictly positive,and µ1 is a deterministic, continuously differentiable function on [0, T]. Byassuming that

2 µ1(t) ≥ ξ21 , for t ∈ [0, T] , (2.2.6)

we enforce that the short rate also almost surely remains strictly positive.

1 See Theorem 4.2 in Maghsoodi (1996).


Given S(0) > 0, the stock price S is assumed to follow the dynamics

S(t) = S(0) +∫ t

0µS(u)S(u)du +

∫ t

0σS(u) S(u)dW(u) ,

in which

µS(t) = r(t) + σS(t)θ(t)′ , σS(t) = (−ξ3ρ√

r(t), 0, −ξ3ρ) ,

where ξ3 is a strictly positive constant, and

θ(t) =(−ψ1(t) ξ−1

1

√r(t), −ψ2(t) ξ−1

2

√λ(t), −ψ3(t) ξ−1

3)

,

with ρ ∈ ]− 1, 1[ and ρ2 = 1− ρ2. The market price of interest rate risk θ1(t)is thus assumed to be proportional to the square root of the short rate (witha time-varying proportionality coefficient) and the market price of mortalityrisk θ2(t) is taken to be proportional to the square root of the mortality rate.The deterministic scaling functions ψi, with i = 1, 2, 3, are required to be con-tinuous and bounded on [0, T].

A zero-coupon bond which pays one unit of currency at its expiration date T1is available for trading in the economy. We assume2 that the expiration date ofthe bond lies beyond the investment horizon of the agent, i.e. T1 > T. The priceprocess P(t, T1) of the zero-coupon bond at an earlier time t ≤ T1 satisfies, seefor example Filipovic (2009, Prop. 5.1),

P(t, T1) = P(0, T1) +∫ t

0P(u, T1)µP(u)du +

∫ t

0P(u, T1)σP(u, T1)dW(u) ,

in which

µP(t) = r(t) + σP(t)θ(t)′ , σP(t) = (−ξ1B1(t, T1)√

r(t), 0, 0) ,

2 This assumption is required to prevent that the volatility of the bond tends to zero near Twhich would lead to a singularity in the optimal asset allocation. For a discussion of the caseT = T1 the reader is referred to Bielecki et al. (2005). Instead of a long-term bond we couldalternatively introduce a ‘rolling bond’ as a tradeable asset in the economy. This correspondsto a self-financing strategy which continuously invests in a bond with fixed time to maturityT2 > 0. Using results in Rutkowski (1999) it can be shown that the price process PR(·, T2) of arolling bond satisfies

PR(t, T2) = PR(0, T2)+∫ t

0PR(u, T2) r(u) du−

∫ t

0ξ1 B1(u, u+ T2) PR(u, T2)

√r(u) dW1(u) ,

in which B1 is the solution to Eq. (2.2.7).

2.2 the model for the economy 15

and the bond duration B1 satisfies the differential equation−∂tB1(t, T1) + (κ1 − ψ1(t)) B1(t, T1) +12 ξ2

1 B1(t, T1)2 = 1 , 0 ≤ t ≤ T1 ,

B1(T1, T1) = 0 .

(2.2.7)

The economy that we wish to study permits agents to mitigate longevity riskvia survival bonds3. Survival bonds are financial securities paying, at their ex-piration date T1, an amount proportional to F(T1) = exp(−

∫ T10 λ(u)du), the

expected fraction of survivors at time T1 among individuals in a (large) pop-ulation with a common mortality rate process λ. Such bonds can be thoughtof as insurance products; they can be used by investors to hedge uncertaintyabout future survival probabilities. The price process F(·, T1) of the survivalbond satisfies

F(t, T1) = F(0, T1) +∫ t

0F(u, T1)µF(u) du +

∫ t

0F(u, T1)σF(u)dW(u) , (2.2.8)

where

µF(t) = r(t) + σF(t)θ(t)′ ,

and

σF(t) = (−ξ1B1(t, T1)√

r(t), −ξ2B2(t, T1)√

λ(t), 0) .

The function B1(·, T1) solves Eq. (2.2.7), and B2(·, T1) is the solution to−∂tB2(t, T1) + (κ2 − ψ2(t)) B2(t, T1) +12 ξ2

2 B2(t, T1)2 = 1 , 0 ≤ t ≤ T1 ,

B2(T1, T1) = 0 .

(2.2.9)

We will show later on in Remark 2.5.2 that this indeed implies that F(·, T1) isan asset price process which replicates the survival fraction at time T1, i.e. thatF(T1, T1) = F(T1).

The investor continuously adjusts his/her portfolio of assets in order to achievea consumption and terminal wealth profile that is optimal in a sense that

3 For a discussion and overview of trading in longevity instruments the reader is reffered toBlake et al. (2013).


will be defined shortly. The vector π(t) = (π1(t), π2(t), π3(t)) denotes theamount invested, at time t ∈ [0, T ∧ τ], in zero-coupon bonds with maturityT1, survival bonds maturing at T1 and stocks, respectively. The remainingwealth π0(t) is invested in the money-market account. This portfolio repre-sents the savings of an agent who starts with initial wealth x > 0 at timet = 0. The nonnegative F (t)-progressively measurable process c, satisfying∫ T

0 c(u)du < ∞ a.s., represents the instantaneous consumption of the agent.

The wealth process Xx,c,π , corresponding to an asset allocation π, a nonnega-tive consumption process c and initial endowment x > 0, is given by

Xx,c,π(t) = x−∫ t

0c(u)du +

∫ t

0π0(u)

dβ(u)β(u)

+∫ t

0π1(u)

dP(u, T1)

P(u, T1)

+∫ t

0π2(u)

dF(u, T1)

F(u, T1)+∫ t

0π3(u)

dS(u)S(u)

.

Expressed in terms of units of the money market account the wealth processsatisfies

Xx,c,π(t)β(t)

= x −∫ t

0

c(u)β(u)

du +∫ t

0

π(u)β(u)

S (u, T1)(

dW(u) + θ(u)′ du)

,

(2.2.10)

where

S (t, T1) = (σP(t), σF(t), σS(t))′ . (2.2.11)

Remark 2.2.1. Due to the extended Feller conditions (2.2.3) and (2.2.6) the volatilitymatrix (2.2.11) is nonsingular almost surely for t ≤ T.

In order to exclude asset allocations leading to infinite wealth in finite timewe restrict our attention to the subset of F (t)-progressively measurable assetallocations π satisfying∫ T

0

∣∣π0(t) + π1(t) + π2(t) + π3(t)∣∣ r(t) dt < ∞ a.s. , (2.2.12)

∫ T

0

∣∣∣∣π(t)S (t, T1)∣∣∣∣2 dt < ∞ a.s. , (2.2.13)

and ∫ T

0

∣∣π(t)S (t, T1)θ(t)′∣∣ dt < ∞ a.s. , (2.2.14)

2.3 the optimal investment and consumption problem 17

An asset allocation for which conditions (2.2.12), (2.2.13) and (2.2.14) are metwill be called a portfolio process.

In the subsequent sections, we will formulate the investment and consumptionoptimisation problem and characterise its solution.

2.3 the optimal investment and consumption problem

In Section 2.2 we introduced a financial market in which the short rate and themortality rate are both stochastic. In this section we will define an optimal in-vestment and consumption problem for an investor whose remaining lifetimeis uncertain.

Definition 2.3.1 (Main problem). Let x > 0 be an amount of initial capital and letT ∈ ]0, T1[. We define the value function of the portfolio and consumption plan max-imizing utility from consumption and terminal wealth on an uncertain time horizonas

V(x) := sup(c,π)∈A

E

[∫ T∧τ

0U1(c(t))dt +U2(Xx,c,π(T))1τ>T

], (2.3.1)

in which, for certain p ∈ (−∞, 1) \ 0 and Υ ∈ [0, ∞),

U1(x) =xp

pand U2(x) = Υ

xp

p, (2.3.2)

or when p = 0,

U1(x) = log x and U2(x) = Υ log x , (2.3.3)

and where

A :=

(c, π) is measurable w.r.t. the collection (F (t))t∈[0,T],

satisfies Eqs. (2.2.12) – (2.2.14), c ≥ 0, and Xx,c,π ≥ 0

,

denotes the set of admissible strategies.

The above definition of admissible strategies ensures that∫ T

0 c(u) du < ∞. In-deed, since wealth is required to be nonnegative on the one hand, and almostsurely finite on the other hand, see the definition of a portfolio process inEqs. (2.2.12)-(2.2.14), it follows that the amount of wealth available for con-sumption is almost surely finite. Moreover, since for every admissible (c, π)the discounted gains are bounded from below, i.e.

P[

1β(t)

(Xx,c,π(t) +

∫ t

0c(u)du

)≥ 0

]= 1 ,


the definition of admissibility excludes doubling strategies on a finite timeinterval, see Karatzas and Shreve (1998, Def. 1.2.4).

The set A of admissible strategies is not empty and contains a strategy whichyields more than a utility of −∞, so V(x) > −∞ for all x > 0. Indeed,consider the strategy which allocates all initial wealth x > 0 to the money-market account, and continuously consumes wealth, at a rate of 1

2 x/T perunit time. This strategy is admissible since U1(

12 x/T) > −∞ for all t ∈ [0, T]

and U2(Xx,c,π(T)) ≥ U2(12 x) > −∞.

Using a dual approach, see for example Cox and Huang (1989) and Karatzasand Shreve (1998), we will solve the problem from Definition 2.3.1. The follow-ing theorem is a direct implication of Propositions 2.5.8 and 2.5.9.

Theorem 2.3.2 (Main result). Assume that the scaling functions for the market priceof risk ψi, i = 1, 2, are analytic on [0, T]. Then the problem from Definition 2.3.1 hasa solution which is unique up to almost-everywhere equivalence when p ≤ 0. Whenp ∈ ]0, 1[ this is still the case if the parameters of the short rate and mortality rateprocesses satisfy

mins∈[0,T]

(κi +

p1−p ψi(s)

)> 0 (2.3.4)

for i = 1, 2 and

p [ (κ1 − ψ1,min)2 +

(ψ21)max−(ψ1,min)

2

1−p ] < κ21 − 2pξ2

1 , (2.3.5)

p [ (κ2 − ψ2,min)2 +


2

1−p ] < κ22 + 2ξ2

2 , (2.3.6)

whereψi,min = min

s∈[0,T]ψi(s), (ψ2

i )max = maxs∈[0,T]

ψi(s)2.

The optimal wealth process X satisfies

d(

X(t) +∫ t

0 c(u)du)

X(t)= η0(t)

dβ(t)β(t)

+ η1(t)dP(t, T1)

P(t, T1)

+ η2(t)dF(t, T1)

F(t, T1)+ η3(t)

dS(t)S(t)

,

in which the observable4 quantities ηi, i = 0, . . . , 3, defined in Eq. (2.5.35), representthe fraction of wealth invested in each of the asset classes. Moreover, the optimalstrategy comprises consumption at the rate c(t) = X(t)/n(t) at each time t duringthe life [0, τ) of the agent, where n, defined in Eq. (2.5.22), is an observable process.

4 A variable is said to be ‘observable’ if its value at any time t in [0, T] is completely determinedby current and historical prices of tradeable assets at that time.

2.4 the laplace transform of a cir process 19

The expression for the optimal asset allocation in Theorem 2.3.2 is explicit upto the solution of a Riccati equation. In the special case where the parametersof the short rate and the mortality rate process are (piecewise) constant, theseRiccati equations admit a closed-form solution, which is given in Lemma 2.4.4.

2.4 the laplace transform of a cir process

To prove our main theorem, we first derive properties of the Laplace transformof an integrated Cox-Ingersoll-Ross process which is scaled by a bounded de-terministic function h. These results will facilitate the computation of semi-closed form expressions for the optimal consumption and investment strate-gies and they enable us to obtain explicit conditions under which the invest-ment problem from Section 2.3 has a unique solution.

We start by establishing in Lemma 2.4.1 that, under appropriate conditions, theLaplace transform of an integrated Cox-Ingersoll-Ross process on a finite timeinterval has an affine representation. A well-known application of this resultis the pricing of bonds in a CIR short rate model, which corresponds to thechoice h ≡ 1 in Lemma 2.4.1. We are however interested in the case where h isnot constant and possibly negative. Moreover, we will not constrain the mean-reversion speed κ to be nonnegative. We thus extend results in Pitman and Yor(1982), Kraft (2003), Wong and Heyde (2006) and Gnoatto and Grasselli (2014)to the case of time-varying parameters and bounded, but possibly negative,mean-reversion speed.

Lemma 2.4.1. Let h, κ, µ and ξ be bounded and continuous functions [0, T] → R

satisfying µ(t) ≥ 0 and ξ(t) > 0 for 0 ≤ t ≤ T. Suppose that r follows a Cox-Ingersoll-Ross process

r(t) = r0 +∫ t

0(µ(s)− κ(s)r(s)) ds +

∫ t

0ξ(s)

√r(s) dW(s) , (2.4.1)

in which W is standard Brownian motion and r0 is a strictly positive constant. Forα ∈ R and 0 ≤ t ≤ T, consider the Laplace transform

g(t, T, x) = E

[exp

−αr(T)−

∫ T

th(s)r(s) ds

∣∣∣∣ r(t) = x]

. (2.4.2)

If the Riccati equation∂tB(t, T) = L(t, B(t, T)) , 0 ≤ t ≤ T ,B(T, T) = α ,

(2.4.3)


in which

L(t, x) = κ(t)x +12

ξ2(t)x2 − h(t) , (2.4.4)

as well as the differential equation∂t A(t, T) = −µ(t)B(t, T) , 0 ≤ t ≤ T ,A(T, T) = 0 ,

(2.4.5)

have bounded solutions A(·, T) and B(·, T), then the Laplace transform g(t, T, r) hasan affine representation

g(t, T, r) = e−A(t,T)−B(t,T)r . (2.4.6)

The previous lemma provides sufficient conditions for finiteness of the Laplacetransform, but it is shown in Korn and Kraft (2004, Prop. 3.2) that in the specialcase where h(t) = h for some h < 0, there exist constants µ and ξ, for anygiven constant mean reversion speed κ, such that the Laplace transform inLemma 2.4.1 is infinite. Conversely, given a set of constant CIR parametersthere is a number h < 0 such that the Laplace transform in Lemma 2.4.1 withh(t) = h is infinite.

Before proving Lemma 2.4.1, we need the following result which, for the caseof nonnegative mean-reversion speed, is due to Shirakawa (2002). We presenta different proof, and we extend the result to a bounded, possibly negative,mean-reversion speed.

Lemma 2.4.2. Let φ : [0, T] → R be a continuous and bounded function and letthe process r be defined as in Eq. (2.4.1). Assume that µ, κ and ξ are continuous andbounded functions and that ξ(t) > 0 and µ(t) ≥ 0 for 0 ≤ t ≤ T. Then the stochasticexponential

Z(t) = exp∫ t

0φ(s)

√r(s)dW(s)− 1

2

∫ t

0φ2(s)r(s) ds

, (2.4.7)

is a martingale for 0 ≤ t ≤ T.

Proof.

First we show that, without loss of generality, we may assume κ > 0. Since κis bounded we have that for all t ∈ [0, T], κ(t) > −m for some m > 0. Definer(t) = e−mtr(t). It follows from Itô’s lemma that

dr(t) = −me−mtr(t)dt + e−mtdr(t)

=[µ(t)− κ(t)r(t)

]dt + ξ(t)

√r(t)dW(t) ,


where µ(t) = µ(t) e−mt, κ(t) = (κ(t) + m) e−mt > 0 and ξ(t) = ξ(t) e−mt areall bounded and continuous on [0, T]. Hence r(t) is a CIR process with strictlypositive mean-reversion speed. Observe that

dZ(t)/Z(t) = φ(t)√

r(t)dW(t) = φ(t)√

r(t)dW(t) ,

in which φ(t) = φ(t)e12 mt is a bounded function on [0, T]. It thus remains to

prove the result for κ > 0.

Suppose that there exists a q < 0 such that, for every u ∈ [0, T], the differentialequations∂tB(t, u) = B(t, u)

(κ(t) + 1

2 ξ2(t)B(t, u))

, 0 ≤ t ≤ u ,

B(u, u) = q ,(2.4.8)

and ∂t A(t, u) = −µ(t)B(t, u) , 0 ≤ t ≤ u ,

A(u, u) = 0 ,(2.4.9)

have continuously differentiable solutions that are bounded uniformly in u ∈[0, T]. We will prove later that there indeed exist solutions to Eqs. (2.4.8)and (2.4.9) with the required properties. For fixed u ∈ [0, T] define f (t, x) =

e−A(t,u)−B(t,u)x . Applying Itô’s lemma to ft := f (t, r(t)) yields

d ft = [−∂t A(t, u)− ∂tB(t, u)r(t)] ft dt− B(t, u) ftdr(t) +12

B(t, u)2 ftd 〈r, r〉t .

Substituting the dynamics of r we obtain

d ft

ft=

[−∂tB(t, u) + κ(t)B(t, u) +

12

ξ2(t)B(t, u)2]

r(t) dt

−[∂t A(t, u) + µ(t)B(t, u)

]dt− B(t, u)ξ(t)

√r(t) dW(t)

= −B(t, u)ξ(t)√

r(t) dW(t) .

We thus find that the process ft = f (t, r(t)) is a nonnegative local martingale,hence a supermartingale. Consequently,

E[e−qr(u)

]= E

[fu

]≤ f0 = e−A(0,u)−B(0,u)r0 . (2.4.10)


From the uniform boundedness of A(·, u) and B(·, u) in u ∈ [0, T] we concludethat there exists a finite constant C such that E

[e−qr(u)

]≤ C for all u ∈ [0, T].

Following Revuz and Yor (1999, p. 338), we will now use the bound Eq. (2.4.10)to show that the process Z, defined by Eq. (2.4.7), has constant expectation.Write Z(t) = E(ζ • W)t

5 in which ζ(t) = φ(t)√

r(t). Since φ is bounded, thereexists a constant M < ∞ such that |φ(t)| ≤ M for 0 ≤ t ≤ T. We have, when0 < ε < −q/M2, that

E[e ε ζ2(v)

]≤ C , for all v ∈ [0, T] .

Fix any t ∈ [0, T] and take 0 < r < s < t such that ε < s − r < 2ε if t ≥ εand s− r = t < ε otherwise. Let ζ(v) = ζ(v)1r<v<s be a truncation of ζ. ByJensen’s inequality we have

E[e

12∫ s

r ζ(v)2dv]≤ E

[1

s− r

∫ s

re

12 (s−r)ζ2(v)dv

]=

1s− r

∫ s

rE[e

12 (s−r)ζ2(v)

]dv ≤ C .

Since ζ satisfies Novikov’s condition it follows that

1 = E(ζ • W)r = E[E(ζ • W)s|Fr

]= E [E(ζ • W)s/E(ζ • W)r|Fr] .

The tower rule for conditional expectation yields

E [E(ζ • W)t] = E[E(ζ • W)t−(s−r)E

[E(ζ • W)t/E(ζ • W)t−(s−r)|Ft−(s−r)

]]= E

[E(ζ • W)t−(s−r)

].

By iteratively applying the above equation we obtain E [E(ζ • W)t] = 1, that is,the process Z has constant expectation. But Z is a nonnegative local martingale,hence a supermartingale. It follows that Z is a true martingale.

The final step is to establish that there exists some q < 0 such that Eqs. (2.4.8)and (2.4.9) admit, for all u ∈ [0, T], continuously differentiable solutions thatare bounded, uniformly in u ∈ [0, T]. From Picard-Lindelöf, see Teschl (2012,Thm. 2.2), and the local Lipschitz continuity of the righthand side of Eq. (2.4.8),

5 We use the following standard notation: Z = X • Y means dZ = XdY with initial value for Zequal to zero unless specified otherwise; and Z(t) = E(X)t solves Z(t) = 1 +

∫ t0 Z(s)dX(s)

with initial value one.


we know that there exists, for every u ∈ [0, T], a unique, continuously dif-ferentiable solution B(·, u) to Eq. (2.4.8) on an open interval (u′, u]. Supposethat (u′, u] is the largest interval on which a solution to Eq. (2.4.8) exists.

Since B(t, u) satisfies Eq. (2.4.8) it cannot change sign, hence

B(t, u) ≤ 0 , t ∈ (u′, u] ,

due to the boundary condition B(u, u) = q < 0. This establishes an upperbound for B(·, u). Define

K0 := 2 κmin/(ξ2)max > 0 ,

in which κmin = inf0≤s≤T κ(s) > 0 and (ξ2)max = sup0≤s≤T ξ(s)2 < ∞. ForB(t, u) ∈ (−K0, 0] we have

κ(t) +12

ξ2(t)B(t, u) ≥ κmin(t) +12(ξ2)maxB(t, u) ≥ 0 ,

and consequently

∂tB(t, u) = B(t, u)(

κ(t) +12

ξ2(t)B(t, u))≤ 0 .

It follows that if B(t, u) starts in (−K0, 0] at time u, then its trajectory will stayin (−K0, 0] for all t ∈ (u′, u].

Since B(·, u) is bounded on (u′, u] when q ∈ (−K0, 0], both a = lim sups↓u′ B(s, u)and b = lim infs↓u′ B(s, u) exist and are finite. Suppose that a > b. There existsequences an ↓ u′ and bn ↓ u′ with B(an, u) → a and B(bn, u) → b. There alsoexists an L ≥ 0 such that |B(t, u)(κ(t) + 1

2 ξ(t)2B(t, u))| ≤ L for all t ∈ (u′, u]because of the continuity of ξ and κ. But then |B(bn, u)− B(an, u)| ≤ L|bn− an|which gives a contradiction for n→ ∞. Hence a = b and

lims↓u′

B(s, u) , (2.4.11)

exists. We can thus apply Picard-Lindelöf again and extend the interval ofexistence beyond (u′, u]. This contradicts the assumption that (u′, u] is thelargest interval of existence. We conclude that if q ∈ (−K0, 0] then B(·, u) iscontinuously differentiable on [0, u] and

−K0 < B(t, u) ≤ 0 , t ∈ [0, u] . (2.4.12)

Moreover,

0 ≤ −A(t, u) = −∫ u

tµ(s)B(s, u) ds ≤ K0

∫ T

0µ(s) ds , (2.4.13)


so we also have that A(·, u) is continuously differentiable on [0, u] and boundedby a constant which does not depend on u.

Choosing any q ∈ (−K0, 0] thus ensures the existence of bounded and contin-uously differentiable functions A(·, u) and B(·, u) satisfying Eqs. (2.4.8) and(2.4.9) on [0, u], and the bounds (2.4.12) and (2.4.13) hold uniformly for all u ∈[0, T].

Using Lemma 2.4.2 we can now proceed to establish the affine representationof the Laplace transform (2.4.2) and prove Lemma 2.4.1.

Proof of Lemma 2.4.1.

By assumption of the lemma there exist bounded functions A(·, T) and B(·, T)satisfying the Riccati equations (2.4.3) and (2.4.5) on the interval [0, T]. Define

f (t, x) = e−∫ t

0 h(s)r(s)ds−A(t,T)−B(t,T)x ,

and observe that

f (T, r(T)) = e−αr(T)−∫ T

0 h(s)r(s)ds .

If the process f (t, r(t)) is a martingale, we have

E [ f (T, r(T)) | Ft] = f (t, r(t)) ,

so it follows that

E[

e−αr(T)−∫ T

0 h(s)r(s)ds∣∣∣ Ft

]= e−

∫ t0 h(s)r(s)ds−A(t,T)−B(t,T)r(t) ,

or, equivalently,

g(t, T, r(t)) = e−A(t,T)−B(t,T)r(t) .

We conclude that, if f (t, r(t)) is a martingale, then the Laplace transform hasthe affine representation as stated in the lemma.

Applying Itô’s lemma to ft := f (t, r(t)) yields

d ft = [−h(t)r(t)− ∂t A(t, T)− ∂tB(t, T)r(t)] ft dt

−B(t, T) ftdr(t) +12

B(t, T)2 ftd 〈r, r〉t .

Substituting the dynamics of r we find

d ft

ft=

[−h(t)− ∂tB(t, T) + κ(t)B(t, T) +

12

ξ2(t)B(t, T)2]

r(t) dt

−[∂t A(t, T) + µ(t)B(t, T)

]dt− B(t, T)ξ(t)

√r(t) dW(t) .


Hence, if B(·, T) solves the Riccati equation (2.4.3) and A(·, T) satisfies (2.4.5)then ft = f (t, r(t)) is a martingale provided that the local martingale solving

d ft

ft= −B(t, T)ξ(t)

√r(t) dW(t) , (2.4.14)

is a true martingale. This follows from Lemma 2.4.2 and the assumption thatB(·, T) and ξ(t) are bounded.

In order to apply Lemma 2.4.1 we need conditions ensuring existence ofa bounded solution to the inhomogeneous Riccati equation (2.4.3). In Lem-mata 2.4.5 and 2.4.6 we will provide such conditions for the case where ξ(t) =ξ for some ξ ∈]0, ∞[. First we need the following comparison result which willbe used to extend a local solution of a Riccati equation to a solution on theinterval [0, T].

Lemma 2.4.3. Let G : [0, ∞[×R→ R be a function which is locally Lipschitz in itssecond argument. Consider, for t ∈ [T′, T], the differential equation

∂ty(t) = G(t, y(t)) , y(T) = 0 ,

and let x(t) and z(t) be bounded functions satisfying the differential inequalities

∂tx(t) ≤ G(t, x(t)) , x(T) = 0 ,

and

∂tz(t) ≥ G(t, z(t)) , z(T) = 0 .

Then z(t) ≤ y(t) ≤ x(t) on [T′, T].

Proof. See Walter (1998, p. 96).

The following lemma provides an explicit solution to the Riccati equations(2.4.3)–(2.4.5) for the case where the functions h and κ are constant. Relatedresults appear in Kraft (2003) and Maghsoodi (1996).

Lemma 2.4.4. Suppose that ξ(t) = ξ, κ(t) = κ, µ(t) = µ and h(t) = h for someξ, κ, µ, h ∈ R such that

−2 ξ2h < κ2 , (2.4.15)

and

κ + a > 0 , (2.4.16)


where a =√

κ2 + 2hξ2 . Then the (real-valued) solution of the Riccati equations(2.4.3)–(2.4.5) is given by

A(t, T) = − µ

ξ2

[(κ − a) (T − t)− 2 log

(1− νe−a(T−t)

1− ν

)], (2.4.17)

B(t, T) = 2h[

κ + a coth(

12

a(T − t))]−1

, (2.4.18)

and ν = (κ − a) / (κ + a) . The functions A(·, T) and B(·, T) are bounded and arecontinuously differentiable with bounded first-order derivatives.

Proof.

It is straightforward to verify that A and B solve Eq. (2.4.3) and (2.4.5). Observethat condition (2.4.15) ensures that A and B are real-valued. Since x 7→ coth(x)is decreasing and bounded from below by 1 on ]0, T], it follows that B ismonotone and bounded on [0, T]. Condition (2.4.16) implies that ν < 1 so A isalso bounded on [0, T]. Since A(·, T) and B(·, T) satisfy Eq. (2.4.3) and (2.4.5),it follows that their first-order derivatives are continuous and bounded.

Set hmin = min0≤s≤T h(s), hmax = max0≤s≤T h(s), and κmin = min0≤s≤T κ(s)which is justified since since h and κ are bounded. Consider the family ofRiccati equations

∂t ϕL(t, u) = κmin ϕL(t, u) + 12 ξ2ϕL(t, u)2 −min(0, hmin) , t ∈ [0, u] ,

ϕL(u, u) = 0(2.4.19)

and the family∂t ϕU(t, u) = κmin ϕU(t, u) + 1

2 ξ2ϕU(t, u)2 −max(0, hmax) , t ∈ [0, u] ,

ϕU(u, u) = 0 ,(2.4.20)

both indexed by u ∈ [0, T]. The next lemma establishes a lower bound and anupper bound for solutions of the Riccati equation (2.4.3) for the case whereα = 0 and ξ(t) = ξ for some ξ ∈ ]0, ∞[. Note that we explicitly allow forthe possibility that the source term h(t) of the Riccati equation (2.4.3) changessign.


Lemma 2.4.5. Suppose that h is bounded and either equals zero or changes signfinitely many times, and that

hmin > 0 or κmin >√−2ξ2hmin . (2.4.21)

If on the interval (T′, T] the function B(·, T) is a solution to the Riccati equation(2.4.3) with α = 0 and ξ(t) = ξ for some ξ ∈ ]0, ∞[, then

ϕL(t, T) ≤ B(t, T) ≤ ϕU(t, T) ,

for t ∈ (T′, T].

Proof.

Condition (2.4.21) implies that Eqs. (2.4.15) and (2.4.16) are satisfied for thevalues h = hmin and κ = κmin. Indeed, h > 0 implies (2.4.15) is trivially satisfiedwhile κ + a = κ +

√κ2 + 2hξ2 > κ + |κ| ≥ 0. If h ≤ 0 then κ + a = κ +√

κ2 + 2hξ2 ≥ κ > 0 so (2.4.16) is also satisfied and (2.4.15) is satisfied byconstruction. It thus follows from Lemma 2.4.4 that the solutions ϕL(t, u) andϕU(t, u) to the Riccati equations (2.4.19) and (2.4.20) exist and are boundedon [0, u] for all 0 ≤ u ≤ T.

If B(·, T) changes sign at t, then ∂tB(t, T) = −h(t). Hence on intervals whereh(t) does not change sign, B(·, T) can change sign only once. Since, by assump-tion, the function h(t) changes sign only finitely many times, it follows thatB(·, T) changes sign only finitely many times. Therefore we may partition theinterval (T′, T] into a finite number of subintervals

(t0, t1], (t1, t2], . . . , (tN−1, tN ] ,

with t0 = T′, tN = T and such that B(ti, T) = 0 while the function B(·, T) hasconstant sign on each of these subintervals.

Take 1 ≤ i ≤ N and consider the interval (ti−1, ti]. Assume that B(·, T) isnegative on (ti−1, ti]. By Eq. (2.4.19) we have that ϕL(t, ti) is also negative onthat interval, so

∂t ϕL(t, ti) ≥ L(t, ϕL(t, ti)) , (2.4.22)

with L as defined in Eq. (2.4.4), while

∂tB(t, T) = L(t, B(t, T)) , (2.4.23)

for t ∈ (ti−1, ti] and B(t, T) = ϕL(t, ti) at the endpoint t = ti. Hence byLemma 2.4.3 the solution B(t, T) is bounded from below by ϕL(t, ti) on (ti−1, ti].


Note that L is locally Lipschitz in its second argument. We also see thatϕL(t, T) is decreasing in T by (2.4.18), so it follows that

ϕL(t, T) ≤ ϕL(t, ti) ≤ B(t, T) ≤ 0 ,

for t ∈ (ti−1, ti].

Now consider the case where B(·, T) is positive on (ti−1, ti]. We have thatϕU(t, ti) ≥ 0 by (2.4.20) so

∂t ϕU(t, ti) ≤ L(t, ϕU(t, ti)) , (2.4.24)

for t ∈ (ti−1, ti]. Hence by Lemma 2.4.3 the solution B(t, T) is bounded fromabove by ϕU(t, ti) on (ti−1, ti]. Since ϕU(t, T) is increasing in T, see Lemma 2.4.4,it follows that

0 ≤ B(t, T) ≤ ϕU(t, ti) ≤ ϕU(t, T) ,

for t ∈ (ti−1, ti].

Since ϕL(t, T) ≤ 0 and ϕU(t, T) ≥ 0 on all intervals (ti−1, ti] we see thatϕL(t, T) ≤ B(t, T) ≤ ϕU(t, T) on all these intervals, which proves the result asstated.

The following lemma establishes conditions under which the Laplace trans-form Eq. (2.4.2) has a finite-valued solution.

Lemma 2.4.6. Suppose that ξ(t) = ξ for some ξ ∈ ]0, ∞[. If the conditions ofLemma 2.4.5 hold, then the (unique) solution B(t, T) to the Riccati equation

∂tB(t, T) = L(t, B(t, T)) , 0 ≤ t ≤ T ,

B(T, T) = 0 ,(2.4.25)

in which L is defined as in Eq. (2.4.4), exists and is bounded on [0, T]. MoreoverB(t, T) is continuously differentiable and ∂tB(t, T) is bounded for t ∈ [0, T].

Proof.

If we can show that a solution B(t, T) exists and is bounded for all t ∈ [0, T],then we immediately have that ∂tB(t, T) is bounded for all such t due toEq. (2.4.25).

The righthand side of the Riccati equation (2.4.25) is continuous (since h and κare continuous) and locally Lipschitz continuous in the second argument witha Lipschitz constant which does not depend on time. From Picard-Lindelöf,

2.5 the optimal consumption and investment strategy 29

see Teschl (2012, Thm. 2.2), we know that there exists a unique solution B(·, T)to the Riccati differential equation on the interval (T′, T] for some T′ < T.

It follows from Lemma 2.4.5 that B(·, T) is bounded on (T′, T]. Hence, byarguments similar to those used in the proof of Lemma 2.4.2, the interval ofexistence can be extended to [0, T].

2.5 the optimal consumption and investment strategy

In this Section we will prove Theorem 2.3.2. First, to show that the marketintroduced in Section 2.2 does not admit arbitrage, we will construct an equiv-alent measure P such that all assets, expressed in units of the money-marketaccount, are P-martingales. We can construct such a measure P using the likeli-hood process defined by the Doléans-Dade exponential of the market price ofrisk, as the next lemma shows. Note that Novikov’s condition is not satisfiedin general for this stochastic exponential, see Kraft (2003, p. 59).

Lemma 2.5.1. The stochastic exponential

Z0(t) = exp−∫ t

0θ(u)dW(u)− 1

2

∫ t

0||θ(u)||2du

(2.5.1)

defines a Radon-Nikodym derivative and the associated measure

P(A) = EP [Z0(T)1A] , for all A ∈ F (T) , (2.5.2)

is a martingale measure for the economy, i.e. β-discounted asset prices are martingalesunder this measure.

Proof. Using Lemma 2.4.2 together with the boundedness of the functionst 7→ ξiBj(t, T1), i, j = 1, 2, and t 7→ ψi(t)/ξi, i = 1, 2, it can be shown thatβ-discounted prices of tradeables are P-martingales. A detailed proof is givenin Section 2.8.

From Lemma 2.5.1 we conclude that the economy does not admit arbitrage;and it follows from Remark 2.2.1 that the economy introduced in Section 2.2is complete so that the martingale measure defined in Lemma 2.5.1 is unique,see Karatzas and Shreve (1998, Thm. 1.4.2, Thm. 1.6.6. and Prop. 1.7.4).

Remark 2.5.2. From a standard result in arbitrage theory6 the price of a survivalbond, which pays the amount F(T1) at time T1, is given by

F(t, T1) = β(t)EP

[β(T1)

−1 F(T1)∣∣∣F (t)] . (2.5.3)

6 See for example Karatzas and Shreve (1998, Prop. 2.2.3).


The survival distribution F is bounded, hence we may apply the Feynman-Kac theo-rem7 to express the price of the survival bond in terms of the solution to the Riccatidifferential equation (2.2.9). We conclude that the price process (2.2.8) of the survivalbond coincides with the replication cost of its payoff.

Applying Itô’s lemma to the product of the state price density H(t) = Z0(t)/β(t)and the wealth process Xx,c,π yields, by Eqs. (2.2.10) and (2.5.1),

H(t)Xx,c,π(t) +∫ t

0H(u)c(u)du

= x +∫ t

0H(u)[S (u, T1)π(u)− Xx,c,π(u)θ(u)]′dW(u) ,

(2.5.4)

for 0 ≤ t ≤ T. The definition of admissible strategies requires the wealthprocess and consumption to be nonnegative. Consequently every admissiblestrategy (c, π) satisfies the budget constraint

E

[H(T)Xx,c,π(T) +

∫ T

0H(u)c(u)du

]≤ x . (2.5.5)

Indeed, since the lefthand side of Eq. (2.5.4) is nonnegative if (c, π) ∈ A , whilethe righthand side is a local martingale, it follows that the righthand side ofEq. (2.5.4) is a supermartingale. Taking expectations in Eq. (2.5.4) thus leads tothe budget constraint (2.5.5); this inequality will play an essential role in theverification of the candidate optimal strategy in the proof of Lemma 2.5.4.

The stopping time τ, which models the time of death of the investor, is notadapted to the filtration F (t) since it depends on both the mortality rate andthe exponentially distributed random variable Θ. Using iterated conditioningthe following alternative representation of the value function can be estab-lished.

Lemma 2.5.3. The value function equals

V(x) = sup(c,π)∈A

E

[ ∫ T

0U1(t, c(t))dt + U2(Xx,c,π(T))

], (2.5.6)

in which U1(t, x) = F(t)U1(x) and U2(x) = F(T)U2(x).

Proof. See Section 2.8.

The following lemma, the proof of which is adapted from Karatzas and Shreve(1998), provides conditions ensuring that the problem from Definition 2.3.1

7 See Karatzas and Shreve (1991, Thm. 5.7.6).


has a unique solution. Let I1 : [0, T]× (0, ∞] → [0, ∞) and I2 : (0, ∞] → [0, ∞),defined by

I1(t, y) = F(t)1

1−p y1

p−1 and I2(y) = Ψ F(T)1

1−p y1

p−1 , (2.5.7)

with Ψ = Υ1

1−p , be the inverses of the marginal utility functions c 7→ ∂c U1(t, c)and x 7→ ∂xU2(x) respectively.

Lemma 2.5.4. Let x > 0 be an amount of initial wealth. There exists an optimalstrategy (c, π) ∈ A for Problem 2.3.1 if there is a v(x) ∈ [0, ∞[ such that

w(0, v(x)) = x , (2.5.8)

where

w(t, y) =1

H(t)E

[∫ T

tH(s)I1(s, yH(s))ds + H(T)I2(yH(T))

∣∣∣∣F (t)] .

The optimal consumption strategy c and optimal wealth process X are given by

c(t) = I1(t, v(x)H(t)) , (2.5.9)

and

X(t) = w(t, v(x)) . (2.5.10)

If V(x) < ∞ then the optimal strategy is unique up to almost-everywhere equivalence.

Proof.

First we will construct an investment strategy π such that the wealth pro-cess Xx,c,π , with consumption c as in Eq. (2.5.9), replicates the (nonnegative)candidate optimal wealth process Eq. (2.5.10) and, in particular, leads to theterminal wealth

ζ := w(T, v(x)) = I2(v(x)H(T)) . (2.5.11)

Let M(t) = H(t)X(t) +∫ t

0 H(u) c(u) du with c(t) and X(t) as in Eqs. (2.5.9)–(2.5.10), and note that

M(t) = E

[∫ T

0H(t)I1(t, v(x)H(t))dt + H(T)I2(v(x)H(T))

∣∣∣∣F (t)] .


Since E|M(t)| = w(0, v(x)) = x < ∞ we conclude that M is a Doob martin-gale. The martingale representation theorem8 guarantees the existence of anadapted process γ satisfying

∫ T0 ||γ(u)||

2du < ∞ and

M(t) = x +∫ t

0γ(u) dW(u) .

The reciprocal of the process Z0 defined in (2.5.1) satisfies

Z−10 (t)θ(t)dW(t) = Z−1

0 (t)θ(t)(dW(t) + θ′(t)dt) =: d(Z−10 (t)) .

Applying Itô’s lemma we find

d(

X(t)β(t)

)= Z−1

0 (t)d(H(t)X(t)) + H(t)X(t)d(Z−10 (t))

+ d⟨

HX, Z−10

⟩t

= Z−10 (t)

(γ(t)dW(t)− H(t)c(t)dt

)+ H(t)X(t)θ(t)Z−1

0 (t)dW(t)

=1

β(t)1

H(t)

[γ(t)+

(M(t)−

∫ t

0H(u)c(u)du

)θ(t)

]dW(t)

− 1β(t)

c(t)dt . (2.5.12)

Comparing this expression to Eq. (2.2.10) we find that if we choose the assetallocation

π(t) =1

H(t)

[γ(t) +

(M(t)−

∫ t

0H(u)c(u)du

)θ(t)

]S (t, T1)

−1 , (2.5.13)

then the corresponding wealth process Xx,c,π satisfies Xx,c,π(t) = X(t), and inparticular Xx,c,π(T) = ζ. It remains to show that the allocation (2.5.13) definesa portfolio process, and that the pair (c, π) is admissible. The paths of θ arealmost surely continuous since we required the functions ψi to be continuous.Therefore we have

∫ T0 ||θ(t)||

2 dt < ∞ and it follows that max0≤t≤T 1/Z0(t) <∞ almost surely. Moreover, the paths of M, β and t 7→

∫ t0 H(u)c(u) du are con-

tinuous so that their supremum is finite almost surely on the interval [0, T]. Re-

8 See Karatzas and Shreve (1991, pp. 182–184).


call that, by the martingale representation theorem, ||γ||22 =∫ T

0 ||γ(u)||2du <

∞. From these estimates it follows that∫ T

0|π(t)S (t, T1)θ

′(t)| dt

=∫ T

0

∣∣∣∣ 1H(t)

γ(t) +

(M(t)−

∫ t

0H(u)c(u)du

)θ(t)

θ′(t)

∣∣∣∣ dt

≤ ||β||∞||Z−10 ||∞

||γ||2||θ||2 + ||θ||22

(||M||∞+

∫ T

0H(u)c(u)du

)< ∞ .

In the last step we used the Cauchy-Schwarz inequality. Likewise,

∫ T

0

||γ(t)||+ ||

(M(t)−

∫ t

0H(u)c(u)du

)θ(t)||

2dt

≤ ||γ||22 +(||M||∞+

∫ T

0H(u)c(u)du

)2||θ||22

+ 2(||M||∞+

∫ T

0H(u)c(u)du

)||γ||2||θ||2 < ∞ .

Hence, by the triangle inequality,∫ T

0||π(t)S (t, T1)||2 dt

=∫ T

0||β(t)Z−1

0 (t)

γ(t) +(

M(t)−∫ t

0H(u)c(u)du

)θ(t)

||2 dt

≤ ||β||2∞||Z−10 ||

2∞

∫ T

0

||γ(t)||+ ||

(M(t)−

∫ t

0H(u)c(u)du

)θ(t)||

2dt

< ∞ .

We conclude that π is a portfolio process. Finally, the pair (c, π) is admissiblesince the wealth process (2.5.10) and the consumption process are nonnegativeby construction.

Next we will establish optimality and uniqueness of the optimal strategy. Dueto Lemma 2.5.3 we may rewrite the problem from Definition 2.3.1 as


E

[ ∫ T

0U1(t, c(t))dt + U2(Xx,c,π(T))

], (2.5.14)


in which U1(t, x) = F(t) xp

p and U2(x) = Υ F(T) xp

p are concave functions of x.Let (c′, π′) be an admissible pair. Due to the concavity of U1(t, ·) and Eq. (2.5.9)we have

U1(t, c′(t))−U1(t, c(t)) ≤ (c′(t)− c(t)) ∂cU1(t, c(t))= (c′(t)− c(t)) v(x)H(t) , (2.5.15)

almost surely for every 0 ≤ t ≤ T. Similarly, due to the concavity of U2,

U2(Xx,c′,π′(T))−U2(ζ) ≤ (Xx,c′,π′(T)− ζ) ∂xU2(ζ)

= (Xx,c′,π′(T)− ζ) v(x)H(T) , (2.5.16)

almost surely. Integrating the inequalities (2.5.15) and (2.5.16) we find

E

[ ∫ T

0U1(t, c′(t))dt + U2(Xx,c′,π′(T))

]−E

[ ∫ T

0U1(t, c(t))dt + U2(ζ)

]≤ v(x)E

[ ∫ T

0(c′(t)− c(t))H(t)dt + (Xx,c′,π′(T)− ζ)H(T)

]≤ 0 ,

by the budget constraint (2.5.5). This proves optimality. Uniqueness (up toalmost-everywhere equivalence) follows from Eq. (2.5.15) and Eq. (2.5.16) pro-vided that V(x) < ∞.

Remark 2.5.5. If w(0, 1) < ∞ then condition (2.5.8) in Lemma 2.5.4 is satisfied for

any initial level of wealth x > 0. Indeed, since w(0, y) = y1

p−1 w(0, 1) , it followsthat if w(0, 1) < ∞ then the function v : ]0, ∞[→ [0, ∞[ defined by

v(x) = xp−1w(0, 1)1−p (2.5.17)

is the inverse of x 7→ w(0, x).

The following auxiliary result gives the multiplicative Doob-Meyer decompo-sition9 of H(s)I1(s, H(s)), for 0 ≤ s ≤ T, and of H(T)I2(H(T)).

Lemma 2.5.6. Assume w(0, 1) < ∞. The optimal consumption and optimal terminalwealth satisfy

H(s)I1(s, H(s)) = Λ(s)m(s) , (2.5.18)

9 See Jamshidian (2007, Prop. 4.2).


for 0 ≤ s ≤ T, and

H(T)I2(H(T)) = ΨΛ(T)m(T) , (2.5.19)

in which

Λ(t) = exp

p

1− p

∫ t

0θ(u)dW(u)− 1

2

(p

1− p

)2 ∫ t

0||θ(u)||2du

(2.5.20)

is a local martingale and

m(t) = exp− 1

1− p

∫ t

0λ(u)du +

p1− p

∫ t

0r(u)du

+p

2(1− p)2

∫ t

0||θ(u)||2du

,

(2.5.21)

is a process of finite variation.

Proof. The identities (2.5.18) and (2.5.19) follow from straightforward computa-tion.

We proceed to derive expressions for the optimal consumption strategy andoptimal terminal wealth which are explicit up to the evaluation of a condi-tional expectation. Later we will show that this conditional expectation has arepresentation as a Laplace transform that can be evaluated using Lemma 2.4.1.

Proposition 2.5.7. Let

n(t) =∫ T

tL(t, u)du + Ψ L(t, T) , (2.5.22)

in which

L(t, s) =1

Λ(t)m(t)E [Λ(s)m(s)|F (t)] . (2.5.23)

If w(0, 1) < ∞ then the optimal consumption and wealth process for the problem inDefinition 2.3.1 are given by

c(t) =1

n(t)X(t) , X(t) =

xH(t)

n(t)n(0)

Λ(t)m(t) . (2.5.24)

The value function is finite and satisfies

V(x) =

1p

n(0)1−p xp , if p 6= 0 ,

−M0 + n(0) log(

xn(0)

), if p = 0 ,

(2.5.25)

in which M0 is a (finite) constant, given by Eq. (2.8.6).



In the special case where the short rate is deterministic and λ ≡ 0 we find thatformula (2.5.22) simplifies to n(t) = 1/m(t)

∫ Tt m(u)du + m(T)/m(t) . We thus

see that Proposition 2.5.7 in this case coincides with the solution of Merton’sclassical portfolio optimization problem.

The following result provides conditions on the parameters of the short rateand the mortality rate process that imply w(0, 1) < ∞; thus ensuring, by Re-mark 2.5.5, the existence of a solution for the problem from Definition 2.3.1.

Proposition 2.5.8. Assume that the functions ψi, i = 1, 2, are analytic on [0, T].The problem 2.3.1 has a solution which is unique up to almost-everywhere equivalencewhen p ≤ 0. When p ∈ ]0, 1[ this is still the case if the parameters of the short rateand mortality rate processes satisfy

mins∈[0,T]

(κi +

p1−p ψi(s)

)> 0 , (2.5.26)

for i = 1, 2 and

p [ (κ1 − ψ1,min)2 +


2

1−p ] < κ21 − 2pξ2

1 , (2.5.27)

p [ (κ2 − ψ2,min)2 +


2

1−p ] < κ22 + 2ξ2

2 , (2.5.28)

whereψi,min = min

s∈[0,T]ψi(s), (ψ2

i )max = maxs∈[0,T]

ψi(s)2.

Furthermore, for given T > 0, L(t, T) has an affine representation which is almostsurely continuous in t and bounded on the domain 0 ≤ t ≤ T.

Proof.

By Lemma 2.4.2 the stochastic exponential Λ in (2.5.20) is a P-martingale on[0, T] since we required the functions ψi(t), i ∈ 1, 2, 3, to be continuous andbounded on [0, T]. Hence we may define the measure

∧P by

d∧P

dP= Λ(T) . (2.5.29)

From Girsanov’s Theorem10 we find that the short rate

r(t) = r0 +∫ t

0(µ1(u)−

∧κ1(u)r(u))du +

∫ t

0ξ1

√r(u) d

∧W1(u) ,

10 See Karatzas and Shreve (1991, Thm. 3.5.1).


in which∧

W1(t) = W1(t) − p1−p

∫ t0 θ′1(s)ds, ∧

κ1(t) = κ1 +p

1−p ψ1(t), and themortality rate

λ(t) = λ0 +∫ t

0(µ2(u)−

∧κ2(u)λ(u))du +

∫ t

0ξ2

√λ(u) d

∧W2(u) ,

in which∧

W2(t) = W2(t) − p1−p

∫ t0 θ′2(s)ds, ∧κ2(t) = κ2 +

p1−p ψ2(t), remain

independent under∧P. Using Bayes’ rule we can rewrite Eq. (2.5.23) as

L(t, T) =1

Λ(t)m(t)EP [Λ(T)m(T)|F (t)] =

1m(t)

E∧P[m(T)|F (t)] .

Hence we find, using Eq. (2.5.21),

L(t, T) = exp−∫ T

th3(u) du

E∧

P

[exp

−∫ T

th1(u)r(u)du

∣∣∣∣ r(t)]

× E∧P

[exp

−∫ T

th2(u)λ(u)du

∣∣∣∣ λ(t)]

, (2.5.30)

where

h1(t) = −1

1− p

(p +

12

p1− p

(ψ1(t)

ξ1

)2)

,

h2(t) = −1

1− p

(−1 +

12

p1− p

(ψ2(t)

ξ2

)2)

,

and

h3(t) = −12

p(1− p)2

(ψ3(t)

ξ3

)2

.

We will now use Lemma 2.4.6 to prove that the conditions of Lemma 2.4.1 aresatisfied for the Laplace transforms in Eq. (2.5.30); it then follows that L(t, T),0 ≤ t ≤ T, has an affine representation. Since ψi, i = 1, 2, are analytic onan open set containing [0, T],11 it follows from the identity theorem for realanalytic functions12 that, for every α ∈ R, either ψi(t) = α on [0, T] or the zeroset t : ψi(t)− α = 0 does not contain an accumulation point. Consequently,the functions hi are either identically zero on [0, T] or change sign only a

11 A function being analytic on a closed set implies that it is analytic on an open covering of thatset, see Rudin (1976, Thm. 8.4).

12 See for example Rudin (1976, Thm. 8.5).


finite number of times. Moreover, the functions hi are continuous and hencebounded on [0, T]. The last factor in (2.5.30) which involves the function h3is therefore always finite. To apply Lemma 2.4.6 to the first two factors inEq. (2.5.30), we need to verify that condition (2.4.21) is satisfied. For p ≤ 0 wesee that hi(t) ≥ 0 for i = 1, 2 so the condition is always satisfied in that case.We therefore only need to consider p ∈ ]0, 1[. For i = 1 we need to check that

mins∈[0,T]

(κ1 +p

1−p ψ1(s))2 > −2ξ21 min

s∈[0,T]

−p1−p (1 +

12(1−p) (ψ1(s)/ξ1)

2)

=2pξ2

11− p

+p

(1− p)2 maxs∈[0,T]

ψ1(s)2.

But since p > 0 and κ1 +p

1−p ψ1(s) > 0 for all s ∈ [0, T], the minimum ofthe expression on the lefthand side must be attained at the point where ψ1 isminimal. We then multiply both sides of the inequality with 1− p to get

κ21(1− p) + 2pκ1ψ1,min +

p2

(1−p)ψ21,min > 2pξ2

1 + p(1−p) (ψ

21)max

which gives

κ21 − 2pξ2

1 > p(κ1 − ψ1,min)2 − pψ2

1,min −p2

(1−p)ψ21,min + p

(1−p) (ψ21)max

= p [ (κ1 − ψ1,min)2 +


2

(1−p) ].

The case i = 2 can be treated in a similar fashion.

We can thus apply Lemma 2.4.1 to the Laplace transforms in Eq. (2.5.30) andwe find that

L(t, T) = e−A1(t,T)−A2(t,T)−B1(t,T)r(t)−B2(t,T)λ(t)−A3(t,T) , (2.5.31)

in which A3(t, T) =∫ T

t h3(u) du, while Ai(·, T) and Bi(·, T), i = 1, 2, are thesolutions to

∂tBi(t, T) =∧κi(t)Bi(t, T) +

12

ξ2i Bi(t, T)2 − hi(t) , (2.5.32)

∂t Ai(t, T) = −µi(t) Bi(t, T) , (2.5.33)

with initial conditions Ai(T, T) = Bi(T, T) = 0, and these functions are boundedand continuously differentiable by Lemma 2.4.6.

Finally, to establish existence of a solution to the problem in Definition 2.3.1we have to verify that the conditions of Lemma 2.5.4 are satisfied. From Re-mark 2.5.5 and the identity w(0, 1) = n(0) it follows that this is equivalent to


proving that n(0) < ∞, which follows from the almost sure boundedness ofL(t, T) on [0, T].

The next proposition characterizes the optimal hedging strategy. Given anyadmissible nonnegative consumption process, there exists a portfolio processwhich replicates the optimal wealth, see Eq. (2.5.13) in the proof of Lemma 2.5.4.Moreover, the choice of the portfolio process only affects the diffusion term ofthe discounted wealth process, while the drift term depends on the consump-tion process. By comparing the dynamics of the optimal wealth process to thedynamics of the three basic tradeables in the (complete) market, we can deter-mine the trading strategy which finances the optimal consumption plan andleads to the optimal distribution of terminal wealth at time T.

Proposition 2.5.9. Under the conditions of Lemma 2.4.5 the dynamics of the optimalwealth and consumption process satisfy

d(

X(t) +∫ t

0 c(u)du)

X(t)= η0(t)

dβ(t)β(t)

+ η1(t)dP(t, T1)

P(t, T1)

+η2(t)dF(t, T1)

F(t, T1)+ η3(t)

dS(t)S(t)

, (2.5.34)

where,

η0(t) = 1−∑3i=1 ηi(t)

η1(t) = 1B1(t,T1)

(1

1−p

(ψ1(t)

ξ21− ρ ψ3(t)

ξ1ξ3ρ

)+ Ξ1(t)

n(t)

)− η2(t),

η2(t) = 1B2(t,T1)

(1

1−pψ2(t)

ξ22

+ Ξ2(t)n(t)

),

η3(t) = 11−p

ψ3(t)ρ ξ2

3,

(2.5.35)

in which,

Ξi(t) = Ψ Bi(t, T)L(t, T) +∫ T

tBi(t, u)L(t, u) du , (2.5.36)

and Bi are the solutions of (2.5.32) and L is given by Eq. (2.5.31).


We conclude from Proposition 2.5.9 that the optimal strategy invests a fractionη1(t) of wealth in bonds, a fraction η2(t) in survival bonds, a fraction η3(t)in stocks and the remaining wealth η0(t) is invested in the money-market


account. The portfolio weights depend on the short rate r and the mortalityrate λ. These rates are observable due to their affine structure; their values canbe inferred from the price of the zero-coupon bond P(·, T1) and the price ofthe survival bond F(·, T1) respectively.

Proposition 2.5.10. The hedging demand (as a fraction of wealth) for all assets in theeconomy is bounded.

Proof. Fix i ∈ 1, 2 and let ϕL(·, u) and ϕU(·, u) be the family, indexed by u ∈[0, T], of lower and upper bounds for Bi(·, u) as constructed in Lemma 2.4.5.Since ϕL(t, u) is decreasing in u, it follows that

ϕL(t, T)− Ξi(t)n(t)

=1

n(t)

(ϕL(t, T)n(t)− Ξi(t)

)=

1n(t)

∫ T

t

(ϕL(t, T)− Bi(t, u)

)L(t, u)du

+ ΨL(t, T)(

ϕL(t, T)− Bi(t, T))

≤ 1n(t)

∫ T

t

(ϕL(t, T)− ϕL(t, u)

)L(t, u)du

≤ 0 .

Similarly, since ϕU(t, u) is increasing in u,

ϕU(t, T)− Ξi(t)n(t)

≥ 1n(t)

∫ T

t

(ϕU(t, T)− Bi(t, u)

)L(t, u)du

≥ 1n(t)

∫ T

t

(ϕU(t, T)− ϕU(t, u)

)L(t, u)du

≥ 0 .

Hence

ϕL(t, T) ≤ Ξi(t)n(t)

≤ ϕU(t, T) .

2.6 economic analysis of the model

Huang et al. (2012) establish a relation between the parameter p of the powerutility functions defined in Eqs. (2.3.2) and (2.3.3), and the optimal initial con-sumption c(0). In Theorem 2.6.1 we show that a similar relation holds in ourmodel setup.

2.6 economic analysis of the model 41

Theorem 2.6.1. Consider two models: the first model has a deterministic mortalityrate λdet, while in the second model the mortality rate λstoch is stochastic. Assumethat ψstoch

2 (t) ≥ 0,13 whereas ψdet2 (t) = 0.14 If the survival probabilities are the same

in both models, that is, if for all s ∈ [0, T]

EP

[e−∫ s

0 λstoch(u)du]

= e−∫ s

0 λdet(u)du , (2.6.1)

then for 0 ≤ p < 1,

cdet(0) ≥ cstoch(0) and Vdet(x) ≤ Vstoch(x) ,

while for p ≤ 0,

cdet(0) ≤ cstoch(0) and Vdet(x) ≥ Vstoch(x) .

We thus find that a stochastic mortality rate, and a nonnegative market priceof longevity risk, lead to a difference in the consumption rate and the valuefunction. The sign of the difference depends on the coefficient of relative riskaversion. Note that in Huang et al. (2012), where it is assumed that longevityrisk is not tradeable, the value function in the stochastic mortality model al-ways exceeds the value function in the deterministic model, regardless of thecoefficient of relative risk aversion. To prove Theorem 2.6.1 we need the fol-lowing Lemma.

Lemma 2.6.2. Assume that ψ2(s) ≥ 0 for all s ∈ [0, T].

If 0 ≤ p < 1, then, for all t ∈ [0, T],

E∧P

[e−∫ T

t λ(u)du∣∣∣ F (t)] ≥ EP

[e−∫ T

t λ(u)du∣∣∣ F (t)] , (2.6.2)

whereas if p ≤ 0, then, for all t ∈ [0, T],

E∧P

[e−∫ T

t λ(u)du∣∣∣ F (t)] ≤ EP

[e−∫ T

t λ(u)du∣∣∣ F (t)] . (2.6.3)


Lemma 2.6.2 relates the survival probability under P to its value under∧P and

shows that their relation depends qualitatively on the risk aversion coefficient.We can now prove Theorem 2.6.1.

13 This corresponds to the case where the drift of the survival bond carries a positive risk pre-mium.

14 If the mortality rate is deterministic, then there is no longevity risk, therefore we haveψdet

2 (t) = 0.


Proof of Theorem 2.6.1.

Fix s ∈ ]0, T] and recall that, by Eq. (2.5.30),

Lstoch(0, s) = cL E∧P

[exp

− 1

1− p

∫ s

0λstoch(u) du

× exp

12

p(1− p)2

∫ s

0||θ2(u)||2 du

],

in which

cL = E∧P

[exp

−∫ s

0h1(u)r(u)du

]exp

−∫ s

0h3(u) du

≥ 0 .

For 0 < p < 1 we find

Lstoch(0, s) ≥ cL E∧P

[φ(

e−∫ s

0 λstoch(u) du)]

,

in which φ(u) = u1

1−p is an increasing, strictly convex function. Hence, byJensen’s inequality

E∧P

[φ(

e−∫ s

0 λstoch(u) du)]≥ φ

(E∧

P

[e−∫ s

0 λstoch(u) du])

. (2.6.4)

Using Lemma 2.6.2 and the fact that φ is an increasing function we concludethat

φ(E∧

P

[e−∫ s

0 λstoch(u) du])≥ φ

(EP

[e−∫ s

0 λstoch(u) du])

,

in which the P-expectation represents the survival probability.

On the other hand, for the deterministic model we find, using the fact thatψdet

2 (t) = 0,

Ldet(0, s) = cL φ(

e−∫ s

0 λdet(u) du)

,

Since, by assumption,

EP

[e−∫ s

0 λstoch(u) du]

= e−∫ s

0 λdet(u) du ,

we conclude that Lstoch(0, s) ≥ Ldet(0, s) for 0 < p < 1 and for all s ∈ ]0, T].Consequently,

nstoch(0) =∫ T

0Lstoch(0, s) ds + Ψ Lstoch(0, T)

≥∫ T

0Ldet(0, s) ds + Ψ Ldet(0, T) = ndet(0) ,

2.7 conclusion 43

and the result for 0 < p < 1 follows from Eqs. (2.5.24) and (2.5.25).

Note that for p = 0 the inequalities above hold as equalities, hence Lstoch(0, s) =Ldet(0, s) for p = 0 and s ∈ ]0, T]. If p < 0 then φ is a concave increasingfunction. In this case all inequalities above are reversed and we find thatLstoch(0, s) ≤ Ldet(0, s) for all s ∈ ]0, T].

The demand for survival bonds in Proposition 2.5.9 has a deterministic compo-nent which earns the risk premium and a stochastic component that representsthe hedging demand. The stochastic part Ξ2(t)/n(t) reduces to a deterministicterm only in the special case where B2 is constant. Due to the boundary con-dition in Eq. (2.5.32) this happens only if B2 ≡ 0. Hence we conclude thatthere will not be a market price of risk function in this model which makesthe demand for longevity derivatives vanish. This means that there is no leastfavourable market completion in the sense of Karatzas et al. (1991). Indeed, inthat paper it is shown that if certain assets are not available for trading, thenthe optimal strategy coincides with the strategy that is optimal for a, fictitious,complete market where the optimal terminal wealth has minimal expectedutility (or, is ‘least favourable’) among all possible market completions. More-over, the optimal terminal wealth in this fictitious completion can be replicatedalmost surely using a strategy that does not invest in the illiquid assets.

2.7 conclusion

In this chapter we have formulated an optimal consumption and investmentproblem with a positive stochastic mortality rate and a time-varying marketprice of risk. We have derived conditions which guarantee that there is noconvex set of admissible investment strategies, or even a trivial strategy whichonly invests in the bank account, which give rise to infinite expected utility ofconsumption or terminal wealth. We can characterize the optimal consump-tion and investment strategy explicitly and show that the hedging demand forall assets in the economy remains bounded.

In our setup we have completed the market by the introduction of an assetwhich allows the investor to earn a risk premium for longevity risk. If such anasset would not be available, the market becomes incomplete in the sense thatnot all mortality-dependent terminal wealth and consumption profiles can begenerated by the investor. A common approach for handling incompletenessin optimal investment problems is to search for the ‘least favourable’ marketcompletion, that is, the market price of risk under which it is optimal not toinvest in a non-tradeable asset such as the survival bond, see Karatzas et al.


(1991). In our market setting no specification of the market price of longevityrisk can make the survival bond redundant, as can easily be seen from ouroptimal portfolio equations. This means that an investor will always benefitfrom the extension of investment opportunities that is provided by longevity-based derivatives.

Our results show that for investors who are only moderately risk averse15,restrictions must be imposed on the risk premium for longevity risk to get awell-posed problem. These restrictions depend on the level of risk aversion, onthe mean reversion speed and on the volatility of the mortality rate process.Some care must therefore be taken if one wants to use utility indifferencepricing methods to extend classical asset pricing theory to survival bonds,longevity swaps and other mortality-dependent financial products.

2.8 proofs


From Lemma 2.4.2 it follows that, for i ∈ 1, 2, 3 and 0 ≤ t ≤ T,

E

[exp

−∫ t

0θi(u)dWi(u)−

12

∫ t

0||θi(u)||2du

]= 1 .

Since W1, W2 and W3 are independent we conclude that Z0 is a martingale,and Eq. (2.5.2) defines by Girsanov’s theorem, see Karatzas and Shreve (1991,Thm. 3.5.1), a measure equivalent to P. Moreover, W(t) = W(t) +

∫ t0 θ(u)du is

a P-Brownian motion.

From Itô’s lemma we obtain that the product of the state price density H(t) =Z0(t)/β(t) and the stock price satisfies

H(t)S(t) = S(0) +∫ t

0[σS(u)− θ(u)] H(u)S(u)dW(u) . (2.8.1)

The product of the bond price and the state price density satisfies

H(t)P(t, T1) = P(0, T1) +∫ t

0[σP(u)− θ(u)] H(u)P(u, T1)dW(u) . (2.8.2)

Similarly, for the survival bond we find

H(t)F(t, T1) = F(0, T1) +∫ t

0[σF(u)− θ(u)] H(u)F(u, T1)dW(u) . (2.8.3)

15 in the sense that our risk aversion parameter p is larger than zero

2.8 proofs 45

Observe that the processes in Eq. (2.8.1), (2.8.2) and (2.8.3) are local martingalesand can be rewritten as stochastic exponentials of the form (2.4.7). Since thefunctions t 7→ ξiBj(t, T1), i, j = 1, 2, and t 7→ ψi(t)/ξi, i = 1, 2, are boundedand continuous, and using the independence of Wi, i = 1, 2, 3, under P, weconclude from Lemma 2.4.2 that the products of S, P(·, T1) and F(·, T1) withthe state price density H are P-martingales. Since Z0(t) is a Radon-Nikodymderivative by Girsanov’s theorem, it follows that the β-discounted prices ofthese tradeables are P-martingales.


Using the tower rule and the fact that U2(Xx,c,π(T)) is F (T)-measurable weobtain

E [U2(Xx,c,π(T))1τ>T ] = E [E [U2(Xx,c,π(T))1τ>T |F (T)]]= E [U2(Xx,c,π(T))E [1τ>T |F (T)]]= E

[U2(Xx,c,π(T))F(T)

].

Using Fubini’s theorem (the integrand is nonnegative) and the fact that U1(c(t))is F (t)-progressively measurable, we can apply the same argument to obtain

E

[ ∫ T

0U1(c(t)) 1τ>t dt

]= E

[ ∫ T

0E [U1(c(t)) 1τ>t | F (t)] dt

]= E

[ ∫ T

0F(t)U1(c(t))dt

].

Consequently,


E

[∫ T∧τ

0U1(c(t))dt +U2(Xx,c,π(T))1τ>T

]= sup

(c,π)∈A

E

[ ∫ T

0U1(c(t)) 1τ>t dt + U2(Xx,c,π(T))1τ>T

]= sup

(c,π)∈A

E

[ ∫ T

0F(t)U1(c(t))dt + F(T)U2(Xx,c,π(T))

].

Lemma 2.8.1. The utility functions U1(t, ·) and U2, defined in Lemma 2.5.3, satisfythe relations

U1(s, I1(s, y)) =

1p

y I1(s, y) , p 6= 0 ,

m(s) log(m(s)/y) , p = 0 ,(2.8.4)


for 0 ≤ s ≤ T and for all y > 0, and

U2(I2(y)) =

1p

y I2(y) , p 6= 0 ,

Ψ m(T) log(Ψ m(T)/y) , p = 0 ,(2.8.5)

for all y > 0.

Proof.

If p 6= 0 then for power utility it holds that p U1(t, x) = x ∂xU1(t, x). Takex = I1(t, y) then p U1(t, I1(t, y)) = I1(t, y) ∂xU1(t, I1(t, y)) = y I1(t, y). Theother identities follow from similar computations.


From the proof of Lemma 2.5.4 we know that the optimal consumption strat-egy and the optimal terminal wealth are given by Eq. (2.5.9) and Eq. (2.5.11)respectively. Observe that the integrand of the optimal wealth process (2.5.10)is positive; hence we may apply the conditional Fubini theorem together withLemma 2.5.6 to obtain

X(t) =1

H(t)E

[∫ T

tH(u)I1(u, v(x)H(u))du + H(T)I2(v(x)H(T))

∣∣∣∣F (t)]=

1H(t)

v(x)1

p−1 Λ(t)m(t)(∫ T

tL(t, u)dt + Ψ L(t, T)

)=

xH(t)

n(t)n(0)

Λ(t)m(t) .

In the last step we used Eq. (2.5.17) and the fact that w(0, 1) = n(0). For theoptimal consumption process Eq. (2.5.9) we find, using Eq. (2.5.18),

c(t) = I1(t, v(x)H(t)) =x

w(0, 1)I1(t, H(t)) =

xw(0, 1)

Λ(t)m(t)H(t)

=X(t)n(t)

.

By substitution of the optimal terminal wealth (2.5.11) and optimal consump-tion strategy (2.5.9) into the value function Eq. (2.5.14) we obtain

V(x) = v(x)p

p−1 E

[∫ T

0U1(t, I1(t, H(t))) dt + U2(I2(H(T)))

].

From Lemma 2.8.1 we have for p 6= 0 that

E [U1(t, I1(t, H(t)))] = Ψ−1E [U2(I2(H(t)))] = p−1L(0, t) .

2.8 proofs 47

Consequently,

V(x) =1p

v(x)p

p−1 n(0) =1p

(x

w(0, 1)

)pn(0) =

1p

n(0)1−p xp < ∞.

The value function for the case p = 0 is given by

V(x) = −M + n(0) log(

xn(0)

),

in which

M0 = E

[∫ T

0F(t) log

(H(t)F(t)

)dt + Ψ F(T) log

(H(T)

ΨF(T)

)], (2.8.6)

is a constant and

log(

H(t)F(t)

)= −

∫ t

0r(u) du+

∫ t

0λ(u) du−

∫ t

0θ(u)dW(u)− 1

2

∫ t

0||θ(u)||2 du .

The terms on the righthand side are bounded in expectation. To check this for

the first term, take κ21

−2ξ21< h < 0 so that the Laplace transform E [e−h

∫ t0 r(u) du]

is finite by Lemma 2.4.1 and Lemma 2.4.6. It follows that∫ t

0 r(u) du has finitemoments of any order. The other terms can be bounded in expectation by sim-ilar arguments. In particular, the stochastic integral over the Brownian motionis a martingale and vanishes in expectation. Thus M < ∞.

Lemma 2.8.2 (Leibniz’ rule for Itô integrals). For all 0 < T < ∞ define the processL(·, T) by

L(t, T) = L(0, T) +∫ t

0µ(u, T) du +

2

∑i=1

∫ t

0ξi(u, T)dWi(u) (2.8.7)

where W(t) = (W1(t), W2(t)) is a Brownian motion, µ is an adapted function[0, T]× [0, T]×Ω→ R and ξ is an adapted function [0, T]× [0, T]×Ω→ R2.

If for i = 1, 2 ,∫ t

0

∫ t

0|µ(u, s)|ds du < ∞ a.s. for all t ∈ [0, T] , (2.8.8)

∫ t

0

ξi(u, s)1u≤s

2du < ∞ a.s. for all t ∈ [0, T] and s ∈ [0, T] , (2.8.9)


∫ t

0

∫ T

0ξi(u, s)1u≤s ds

2

du < ∞ a.s. for all t ∈ [0, T] , (2.8.10)

and

t 7→∫ T

0

∫ t

0ξi(u, s)1u≤s dWi(u)

2ds (2.8.11)

is almost surely continuous then, for fixed T,

d(∫ T

tL(t, s)ds

)=

−L(t, t) +

∫ T

tµ(t, s)ds

dt

+2

∑i=1

∫ T

tξi(t, s)ds

dWi(t) .

Proof.

A proof can be found in Munk and Sørensen (2000, Section A). The conditionsof the Lemma are taken from Heath et al. (1992) and are required to justify theinterchange of a stochastic integral and a Lebesgue integral.


Applying Itô’s lemma to the optimal wealth process in Eq. (2.5.24) gives

X(t)β(t)

=x

n(0)Λ(t)Z0(t)

m(t)n(t)

= x +∫ t

0α0(u) du +

∫ t

0

x m(u)n(u)n(0)

d(Λ(u)Z−1

0 (u))

+∫ t

0

x m(u)Λ(u)n(0)Z0(u)

dn(u)

= x +∫ t

0α1(u) du +

∫ t

0

11− p

X(u)β(u)

θ(u)dW(u)

+∫ t

0

1n(u)

X(u)β(u)

dn(u) , (2.8.12)

for some processes α0 and α1. Notice that the last stochastic integral on therighthand side has a diffusion component and a component of finite varia-tion. To determine the dynamics of the process n(t) =

∫ Tt L(t, u)du + ΨL(t, T)

2.8 proofs 49

we apply Lemma 2.8.2. First, we verify that the conditions of this lemma aresatisfied. From Eq. (2.5.31) and Itô’s lemma we obtain

L(t, T) = L(0, T)−∫ t

0

[r(u)∂tB1(u, T) + λ(u)∂tB2(u, T)

+ ∑3i=1 ∂t Ai(u, T)

]L(u, T) du −

∫ t

0B1(u, T)L(u, T) dr(u)

−∫ t

0B2(u, T)L(u, T) dλ(u) +

12

∫ t

0B1(u, T)2d 〈r, r〉u

+12

∫ t

0B2(u, T)2d 〈λ, λ〉u . (2.8.13)

Observe that Bi(·, s) is bounded, uniformly in s, for i = 1, 2 by Lemma 2.4.5.Consequently the same holds for Ai(·, s), for i = 1, 2, and hence for L(·, s).Moreover, the paths of r are almost surely continuous, hence finite on [0, T]. Itfollows that for all t ∈ [0, T] and all s ∈ [0, T]∫ t

0

L(u, s)B1(u, s)ξ1

√r(u)

2du < ∞, a.s., (2.8.14)

and, for all t ∈ [0, T]∫ t

0

∫ T

0L(u, s)B1(u, s)ξ1

√r(u) ds

2

du < ∞, a.s. (2.8.15)

Similarly, from the continuity of λ we have, for all t ∈ [0, T] and all s ∈ [0, T]∫ t

0

L(u, s)B2(u, s)ξ2

√λ(u)

2du < ∞, a.s., (2.8.16)

and, for all t ∈ [0, T]∫ t

0

∫ T

0L(u, s)B2(u, s)ξ2

√λ(u) ds

2

du < ∞, a.s. (2.8.17)

Hence conditions (2.8.9) and (2.8.10) are met. It follows from Eqs. (2.5.32)–(2.5.33), Lemma 2.4.5 and Eqs. (2.4.17)–(2.4.18) that the derivatives ∂t Ai(·, s),∂tBi(·, s), i = 1, 2, are bounded uniformly in s ∈ [0, T]. Therefore, the driftterm in Eq. (2.8.13) satisfies condition (2.8.8). Thus we can apply Lemma 2.8.2and conclude that, for some process α2,

d(∫ T

tL(t, u)du

)= α2(t) dt −

∫ T

tB1(t, u)L(t, u)du dr(t)

−∫ T

tB2(t, u)L(t, u)du dλ(t) . (2.8.18)


For Ξ1 and Ξ2 as in Eq. (2.5.36), we obtain

dn(t) = d(∫ T

tL(t, u)du + ΨL(t, T)

)= α3(t) dt − Ξ1(t) dr(t) − Ξ2(t) dλ(t) , (2.8.19)

for a certain process α3. Substituting (2.8.19) into (2.8.12) we find, for certainprocesses α4 and α5,

X(t)β(t)

= x +∫ t

0α4(u) du

+∫ t

0

X(u)β(u)

1

1− p

(−ψ1(u)

ξ1

√r(u)dW1(u)−

ψ2(u)ξ2

√λ(u)dW2(u)

−ψ3(u)ξ3

dW3(u))− 1

n(u)

(Ξ1(u)dr(u) + Ξ2(u)dλ(u)

)= x +

∫ t

0α5(u) du−

∫ t

0

X(u)β(u)

(1

1− pψ3(u)ξ2

3 ρ

)ξ3 ρ dW3(u)

+∫ t

0

X(u)β(u)

(1

1− p−ψ2(u)

ξ22− Ξ2(u)

n(u)

)ξ2

√λ(u) dW2(u)

+∫ t

0

X(u)β(u)

(1

1− p−ψ1(u)

ξ21− Ξ1(u)

n(u)

)ξ1

√r(u) dW1(u) . (2.8.20)

On the other hand, if η is the vector of fractions of wealth invested in each ofthe asset classes, then from Eq. (2.2.10) we have

X(t)β(t)

= x−∫ t

0

c(u)β(u)

du +∫ t

0

X(u)β(u)

η(u)S (u, T1) dW(u) . (2.8.21)

The optimal portfolio weights are obtained when we use martingale represen-tation to equate the diffusion terms in Eq. (2.8.20) and Eq. (2.8.21).


From Lemma 2.4.1 we know that the survival probability, which is calculatedas an expectation under P, satisfies

EP

[e−∫ T

t λ(u)du∣∣∣ F (t)] = e−A2(t,T)−B2(t,T) λ(t) ,

where B2(·, T) solves∂tB2(t, T) = κ2B2(t, T) + 1

2 ξ22 B2(t, T)2 − 1 ,

B2(T, T) = 0 ,

2.8 proofs 51

and A2(t, T) =∫ T

t µ2(u)B2(u, T) du .

Similarly, under∧P we have

E∧P

[e−∫ T

t λ(u)du∣∣∣ F (t)] = e−A2(t,T)−B2(t,T) λ(t) ,

where B2(·, T) solves∂tB2(t, T) =

∧κ2(t)B2(t, T) + 1

2 ξ22 B2(t, T)2 − 1 ,

B2(T, T) = 0 ,

and A2(t, T) =∫ T

t µ2(u)B2(u, T) du .

Further note that B2(t, T) ≥ 0 and B2(t, T) ≥ 0 for all t ∈ [0, T] due to thenegative source term and the terminal condition.

Consider first the case where 0 ≤ p < 1. Since ψ2(t) ≥ 0 by assumption, wehave

∧κ2(t) = κ2 +

p1− p

ψ2(t) ≥ κ2 ,

for all t. By Lemma 2.4.3 we thus obtain that 0 ≤ B2(t, T) ≤ B2(t, T). Thisimplies that e−B2(t,T)z ≥ e−B2(t,T)z for all z ≥ 0. Moreover,

A(t, T) =∫ T

tµ2(u)B(u, T) du ≤

∫ T

tµ2(u)B(u, T) du = A(t, T) ,

hence

e−A2(t,T)−B2(t,T)z ≥ e−A2(t,T)−B2(t,T)z ,

for all z ≥ 0.

Inequality (2.6.3) is obtained by similar reasoning. Note that if p ≤ 0 then∧κ2(t) ≤ κ2 for all t, hence the direction of the inequalities are reversed.

Part II

L O N G - T E R M I N T E R E S T R AT E R I S K

3 C O N V E R G E N C E O F T H E L O N G - T E R M I N T E R E S T R AT EI N H E AT H - J A R R O W- M O RT O N M O D E L S

In this chapter we discuss the behaviour of long-term interest rates in HeathJarrow-Morton (HJM) models. Given a mode of convergence, the long rateis defined to be the asymptotic interest rate in the limit when the expirationdate tends to infinity, provided that this limit exists. If the interest rate isa forward rate, then this quantity is commonly referred to as the ultimatefoward rate (UFR).

In Section 3.1, the HJM framework is introduced, which is a general interestrate model that allows for a large range of forward rate dynamics. In Subsec-tion 3.3.1 and 3.3.2 we review existing results on ucp and almost sure conver-gence of long-term rates, some of which have appeared earlier in Deelstra andDelbaen (1995), El Karoui et al. (1998), Yao (1999a) and Biagini et al. (2016), andwe present those results in a unified framework. In Subsection 3.3.3 we studythe existence of the long rate as a limit in the S1 norm, which is stronger thanucp convergence, and we provide necessary and sufficient conditions suchthat these limits exist, formulated in terms of the HJM volatility function.

3.1 the hjm framework

Consider a probability space (Ω,F , P) on which W = (W1, . . . , Wd)′ is a d-

dimensional Brownian motion, with independent components, under P. Letp(t, T) denote the time t ≥ 0 price of a zero-coupon bond that pays 1 unit ofcurrency at time T ≥ t. The instantaneous forward rate f (t, T) is the forwardrate at time t corresponding to the time interval [T, T + δ[ with δ ↓ 0, that is,

f (t, T) = limδ↓0

1δ

logp(t, T)

p(t, T + δ)= −∂T log p(t, T) .

Consequently, bond prices can be expressed as p(t, T) = exp(−∫ T

t f (t, s)ds).The exponential (or continuously compounded) rate is defined by

z(t, T) =1

T − t

∫ T

tf (t, u) du , 0 ≤ t ≤ T . (3.1.1)

55

56 convergence of the long-term interest rate in hjm models

In the Heath-Jarrow-Morton (HJM) framework the instantaneous forward ratef (·, T), for given f (0, T) with T ≥ 0, is assumed to satisfy

f (t, T) = f (0, T) + A(t, T) + M(t, T) , (3.1.2)

in which

t 7→ A(t, T) =∫ t

0α(u, T)du ,

is a finite variation process and

t 7→ M(t, T) =∫ t

0σ(u, T)dW(u) ,

a local martingale, where σ : R+×R+×Ω→ R1×d and α : R+×R+×Ω→ R

are such that, for all T > 0,∫ T

0

∫ T

0|α(s, u)| ds du < ∞ , and sup

0≤s≤T||σ(s, t)|| < ∞ . (3.1.3)

We further assume that α and σ are Rd-valued, progressively measurable pro-cesses. That is, for every 0 ≤ t ≤ T the map (s, ω) 7→ σ(s, T, ω) on [0, t]×Ωand the map (s, ω) 7→ α(s, T, ω) on [0, t] × Ω are measurable with respectto the sigma algebra generated by the Cartesian product of sets in B[0, t]and F (t), where B[0, t] denotes the Borel σ-algebra of [0, t]

As a standing assumption, we impose that no arbitrage possibilities exist. Weadopt the same notion of no-arbitrage as in Hubalek et al. (2002). Thus, weassume that there exists a measure Q, locally equivalent to P, such that, forall t ≥ 0 and h ≥ 0, the quotient B(t)/B(t + h) is Q-integrable where B(t) :=exp(

∫ t0 f (s, s) ds), and such that discounted bond prices B(t)−1 p(t, T) are Q-

martingales for 0 ≤ t ≤ T.

Due to Eq. (3.1.3) we can write1

d log p(t, T) = f (t, t)dt−∫ T

td f (t, u)du

= f (t, t)dt− dt∫ T

tα(t, u)du− dW(t)

∫ T

tσ(t, u)du

= r(t)dt− b(t, T)dt− Σ(t, T)dW(t) , (3.1.4)

with r(t) := f (t, t) the short rate and

b(t, T) =∫ T

tα(t, u)du, Σ(t, T) =

∫ T

tσ(t, u)du .

1 See Leibniz’ rule for Itô integrals, Lemma 2.8.2 in Chapter 2.

3.1 the hjm framework 57

Because Q is locally equivalent to P, it follows from Björk (2004, Thm. A.52,Prop. B.39) that there exists, for every t ≥ 0, a martingale L such that dP/dQ =L(s) on F (s) for 0 ≤ s ≤ t. The martingale representation theorem (Karatzasand Shreve, 1991, p. 182–184) gives us a process ϕ, called the market price ofrisk, such that dL(s) = ϕ(s)L(s)dW(s), 0 ≤ s ≤ t.

The process t→ p(t, T) for t ∈ [0, T] is the price of a tradeable for every T > 0and its discounted value must thus be a martingale under Q. By Itô’s lemma

dp(t, T)p(t, T)

= d log p(t, T) + 1/2 d 〈log p(·, T)〉t

= r(t)dt− Σ(t, T)dW(t)−(

b(t, T)− 1/2‖Σ(t, T)‖2)

dt ,

in which ||Σ(t, T)||2 = Σ(t, T)Σ(t, T)′, and we conclude that B(t)−1 p(t, T) be-comes a Q-martingale if and only if

−ϕ(t)Σ(t, T)′ = b(t, T)− 1/2‖Σ(t, T)‖2 ,

which gives after differentiation with respect to T the well-known HJM equa-tion:

α(t, T) = σ(t, T)(∫ T

tσ(t, u)du− ϕ(t)

)′. (3.1.5)

We will discuss two well-known examples that we will need later on.

Example 3.1.1. The short rate r(t) in the Vasicek model satisfies

dr(t) = [µ− κr(t)] dt + σ0 dW(t) ,

for given values r(0) ∈ R and σ0, κ ∈ R+, µ ∈ R. The price of a zero-coupon bondhas a representation as log p(t, T) = D(t, T)− η(t, T)r(t) in which η and D satisfythe Riccati equations

∂tη(t, T) = κη(t, T)− 1, η(T, T) = 0 ,

∂tD(t, T) = µη(t, T)− 12

σ20 η2(t, T), D(T, T) = 0 .

By Itô’s lemma and Fubini’s theorem

d f (t, T) = d (−∂T log p(t, T))= [−∂T∂tD(t, T) + ∂T∂tη(t, T)r(t) + (µ− κr(t))∂Tη(t, T)] dt

+σ0∂Tη(t, T)dW(t)= σ2

0 η(t, T)∂Tη(t, T) dt + σ0∂Tη(t, T)dW(t)


Solving the first Riccati equation thus gives the following HJM drift and diffusioncoefficients for the Vasicek model

σ(t, T) = σ0 ∂Tη(t, T) = σ0 e−κ(T−t) ,

and

α(t, T) = σ(t, T)∫ T

tσ(t, u) du =

1κ(σ0 σ(t, T)− σ2(t, T)) .

Example 3.1.2. In the Cox-Ingersoll-Ross model the short rate r(t) with r(0) > 0satisfies

dr(t) = [µ− κr(t)] dt + σ0

√r(t)dW(t) ,

in which κ ≥ 0 and σ0 > 0. The price of a zero-coupon bond has a representation aslog p(t, T) = ψ(t, T)− B(t, T)r(t) in which ψ and B satisfy the Riccati equations

∂tψ(t, T) = µB(t, T), ψ(T, T) = 0 ,

∂tB(t, T) = κB(t, T) +12

σ20 B2(t, T)− 1, B(T, T) = 0 .

An application of Itô’s lemma yields

d f (t, T) = d (−∂T log p(t, T))= [−∂T∂tψ(t, T) + ∂T∂tB(t, T)r(t) + (µ− κr(t))∂TB(t, T)] dt

+σ0∂TB(t, T)√

r(t)dW(t)

= σ20 B(t, T)∂TB(t, T)r(t) dt + σ0∂TB(t, T)

√r(t)dW(t)

Solving the Riccati equations thus gives the following HJM drift and diffusion coeffi-cients for the CIR model

σ(t, T) = σ0∂TB(t, T)√

r(t)

= σ0γ2[κ sinh(1/2 γ(T − t)) + γ cosh(1/2 γ(T − t))]−2√

r(t) ,

and

α(t, T) = σ(t, T)∫ T

tσ(t, u) du =

12

σ20 ∂TB2(t, T)r(t) ,

in which γ = (κ2 + 2σ20 )

12 .

3.2 almost sure convergence on a finite horizon 59

3.2 almost sure convergence on a finite horizon

In this section we consider almost sure convergence of the instantaneous for-ward rate f (t, T) in an HJM model to a limit f∞ ∈ R on a finite horizon. Notethat it is not natural to impose that the exponential yield z(t, T) tends to a con-stant in finite time due to its definition as an average instantaneous forwardrate.

Definition 3.2.1. An HJM model is said to have an ultimate forward rate f∞ withpoint of convergence tu ∈ [0, ∞) if f (t, t + x) = f∞ almost surely for all t ≥ 0 andevery x ≥ tu.

For given x > tu, Definition 3.2.1 requires that f (t, t + x) = f∞ holds for allt ≥ 0, rather than for t = 0 only. Hence we achieve that having an ultimateforward rate is not merely a property of the initial forward curve T 7→ f (0, T),but that this property is present in all future curves generated by the dynamicsof the forward rate.

Our definition of the ultimate forward rate (UFR) is closely related to theone used by the European regulator EIOPA. If the 1-year forward Libor rateRt,T := p(t, T)/p(t, T + 1)− 1 beyond maturity tu = 60 is equal to R∗, thenwe say that the ultimate forward rate in the sense of EIOPA equals R∗. On theother hand, if the UFR in the sense of Definition 3.2.1 exists and is equal tof∞ = log(1 + R∗), then

limT→∞

log(1 + Rt,T) = limT→∞

∫ T+1

Tf (t, t + s)ds = log(1 + R∗) .

Hence, convergence of the instantaneous forward rate implies convergence ofthe 1-year forward Libor rate.

The following proposition characterises the existence of the ultimate forwardrate in terms of the volatility process of an HJM model.

Proposition 3.2.2. An HJM model has ultimate forward rate f∞ with point of con-vergence tu ∈ [0, ∞), i.e.

f (t, t + x) = f∞ a.s. for all t ≥ 0 and every x ≥ tu ,

if and only if

f (0, x) = f∞ , for x ≥ tu , (3.2.1)

and

σ(t, t + x) = 0 , for x > tu . (3.2.2)


Proof.

(If)The result follows directly by substituting Eq. (3.2.1) and (3.2.2) into the gen-eral HJM forward rate dynamics. This yields that, for all x > tu,

0 = f (t, t + x)− f (0, t + x)

=∫ t

0α(u, t + x)du +

∫ t

0σ(u, t + x)′dW(u) , (3.2.3)

and σ(t, t + x) = 0 implies α(t, t + x) = 0 by Eq. (3.1.5).

(Only if)Suppose that f (t, t + x) = f∞ a.s. for all t ≥ 0 and all x ≥ tu. In particular, fort = 0, we have f (0, x) = f∞ for all x ≥ tu which implies Eq. (3.2.1). Takingquadratic variations on the lefthand side and the righthand side in Eq. (3.2.3)yields, for all t ≥ 0 and all x ≥ tu,

0 = E

[∫ t

0σ2(u, t + x)du

].

Hence Eq. (3.2.2) is also satisfied.

To create UFR-consistent models we can thus take any forward rate modelspecified in terms of its volatility term structure function σ(t, T) and define

f U(t, T) = f (0, T) +∫ t

0αU(s, T)ds +

∫ t

0σU(s, T)dW(s) ,

with

σU(t, T) = σ(t, T)g(T − t),

αU(t, T) = g(T − t)σ(t, T)(∫ T

tg(u− t)σ(t, u)du− ϕ(t)

),

where the maturity-dependent scalar function g must satisfy g(t) = 0 fort ≥ tu. If the UFR is required to leave bond prices unchanged for T − t ≤ tlthen we must have g(t) = 1 for t ≤ tl . The maturity tl is called ‘last liquidpoint’.

We thus find a general formula for an UFR-consistent dynamic term structuremodel with maturity-dependent interpolation:

p(t, T) = exp[−∫ T

tdv

f (0, v) +∫ t

0g(v− s)σ(s, v)

(dW(s)

−φ(s)′ ds +∫ v

sg(u− s)σ(s, u)′ du ds

)],

with g(t) = 1t<tl + g(t)1t∈[tl ,tu] for some scalar function g.

3.3 convergence on an infinite horizon 61

3.3 convergence on an infinite horizon

Contrary to the setting discussed in the previous section, in which the ultimateforward rate was attained before a certain finite expiration date, yield curveextrapolation methods such as the Smith-Wilson procedure prescribed by theEuropean regulator EIOPA imply that the term structure approaches such alimit asymptotically as the maturity date tends to infinity. In this section wediscuss whether the result of Proposition 3.2.2 can be extended to HJM modelson an infinite horizon in which the long rate converges.

3.3.1 Ucp convergence of long-term rates

In this subsection, we study convergence in ucp (uniform in probability oncompact intervals) of long-term interest rates. A sequence of processes Xn issaid to converge to X in ucp if P(sups<t |Xn(s)−X(s)| > ε)→ 0 as n→ ∞ forall ε > 0 and all t > 0.

A variant of the following proposition, without the conditions required to en-sure ucp convergence, first appeared in El Karoui et al. (1998) and Yao (1999a).A more general and formal result, for HJM models on the state space S+

d ofsymmetric positive semidefinite matrices, can be found in Biagini et al. (2016).We will give an alternative proof here for the special case of an HJM model.The main tool that is used in the proof is a dominated convergence theoremfor Itô integrals, see Protter (2005, Thm. 32, Chapter IV). This theorem will beused to establish that the Itô integral M in Eq. (3.1.2) converges to zero in ucp.

Proposition 3.3.1. Let 0 ≤ q ≤ 1 and assume that, eventually2 in T,

||ϕ(t)||2 + ||Σ(t, T)||2 ≤ G2(t) Tq , 0 ≤ t ≤ T , (3.3.1)

almost surely, in which G is a process satisfying E∫ n

0 G2(s) ds < ∞ for every n > 0.Then the long zero coupon rate satisfies limT→∞ z(t, T) = z∞(t) in ucp, with

z∞(t) = z∞(0) +12

∫ t

0δ(s) ds , (3.3.2)

if

z∞(0) := limT→∞

z(0, T) and δ(s) := limT→∞

T−1||Σ(s, T)||2 ,

2 We say that a property holds eventually in T if there exists a finite constant T0 such that theproperty holds for all T ≥ T0.


exist almost surely. Moreover, if Eq. (3.3.1) holds with q < 1 then δ(t) = 0 for allt ≥ 0.

Proof.

Using the HJM drift condition Eq. (3.1.5) we obtain from Eq. (3.1.4) that

log p(t, T) = log p(0, T) +∫ t

0

[r(s)− 1

2||Σ(s, T)||2 + ϕ(s)Σ(s, T)′

]ds

−∫ t

0Σ(s, T)dW(s) ,

in which r(s) = f (s, s) denotes the short rate, while ϕ(s)Σ(s, T)′ is commonlyreferred to as the term premium. Since z(t, T) = − 1

T−t log p(t, T) it followsthat

z(t, T) =T

T − tz(0, T)− 1

T − t

∫ t

0

[r(s)− 1

2||Σ(s, T)||2 + ϕ(s)Σ(s, T)′

]ds

+1

T − t

∫ t

0Σ(s, T) dW(s) . (3.3.3)

By Eq. (3.3.1) we have that, for sufficiently large T > 0,

T−1||Σ(t, T)||2 ≤ G2(t) ,

almost surely, and due to the assumptions of the proposition, G satisfies∫ t

0G2(s) ds < ∞ .

It follows from the dominated convergence theorem that, almost surely,

0 = limT→∞

∫ t

0

∣∣∣∣ ||Σ(s, T)||2T

− δ(s)∣∣∣∣ ds

= limT→∞

∫ t

0

∣∣∣∣ ||Σ(s, T)||2T − t

− TT − t

δ(s)∣∣∣∣ ds .

By the triangle inequality∣∣∣∣ ||Σ(s, T)||2T − t

− δ(s)∣∣∣∣ ≤ ∣∣∣∣ ||Σ(s, T)||2

T − t− T

T − tδ(s)

∣∣∣∣+ ∣∣∣∣ TT − t

δ(s)− δ(s)∣∣∣∣ .

Consequently,

limT→∞

∫ t

0

∣∣∣∣ ||Σ(s, T)||2T − t

− δ(s)∣∣∣∣ ds = 0 , a.s. (3.3.4)


However, what we need is ucp convergence. Observe that

supw<t

∣∣∣∣∫ w

0

[||Σ(s, T)||2

T − t− δ(s)

]ds∣∣∣∣ ≤ sup

w<t

∫ w

0

∣∣∣∣ ||Σ(s, T)||2T − t

− δ(s)∣∣∣∣ ds

=∫ t

0

∣∣∣∣ ||Σ(s, T)||2T − t

− δ(s)∣∣∣∣ ds .

Since the right hand side tends to zero almost surely for T → ∞ we also haveconvergence in probability, that is, for all t > 0 and every ε > 0,

P(

supw<t

∣∣∣∣∫ w

0

[||Σ(s, T)||2

T − t− δ(s)

]ds∣∣∣∣ > ε

)→ 0 as T → ∞ .

From the Cauchy-Schwarz inequality and Eq. (3.3.1) it follows that, for suffi-ciently large T > 0,

T−1|ϕ(t)Σ(t, T)′| ≤ T−1||ϕ(t)|| ||Σ(t, T)|| ≤ G2(t) ,

hence by the dominated convergence theorem, almost surely,

limT→∞

1T − t

∫ t

0µP(s, T) ds = lim

T→∞

1T − t

∫ t

0r(s) ds

+∫ t

0lim

T→∞

1T − t

ϕ(s)Σ(s, T)′ ds = 0. (3.3.5)

The last step follows since limT→∞ T−1|Σ(t, T)| = 0 and neither ϕ(t) nor r(t)depends on T. We have thus established ucp convergence of the finite variationpart of the exponential yield dynamics. It remains to show that the Itô integralalso converges.

Fix j ∈ 1, . . . , d. By Eq. (3.3.1), for T > 0, eventually in T,

T−1|Σj(t, T)| ≤ T−12 |Σj(t, T)| ≤ T−

12 ||Σ(t, T)|| ≤ G(t) ,

in which E∫ n

0 G2(s) ds < ∞ for all n > 0. Moreover, for all t ≥ 0,

limT→∞

T−1|Σj(t, T)| ≤ limT→∞

T−12 G(t) = 0.

By the dominated convergence theorem for stochastic integrals, see Theo-rem 32 in Protter (2005, Chapter IV), it follows that, for all t ≥ 0 and everyε > 0,

P(

supw<t

∣∣∣∣ 1T

∫ w

0Σj(s, T) dWj(s)

∣∣∣∣ > ε

)→ 0 as T → ∞ .


Since this holds for every j ∈ 1, . . . , d we find, by Minkowski’s inequality,that the diffusion term in Eq. (3.3.3) vanishes in ucp.

Finally, if Eq. (3.3.1) holds with q < 1 then δ(t) = limT→∞ T−1||Σ(t, T)||2 ≤limT→∞ G2(t)Tq−1 = 0.

Proposition 3.3.1 states that if the squared Euclidean norm ||Σ(t, T)||2 of thebond volatility is bounded from above by G2(t)Tq for some q ∈ [0, 1[, in whichG is a suitably integrable function, then the exponential yield converges to aconstant long rate in ucp.3 On the other hand, if ||Σ(t, T)||2 ≥ G2(t)Tq forsome q > 1 and

∫ t0 G2(s) ds > 0 then

limT→∞

1T − t

∫ t

0

12||Σ(s, T)||2 ds ≥ 1

2lim

T→∞

Tq

T − t

∫ t

0G2(s) ds = ∞ .

Hence in this case the exponential long rate, i.e. the limit T → ∞ in Eq. (3.3.3),is infinite. In the intermediate situation, where ||Σ(t, T)|| asymptotically be-haves as

√T, the long rate can be a finite-valued non-constant process, pro-

vided that the conditions of Proposition 3.3.1 are satisfied.

Since δ(t) = limT→∞ T−1||Σ(t, T)||2 ≥ 0 we obtain that the exponential rateis nondecreasing. This conclusion is similar to that of the Dybvig-Ingersoll-Ross theorem, see Theorem 3.3.7 in Section 3.3.2, and this has been notedearlier in Yao (1999a) and Biagini et al. (2016). We stress that if z(t, T) tendsto z∞(t) in ucp, then we cannot conclude, based on Lemma 3.3.1, that z∞(t)is nondecreasing unless a bounding process G is given which satisfies theproperties stated in the lemma. Summarising, we have the following corollary.

Corollary 3.3.2. Under the conditions of Proposition 3.3.1, the exponential long ratez∞ is nondecreasing.

In EIOPA (2016) the European insurance regulator invited stakeholders to dis-cuss their proposal for the estimation of the ultimate forward rate for insurers.In this consultation paper it is proposed to exclude the term premium fromthe UFR estimate. The following remark shows that this is consistent with longterm interest rate behaviour for HJM models.

Remark 3.3.3. It follows from Eq. (3.3.5) in the proof of Proposition 3.3.1 that theterm premium ϕ(·)Σ(·, T)′, i.e. the premium for the risk of holding fixed incomeinvestments, disappears from the dynamics of the long zero coupon rate. A relatedobservation was made in Yao (1999a, Remark 3.1).

3 In particular, notice that if ||Σ(t, T)||2 ∈ O((T − t)q) as T → ∞ for some 0 ≤ q ≤ 1, then theconditions of Proposition 3.3.1 are satisfied since (T− t)q ≤ Tq for q ≥ 0 and 0 ≤ t ≤ T. Here,f (x) ∈ O(g(x)) as x → ∞ means that eventually | f (x)| ≤ C|g(x)|.


The next lemma shows that Corollary 3.3.2 and Remark 3.3.3 also hold for thelong instantaneous forward rate.

Lemma 3.3.4. Assume that the instantaneous forward rate f (t, T) and the long in-stantaneous forward rate f∞(t) are continuous in both t and T. If f (·, T) → f∞ inucp as T → ∞, then also z(·, T)→ f∞ in ucp as T → ∞.


We discuss two examples to illustrate Proposition 3.3.1.

Example 3.3.5. We show that the Vasicek model from Example 3.1.1 satisfies theconditions of Proposition 3.3.1. The volatility of the bond price satiesfies Σ(t, T) =∫ T

t σ(t, u) du = σ0κ (1− e−κ(T−t)). Consequently, T−

12 |Σ(t, T)| = σ0

κ T−12 (1−

e−κ(T−t)). Observe that limT→∞ T−12 |Σ(t, T)| = 0. Also, limT→t T−

12 |Σ(t, T)| = 0

for all t ≥ 0. Therefore T−12 |Σ(t, T)| can be bounded by a function G(t) which is

locally square integrable.

Example 3.3.6. In this example, we show that the Cox-Ingersoll-Ross model from Ex-ample 3.1.2 satisfies the conditions of Proposition 3.3.1. The volatility of the bond pricesatisfies Σ(t, T) =

∫ Tt σ(t, u) du = σ0B(t, T)

√r(t). Consequently, T−

12 |Σ(t, T)| =

σ0 T−12 B(t, T)

√r(t). The function B(t, T) is bounded by a constant independent of t

and T, hence limT→∞ T−12 |Σ(t, T)| = 0. Moreover, limT→t T−

12 |Σ(t, T)| = 0 since

B(t, t) = 0. Therefore T−12 |Σ(t, T)| can be bounded by a function G(t) which satisfies

the square integrability condition of Proposition 3.3.1.

3.3.2 Almost sure convergence of long-term rates

This section is concerned with almost sure convergence of long-term rates. Thelong instantaneous forward rate is defined by f∞(t) = limx→∞−∂x log p(t, t +x), whenever the limit exists, while the exponential long rate is given byz∞(t) = limx→∞−x−1 log p(t, t + x), provided that the limit exists. Our mo-tivation to study this type of convergence stems from the following theorem,which gives an important implication for the dynamics of the long rate, if suchrate exists as an almost sure limit.

Theorem 3.3.7 (Dybvig-Ingersoll-Ross). If z∞(t) and z∞(s) exist almost surely for0 ≤ t ≤ s, then the exponential long rate z∞ is nondecreasing almost surely.

Proof. See Theorem 3.1 in Hubalek et al. (2002).


The Dybvig-Ingersoll-Ross theorem states that the exponential long rate (ifit exists) must be a nondecreasing process. The following lemma shows thatTheorem 3.3.7 also holds for the long instantaneous forward rate.

Lemma 3.3.8. If the long instantaneous forward rate exists almost surely for somet ≥ 0, then the exponential long rate also exists and f∞(t) = z∞(t).

Proof. For all ω ∈ Ω such that f∞(t) = c for some constant c 6= 0 we have byl’Hôpital’s rule that

z∞(t) = limT→∞

∫ Tt f (t, u) du

T − t= lim

T→∞

∂T∫ T

t f (t, u) du∂T(T − t)

= limT→∞

f (t, T) = f∞(t) .

For ω ∈ Ω such that f∞(t) = 0 it also holds that z∞(t) = 0.

The converse of Lemma 3.3.8 does not hold; indeed, a counterexample is ob-tained by taking f (t, T) = 1 + sin(T).

The following proposition provides sufficient conditions such that the termstructure almost surely converges to an exponential long rate. In addition tothe conditions imposed in Proposition 3.3.1, we require that the mean of thesquared bond volatility is of order Tq for some 0 ≤ q < 1. Notice that theseconditions are thus too strong to allow non-constant behaviour of the longrate, because in that case we would need the result to hold for q = 1.

Proposition 3.3.9. Suppose that the conditions of Proposition 3.3.1 hold. Further,assume that there exist constants q ∈ [0, 1[ and C ≥ 0 such that, given any fixedt ≥ 0, eventually4 in T,

E||Σ(t, T)||2 ≤ C Tq . (3.3.6)

Then limT→∞ z(t, T) = z∞(0) almost surely for every t ≥ 0.

Proof.

Fix t ≥ 0 and assume that Eq. (3.3.6) holds for all T ≥ T0 for some T0 > t + 1.Consider the estimate

∞

∑u=T0

E

[∫ t

0

||Σ(s, u)||2(u− t)2 ds

]≤∫ ∞

T0−1

∫ t

0

Cuq

(u− t)2 ds du =∫ ∞

T0−1

Ctuq

(u− t)2 du .

Since∫ ∞

T0−1 uq−2 du < ∞, for 0 ≤ q < 1, it follows from the limit comparisontest that the integral on the righthand side is finite. Itô isometry yields that,for every t ≥ 0,

∞

∑u=T0

E

∣∣∣∣∫ t

0

1u− t

Σ(s, u)dW(s)∣∣∣∣2 =

∞

∑u=T0

E

[∫ t

0

||Σ(s, u)||2(u− t)2 ds

]< ∞ .

4 See footnote 2.


Applying Fubini’s theorem gives

E

[∞

∑u=T0

∣∣∣∣∫ t

0

1u− t

Σ(s, u)dW(s)∣∣∣∣2]

< ∞ .

Hence, almost surely,

∞

∑u=T0

∣∣∣∣∫ t

0

1u− t

Σ(s, u)dW(s)∣∣∣∣2 < ∞ .

Since the sum converges, the summand must almost surely vanish in the limit,i.e.

limu→∞

∫ t

0

1u− t

Σ(s, u)dW(s) = 0 a.s.

By the same arguments as in the proof of Proposition 3.3.1, see Eq. (3.3.4), thefinite variation part of the exponential yield process in Eq. (3.3.3) convergesalmost surely with δ(t) = 0 for all t ≥ 0.

Based on Proposition 3.3.9 it is easy to prove almost sure convergence of long-term rates in the Vasicek and CIR models as we will show in Examples 3.3.10

and 3.3.11. Related results are presented in Deelstra and Delbaen (1995). Theanalysis in that paper focuses on the almost sure convergence of the long-term averaged short rate, which is modelled by an extended CIR process, andconvergence of the Itô integral is established based on Kronecker’s lemma.Zhao (2009) and Bao and Yuan (2013) have further extended these results totwo-dimensional CIR processes with jumps.

Example 3.3.10. In case of the Vasicek model, the bond volatility Σ(t, T) = σ0/κ (1−e−κ(T−t)) is deterministic and bounded. It follows from Proposition 3.3.9 that theexponential long rate in the Vasicek short rate model is constant.

Example 3.3.11. For the CIR model it holds that Σ(t, T) = σ0B(t, T)√

r(t), hence

E||Σ(t, T)||2 = σ20 B(t, T)2E|r(t)| ,

Since E|r(0)| = r(0) and limt→∞ E|r(t)| = µ/κ, the righthand side is boundeduniformly in t and T. It follows from Proposition 3.3.9 that the exponential long ratein the CIR short rate model is constant.

3.3.3 S1 convergence of long-term rates

In Section 3.2 we provided a characterisation of HJM models for which thelong rate exists as an (almost sure) limit which is already attained for a finite


maturity. In this section our objective is to extend this result to the case ofan infinite horizon. In addition to the sufficient conditions discussed in Sec-tions 3.3.1 and 3.3.2 for ucp and almost sure convergence, we thus now alsoseek to formulate necessary conditions for convergence to a long rate. Thisraises the question which type of convergence could give us such a result.

In the current section we consider uniform convergence in L1, which for semi-martingales is commonly referred to as S1 convergence. Given an adaptedprocess X, the S1 norm is defined by, see Protter (2005, Section V.2),

||X||S1 = ||X∗||L1 , X∗ = supt≥0|X(t)| .

The L1 norm in this chapter is always with respect to the probability mea-sure P. Convergence in probability is weaker than convergence in L1(P), anduniform convergence on compacts, discussed in Subsection 3.3.1, is weakerthan uniform convergence on [0, ∞[. Therefore, convergence with respect to S1

is stronger than ucp convergence.

To formulate our results, we must also introduce the H1 norm for semimartin-gales. For a finite variation process A this norm is, see Protter (2005),

||A||H1 = E

[∫ ∞

0|dA(s)|

],

in which∫ ∞

0 |dA(s)| denotes the first order variation5 of A on [0, ∞[, while fora local martingale M it holds that

||M||H1 = E

[[M, M]

12∞

],

where [·, ·]∞ denotes the quadratic covariation on the interval [0, ∞[. For gen-eral semimartingales, the H1 norm is defined as an infimum of such expres-sions over all possible decompositions, but we will not need such generality inour context, since the instantaneous forward rate process (3.1.2) is a so calledspecial semimartingale which has a unique decomposition6.

The following result is adapted from Barlow and Protter (1990). This lemmawill allow us to characterise the existence of a, possibly non-constant, long ratein HJM models in terms of the drift and volatility function of the instantaneousforward rate process.

5 The first-order variation of a function f : [a, b]→ R is defined by supΠ ∑nΠ−1j=1 | f (xj+1)− f (xj)|,

where the supremum is taken over all partitions Π = a = x1, x2, . . . , xnΠ = b.6 A semimartingale X with decomposition X(t) = X(0) + A(t) + M(t) is said to be special if

A(0) = M(0) = 0 and A is predictable. The decomposition of a special semimartingale isunique, see Protter (2005, Chapter III, Thm. 30).


Lemma 3.3.12. Let X(t, T)T≥0 be a family of special semimartingales with canon-ical decomposition X(t, T) = X(0, T) + A(t, T) + M(t, T), satisfying, for some con-stant K,

||A(·, T)||H1 ≤ K , for all T ≥ 0 . (3.3.7)

If the process X satisfies

limT→∞

||X(·, T)− X||S1 = 0 , (3.3.8)

and its canonical decompostion is X(t) = X(0) + A(t) then

limT→∞

||A(·, T)− A||S1 = 0 , (3.3.9)

and

limT→∞

||M(·, T)||H1 = 0 , (3.3.10)

as T → ∞. Conversely, if Eqs. (3.3.9) and (3.3.10) are satisfied, then Eq. (3.3.8) holdswith X(t) = X(0) + A(t).

Proof. The proof is an application of Corollary 2 in Barlow and Protter (1990),for details see Section 3.4.

In the subsequent analysis we do not impose convergence of the instantaneousforward rate process f (t, T) uniformly for t ∈ [0, T], because this would implythat the future short rate f (T, T) and forward rates at longer times to maturitymust converge to the same limiting process. Instead, in Proposition 3.3.13, weexclude forward rates with short maturities.

To achieve this, we apply Lemma 3.3.12 to a restriction of the family of forwardrate processes f (·, T)T≥0 to the domain (t, T) | t ≤ h(T), in which h is acontinuous bijection from [0, ∞[ onto itself satisfying h(t) < t. Notice thath(t) = t would yield the triangular subset of R2 that commonly supportsthe instantaneous forward rate. For fixed t ≥ 0, we thus retain the behaviourof f (t, T) for large T. But for values of t that are close to T, we do not requireconvergence of the forward rate.

Without loss of generality we take ϕ = 0. Indeed, if the S1 limit exists, thenthis implies convergence to the same long rate in ucp for every fixed t ≥ 0,and we know that the long rate, as a ucp limit, does not depend on the marketprice of risk by Remark 3.3.3 and Lemma 3.3.4.


Proposition 3.3.13. Let f∞ be a finite variation process satisfying f∞(t) = f∞(0) +A∞(t) with A∞(0) = 0, and suppose that

E

[∫ ∞

0|d f∞(s)|

]< ∞ . (3.3.11)

Consider the family of instantaneous forward rate processes f (·, T)T≥0 with canon-ical decomposition given by Eq. (3.1.2) for every fixed T ≥ 0. Let h be a continuousbijection from [0, ∞[ onto itself satisfying h(s) < s, and asssume that, for some con-stant K,

E

[∫ h(T)

0|σ(s, T)Σ(s, T)′| ds

]≤ K , for all T ≥ 0 . (3.3.12)

Then

limT→∞

E[

sup0≤t≤h(T)

∣∣ f (t, T)− f∞(t)∣∣ ] = 0 , (3.3.13)

if and only if

limT→∞

E[

sup0≤t≤h(T)

∣∣A(t, T)− A∞(t)∣∣ ] = 0 , (3.3.14)

and

limT→∞

E

(∫ h(T)

0||σ(s, T)||2 ds

) 12

= 0 . (3.3.15)

In particular, if σ ≥ 0 and A∞(t) = 0 for t ≥ 0, then condition (3.3.14) is equivalentto

limT→∞

E

[∫ h(T)

0σ(s, T)Σ(s, T)′ ds

]= 0 . (3.3.16)

Proof. The proof follows by applying Lemma 3.3.12 to the family of semi-martingales

f (t, T) = f (t ∧ h(T), T) + 1t>h(T) ( f∞(t)− f∞(h(T))) , t ≥ 0 ,

indexed by T ≥ 0. See Section 3.4 for details.

If the long rate is a nondecreasing process, and its mean is bounded uniformlyin t ≥ 0, then condition (3.3.11) holds. Indeed, in this case,

E

∫ ∞

0|d f∞(s)| = lim

t→∞E [ f∞(t)]− f∞(0) < ∞ .


It is shown in El Karoui et al. (1998) and Biagini et al. (2016) that the longrate converges in ucp provided that the HJM volatility function σ(t, T) is even-tually bounded by a constant multiple of 1/√T − t for sufficiently large T. InCorollary 3.3.15 we will provide a similar statement for S1 convergence. No-tice that Proposition 3.3.13 depends on the choice of the function h. To estab-lish convergence it is sufficient to find an h for which the conditions of thisproposition are satisfied, whereas to rule out convergence we must show thatthey are violated for any h. Therefore, we first need the following lemma.

Lemma 3.3.14. Let H denote the set of continuous bijections from [0, ∞[ onto itselfthat satisfy h(t) < t for all t ≥ 0. Then

1. If α ≥ 1 then lim supT→∞ Tα − (T − h(T))α = ∞ for all h ∈H ;

2. If α < 1 then limT→∞ Tα − (T − h(T))α = 0 for some h ∈H .


The next corollary states that, if the HJM volatility coefficient is bounded fromabove by a deterministic function of time to maturity which falls off suffi-ciently fast, then the long rate is constant.

Corollary 3.3.15. If for some C ≥ 0 and r ∈ ]−1,−1/2[ ,

|σ(t, T)| ≤ C(T − t)r ,

then the instantaneous forward rate in the corresponding HJM model converges, in S1,to a constant long rate.

Proof. Let r ∈ ]−1,−1/2[, then by Lemma 3.3.14 there exists an h ∈ H suchthat limT→∞ T2+2r − (T − h(T))2+2r = 0. Hence

limT→∞

∫ h(T)

0σ2(s, T) ds ≤ C2 lim

T→∞

T1+2r − (T − h(T))1+2r

1 + 2r= 0 ,

and

limT→∞

∫ h(T)

0σ(s, T)Σ(s, T) ds ≤ C2 lim

T→∞

T2+2r − (T − h(T))2+2r

2(1 + r)2 = 0 .

The result follows by Proposition 3.3.13.

From Lemma 3.3.14 and the proof of Corollary 3.3.15 it is also clear that ifσ(t, T) ≥ C(T − t)r for some C > 0 and r ∈ [−1/2, 0], then condition (3.3.12)of Proposition 3.3.13 is not satisfied regardless of the choice of h. Hence, in


contrast to Proposition 3.3.1 on ucp convergence7, the previous corollary doesnot give us information about the case where r = −1/2. In Example 3.3.18 wewill show that it is nevertheless possible for the instantaneous forward rate toconverge in S1 to a non-constant long rate.

The next examples serve to illustrate that the Vasicek and Cox-Ingersoll-Rossshort rate models converge in S1 to a constant long rate, which is in linewith our findings in previous sections for ucp and almost sure convergence.Example 3.3.17 shows that the exclusion of short maturities is essential.

Example 3.3.16. Consider the Vasicek short rate model from Example 3.1.1. For everyt > 0,

α(t, T) = σ20 e−κ(T−t)

∫ T

te−κ(u−t) du =

σ20

κe−κ(T−t)[1− e−κ(T−t)] .

It follows that, for all t > 0,

E

∫ t

0α(u, T) du =

12

σ20 κ−2e−2κT(eκt − 1)(2eκT − eκt − 1) .

Take h(t) = εt, for some 0 < ε < 1, then the lefthand side of Eq. (3.3.12) equals

E

∫ εT

0α(u, T) du =

12

κ−2σ20 (2e−κ(1−ε)T − e−2κ(1−ε)T − 2e−κT + e−2κT),

which tends to 0 as T → ∞ so Eq. (3.3.14) is satisfied for A∞ = 0. Similarly, since∫ εT

0σ2(s) ds =

12

κ−1σ20 (e−2κ(1−ε)T − e−2κT) ,

it follows that Eq. (3.3.15) is satisfied:

E

[(∫ εT

0σ2(s, T) ds

) 12]

=12

√2 σ0κ−1/2

(e−2κ(1−ε)T − e−2κT

)1/2

,

because the right hand side vanishes as T → ∞. Eq. (3.3.11) is trivially satisfied forA∞ = 0. Hence, we conclude that the instantaneous forward rate in the Vasicek modelconverges to a constant.

7 Notice that, due to Σ(t, T) =∫ T

t σ(s, T) ds, the choice r = −1/2 in Corollary 3.3.15 correpondsto q = 1 in Proposition 3.3.1.


Example 3.3.17. Consider the CIR short rate model from Example 3.1.2 and observethat we have the following almost sure bound on the HJM diffusion coefficient

0 ≤ σ(t, T) = σ0γ2√

r(t)(κ sinh(1/2 γ(T − t)) + γ cosh(1/2 γ(T − t)))−2

≤ σ0

√r(t) cosh−2(1/2 γ(T − t)) ≤ 4σ0

√r(t) e−γ(T−t) .

We thus find that σ(t, T) is bounded by 4√

r(t) times the volatility function of theVasicek model. From here on, the computations are therefore similar to those in Exam-ple 3.3.16.

Set h(t) = εt, for some 0 < ε < 1, then the lefthand side of Eq. (3.3.12) admits thefollowing almost sure bound:

E

∫ εT

0α(u, T) du ≤ 8γ−2σ2

0(2e−2γ(1−ε)T − e−2γ(1−ε)T

−2e−γT + e−2γT) sups≤εT

E|r(s)| .

The righthand side tends to 0 as T → ∞ so Eq. (3.3.14) is satisfied for A∞ = 0.Similarly, since

E

∫ εT

0σ2(s) ds ≤ 8γ−1σ2

0 (e−2γ(1−ε)T − e−2γT) sup

s≤εTE|r(s)| ,

it follows from Jensen’s inequality that Eq. (3.3.15) is satisfied:

E

[(∫ εT

0σ2(s, T) ds

) 12]≤

(E

[∫ εT

0σ2(s, T) ds

]) 12

=

(8γ−1 σ2

0 (e−2γ(1−ε)T − e−2γT) sup

s≤εTE |r(s)|

) 12

,

because the right hand side vanishes as T → ∞. Eq. (3.3.11) is trivially satisfied ifA∞ = 0. We thus conclude that the instantaneous forward rate in the CIR modelconverges in S1 to a constant.

The exclusion of short maturities, enforced by taking ε < 1, cannot be omitted for theCox-Ingersoll-Ross model. To see this, consider the estimate

γ cosh(1/2γ(T − t)) + κ sinh(1/2γ(T − t)) ≤ (γ + κ) cosh(1/2γ(T − t))

≤ (γ + κ)e1/2γ(T−t) .


We thus have the following lower bound for the Cox-Ingersoll-Ross forward ratevolatility

σ(t, T) ≥ σ0 γ2(γ + κ)−2 e−γ(T−t)√

r(t) .

Consequently,

α(t, T) = σ(t, T)∫ T

tσ(t, u) du ≥ σ2

0 γ3 (γ+ κ)−4 e−γ(T−t) [eγ(T−t)− 1] r(t) .

Using Fubini we obtain

E

∫ T

0α(u, T) du ≥ 1

2σ2

0 γ2 (γ + κ)−4(1− e−γT)2 infs<T

E|r(s)| .

The righthand side tends to 12 σ2

0 γ2 (γ + κ)−4 infs>0 E|r(s)| almost surely as T →∞. This limit is strictly positive if r(0) > 0, so (3.3.16) is not satisfied for h(T) = T.

In the following example we construct an HJM model in which the term struc-ture converges in S1 to a non-deterministic and non-constant long rate.

Example 3.3.18. Let the HJM volatility function be given by

σ(t, T) =12

√2

(T − t)−1/2 z(t)1/4, if 0 ≤ t < T,

0 , otherwise,(3.3.17)

in which z(t) = 1 + 2√

2∫ t

0 z(s) dW(s) is a geometric Brownian motion.

From the HJM drift condition it follows that α(t, T) = z(t)1/2, for 0 ≤ t < T, andzero otherwise. The semimartingale decomposition for the instantaneous forward ratethus satisfies

M(t, T) =∫ t

0

z(s)1/4

(T − s)1/2dW(s) , A(t, T) =

∫ t

0z(s)1/2 ds .

Set h(t) = εt, for some 0 < ε < 1. Since z(t)1/2 is a supermartingale with exponen-tially decaying mean we find,

E

[∫ εT

0|α(s, T)| ds

]=

∫ εT

0E|z(s)1/2| ds =

∫ εT

0e−s ds = 1− e−εT ≤ 1 .

Define f∞(t) = f∞(0) + A∞(t) with A∞(t) =∫ t

0 z(s)1/2 ds. This process satisfies

E

∫ ∞

0|d f∞(s)| = E

∫ ∞

0z(s)1/2 ds = 1 .


For T > 1/ε we find, using Jensen’s inequality and Fubini’s theorem,

E

[(∫ εT

0|σ(s, T)|2 ds

)1/2]≤(E

[∫ εT

0|σ(s, T)|2 ds

])1/2

=

(12E

[∫ εT

0

z(s)1/2

T − sds

])1/2

=

(12

∫ εT

0

e−s

T − sds)1/2

≤(

12(1− ε)T

∫ εT

0e−s ds

)1/2

=

(1− e−εT

2(1− ε)T

)1/2

.

The right-hand side vanishes as T → ∞. Finally, A(t, T)− A∞(t) = 0 for every T ≥0. Hence supt<εT | f (t, T)− f∞(t)| converges to zero in mean by Proposition 3.3.13with h(t) = εt for any 0 < ε < 1.

If we had taken σ(t, T) = 12

√2 (T − t)−1/2 z(t)1/2 in Example 3.3.18, then the

no-arbitrage drift coefficient α(t, T) = z(t) would be a geometric Brownianmotion. In this case condition (3.3.11) of Proposition 3.3.13, which states thatthe long rate should be finite in expectation, does not hold.

For completeness, we note that the above analysis can also be applied to theexponential yield z(t, T). Using Itô’s lemma one can show that z(t, T) has thesemimartingale decomposition z(t, T) = z(0, T) + A(t, T) + M(t, T) for t < Tin which

A(t, T) =∫ t

0

1T − s

(−r(s) +

12||Σ(s, T)||2 + z(s, T)

)ds , (3.3.18)

while

M(t, T) =∫ t

0

1T − s

Σ(s, T) dW(s) , (3.3.19)

and where we assumed as before that ϕ = 0.

Proposition 3.3.19. Let z∞ be a finite variation process satisfying z∞(t) = z∞(0) +A∞(t) with A∞(0) = 0, and suppose that

E

[∫ ∞

0|dz∞(s)|

]< ∞ . (3.3.20)

Consider the family of zero coupon yield processes z(·, T)T≥0 with canonical de-composition given by Eqs. (3.3.18) and (3.3.19) for every fixed T ≥ 0. Let h be a


continuous bijection from [0, ∞[ onto itself satisfying h(s) < s, and asssume that, forsome constant K,

E

[∫ h(T)

0|dA(s, T)|

]≤ K , for all T ≥ 0 . (3.3.21)

Then

limT→∞

E[

sup0≤t≤h(T)

∣∣z(t, T)− z∞(t)∣∣ ] = 0 , (3.3.22)

if and only if

limT→∞

E[

sup0≤t≤h(T)

∣∣A(t, T)− A∞(t)∣∣ ] = 0 , (3.3.23)

and

limT→∞

E

(∫ h(T)

0

1(T − s)2 ||Σ(s, T)||2 ds

) 12

= 0 , (3.3.24)

with A(t, T) as defined in Eq. (3.3.18).

Proof. Apply Lemma 3.3.12 to the family of (continuous) semimartingales

z(t, T) = z(t ∧ h(T), T) + 1t>h(T) (z∞(t)− z∞(h(T))) , t ≥ 0 ,

indexed by T ≥ 0.

3.4 proofs


For 0 ≤ s ≤ t < T consider the estimate

|z(s, T)− f∞(s)| =

∣∣∣∣ 1T − s

∫ T

sf (s, u) du− f∞(s)

∣∣∣∣≤ 1

T − s

∫ T

s| f (s, u)− f∞(s)| du

=1

T − s

∫ √T

s| f (s, u)− f∞(s)| du +

∫ T√

T| f (s, u)− f∞(s)| du

≤√

TT − t

supu≤√

T| f (s, u)− f∞(s)|+ T

T − tsup√

T≤u≤T| f (s, u)− f∞(s)| .

3.4 proofs 77

Fix ε > 0 and t ≥ 0. From our previous estimate we obtain

P(

sups<t|z(s, T)− f∞(s)| > ε

)≤ P

( √T

T − tsups<t

supu≤√

T| f (s, u)− f∞(s)|

+T

T − tsups<t

sup√T≤u≤T

| f (s, u)− f∞(s)| > ε

)

≤ P

( √T

T − tsups<t

supu≤√

T| f (s, u)− f∞(s)| > ε

2

)

+ P

(T

T − tsups<t

sup√T≤u≤T

| f (s, u)− f∞(s)| > ε

2

). (3.4.1)

By the extreme value theorem

Kt,T := argmax(s,u)∈[0,t]×[0,√

T] | f (s, u)− f∞(s)| ,

is nonempty and compact due to the continuity of f and f∞. Hence, for all0 ≤ t ≤ T, we can choose

v(t, T) ∈ argmax(η,ρ)∈Kt,Tρ ,

that is, we take v(t, T) to be an element in the compact set Kt,T for which thesecond coordinate v2(t, T) is maximal.

Let A denote the set of those elements in Ω for which there exists an M > 0such that v2(t, T) ≤ M eventually in T. Since T 7→ v2(t, T) is nondecreasingby construction of v, we can write

A =⋃

M∈Nv2(t, T) ≤ M eventually in T =

⋃M∈N

⋂n∈Nv2(t, n) ≤ M .

Moreover,

v2(t, n) ≤ M =⋂

a,b∈Qa≤t, b≤

√n

⋃a′,b′∈Q

a′≤t, b′≤M

| f (a, b)− f∞(a)| ≤ | f (a′, b′)− f∞(a′)| .

Hence A is measurable.

By the continuity of f and f∞ there exists a random variable C such that, onthe set A,

| f (v1(t, T), v2(t, T))− f∞(v1(t, T))| < C , eventually in T.


Consequently, on A,

limT→∞

√T

T − tsups<t

supu≤√

T| f (s, u)− f∞(s)|

= limT→∞

√T

T − t| f (v1(t, T), v2(t, T))− f∞(v1(t, T))|

≤ C limT→∞

√T

T − t= 0 .

On the other hand, we have v2(t, T) → ∞ as T → ∞ on the set Ac = Ω \ A,hence

limT→∞

P

(Ac ∩

√T

T − tsups<t

supu≤√

T| f (s, u)− f∞(s)| > ε

2

)

≤ limT→∞

P

(Ac ∩

sups<t

supu≤√

T| f (s, u)− f∞(s)| > ε

2

)

= limT→∞

P(

Ac ∩

sups<t| f (s, v2(t, T))− f∞(s)| > ε

2

)≤ lim

v→∞P(

sups<t| f (s, v)− f∞(s)| > ε

2

)= 0 .

It follows that the first term on the righthand side in Eq. (3.4.1) vanishes asT → ∞.

Similarly, by the continuity of f (t, ·) and the extreme value theorem we cantake, for all 0 ≤ s ≤ T, w(s, T) ∈ argmax√T≤u≤T | f (s, u)− f∞(s)| . Notice thatw(s, T) ≥

√T → ∞ as T → ∞ hence the second term on the righthand side in

Eq. (3.4.1) also vanishes by the assumption of the lemma.


Let An(t) = A(t, xn) and Mn(t) = M(t, xn) for some sequence xn∞n=0 such

that xn → ∞ as n→ ∞. If we set Xn(t) = X(t, xn) then we can write Xn(t) =Xn(0) + An(t) + Mn(t). By Eq. (3.3.8) we have

||Xn − X||S1 → 0 , as n→ ∞ .

It follows from Corollary 2 in Barlow and Protter (1990) that

||An − A||S1 → 0 and ||Mn||H1 → 0 ,

3.4 proofs 79

as n → ∞. Since this holds for any sequence xn∞n=0 such that xn → ∞ as

n→ ∞, we obtainEqs. (3.3.9) and (3.3.10).

It remains to prove the converse. Let xn∞n=0 be any sequence such that xn →

∞ as n → ∞. Write Xn(t) = Xn(0) + An(t) + Mn(t) with An(t) = A(t, xn)and Mn(t) = M(t, xn). We must show that

0 = limn→∞

||(Xn − X)∗||L1(P)

= limn→∞

||(Xn(0)− X(0) + An − A∞ + Mn)∗||L1(P) .

Because supt |X + Y| ≤ supt |X|+ supt |Y|, we obtain

||(Xn(0)− X(0) + An − A∞ + Mn)∗||L1(P)

≤ ||(Xn(0)− X(0))∗ + (An − A∞)∗ + (Mn)∗||L1(P) .

Using Minkowski’s inequality we find

||(Xn(0)− X(0))∗ + (An − A∞)∗ + (Mn)∗||L1(P)

≤ ||Xn(0)− X(0)||L1(P) + ||(An − A∞)∗||L1(P) + ||(Mn)∗||L1(P) .

From Protter (2005, Chapter V, Thm. 2) we have that there exists a universal(i.e. independent of the semimartingale) constant c1 such that

||(Mn)∗||L1(P) = ||Mn||S1 ≤ c1||Mn||H1 .

Hence we arrive at

||(Xn − X)∗||L1(P) ≤ ||Xn(0)− X(0)||L1(P)

+ ||(An − A∞)∗||L1(P) + c1||Mn||H1 .

The righthand side vanishes in the limit as n→ ∞.

Proof of Proposition 3.3.13.

Consider the family of semimartingales f (t, T) = f (0, T) + A(t, T) + M(t, T),indexed by T ≥ 0, in which

A(t, T) = A(t ∧ h(T), T) + 1t>h(T) (A∞(t)− A∞(h(T))) ,

is a process of finite variation, and

M(t, T) = M(t ∧ h(T), T) ,


is a local martingale. For given T ≥ 0, the semimartingale f (·, T) is special,hence its decomposition is unique8. Note that for fixed t ≥ 0 and sufficientlylarge T, it holds that f (t, T) = f (t, T).

The rest of the proof follows from an application of Lemma 3.3.12 to the familyof semimartingales f (·, T)T≥0. Equation (3.3.8) can be written as

|| f (·, T)− f∞||S1 = E[sup

t| f (t, T)− f∞(t)|

]= E

[sup

0≤t≤h(T)| f (t, T)− f∞(t)|

].

Similarly, Eq. (3.3.9) can be expressed as

||A(·, T)− A∞||S1 = E[sup

t|A(t, T)− A∞(t)|

]= E

[sup

0≤t≤h(T)|A(t, T)− A∞(t)|

],

or, if σ ≥ 0 and A∞(t) = 0 for all t ≥ 0,

||A(·, T)||S1 = || sup0≤t≤h(T)

∣∣∣∣∫ t

0α(s, T) ds

∣∣∣∣ ||L1(P) = E

[∫ h(T)

0α(s, T) ds

],

with α(t, T) = σ(t, T)Σ(t, T)′ by Eq. (3.1.5). For Eq. (3.3.10) we obtain,

||M(·, T)||H1 = E

[[M(·, T), M(·, T)]

12∞

]

= E

(∫ h(T)

0||σ(s, T)||2 ds

) 12

.

Finally, given T ≥ 0 and K ≥ 0, condition (3.3.7) satisfies

K ≥ ||A(·, T)||H1 = E

[∫ ∞

0|dA(s, T)|

]= E

[∫ h(T)

0|α(s, T)| ds +

∫ ∞

h(T)|d f∞(s)|

].

This condition is met for every T ≥ 0 if Eqs. (3.3.11) and (3.3.12) hold.

8 See footnote 6.

3.4 proofs 81


Part 1: If there is no largest T for which T− h(T) ≤ α1

1−α , with α ≥ 1, then thestatement of the lemma clearly holds since in that case there is a subsequencethat converges to infinity. We may thus assume that eventually T − h(T) >

α1

1−α . Observe that if x > α1

1−α then the derivative of the function x 7→ xα

exceeds 1. Hence for T sufficiently large, the increment of this function on theinterval [T − h(T), T] is at least linear, that is,

limT→∞

Tα − (T − h(T))α ≥ limT→∞

h(T) = ∞ .

Part 2: For α ≤ 0 the statement holds for any h. Assume therefore 0 < α < 1.By the mean value theorem there exists a u ∈ [T − h(T), h(T)] such that

Tα − (T − h(T))α = h(T)αuα−1 ≤ αh(T)(T − h(T))α−1 .

Hence, h(T) = Tβ with β satisfying 0 < β < 1 and β+ α− 1 < 0 will work.

4 O N T H E L O N G R AT E I N FA C T O R M O D E L S O F T H ET E R M S T R U C T U R E

This chapter is based on J. de Kort, A note on the long rate in factor models of theterm structure, To appear in: Mathematical Finance. doi: 10.1111/mafi.12151.

In this chapter, we consider factor models of the term structure based on aBrownian filtration. We show that the existence of a non-deterministic longrate in a factor model of the term structure implies, as a consequence of theDybvig-Ingersoll-Ross theorem, that the model has an equivalent representa-tion in which one of the state variables is non-decreasing. For two-dimensionalfactor models, we prove moreover that if the long rate is non-deterministic,the yield curve flattens out, and the factor process is asymptotically non-deterministic, then the term structure is unbounded. Finally, following up onan open question in El Karoui et al. (1998), we provide an explicit example of athree-dimensional affine factor model with a non-deterministic yet finite longrate in which the volatility of the factor process does not vanish over time.

4.1 introduction

A theorem by Dybvig, Ingersoll, and Ross (1996) states that the long-term in-terest rate in an arbitrage-free term structure model cannot decrease over time.Nevertheless, the long rate can be a non-deterministic process. It was shownin El Karoui et al. (1998) that a Heath-Jarrow-Morton model may generate anon-deterministic long rate if, for large maturities, the volatility of the instan-taneous forward rate vanishes sufficiently fast to make the long rate finite, andsufficiently slow to ensure that it is nonconstant. In this chapter we considerthe question whether the subclass of term structure models known as factormodels is rich enough to admit a finite stochastic long rate if we additionallyrequire that the volatility of the factor process does not vanish asymptoticallyover time. We show that this question can be answered affirmatively, but thatat least three state variables are required.

The result of Dybvig, Ingersoll and Ross has been extended in a number of di-rections. For example, for weaker notions of convergence of the long rate, see

83

84 on the long rate in factor models of the term structure

Goldammer and Schmock (2012). Kardaras and Platen (2012) considered theconvergence speed of the zero-coupon yields at different times as the maturitydate tends to infinity. Recently, Biagini et al. (2016) studied the behaviour ofthe long rate in HJM models where instantaneous forward rates are drivenby affine processes on a state space of symmetric positive semidefinite matri-ces. In another interesting paper, Biagini and Härtel (2014) look at interest ratemodels driven by Lévy processes and find that if the driving process has pathsof finite variation and negative jumps only then, contrary to the diffusion case,volatility does not necessarily need to vanish for large maturities in order forthe long rate to be finite. Dybvig et al. (1996) also proved that the long ratecannot increase almost surely if the state space is finite. Schulze (2008) gener-alized this result to infinite state spaces. In other recent work, Zhao (2009) andBao and Yuan (2013) characterize almost sure convergence to a constant longrate in two-factor Cox-Ingersoll-Ross (CIR) models with Lévy jumps. To builda model in which the long Libor-rate can randomly move up and down, with-out violating no-arbitrage, Brody and Hughston (2016) use discount functionsof the generalized hyperbolic type; these are discount functions for which therelation between rates and bond prices is not asymptotically exponential.

Long-term fixed income returns in models that admit a factor representationhave previously been studied by Deelstra and Delbaen (1995). They consid-ered a single-factor generalized CIR process and proved that the long rateconverges almost surely to a random variable which is proportional to the re-version level of the CIR process. Based on these results, Deelstra et al. (2000)suggests a modification of a three-factor short rate model introduced by Ticeand Webber (1997) such that the long rate converges to a stochastically time-varying mean-reversion level. The present chapter is concerned with modelsin which not only the short rate but also the yield or the instantaneous for-ward rate admits a factor representation. In models where the short rate hasa factor structure, the instantaneous forward rate inherits this factor structureonly if the bond price has an exponential affine representation. Yao (1999b),using an instantaneous forward rate specification with separable volatilityfunction and building on results in Ritchken and Sankarasubramanian (1995),constructs a factor model with two state variables in which the long rate isnon-deterministic. Both the factor process and the yield curve parameteriza-tion in that model are time-inhomogeneous, and the volatility of the factorprocess vanishes over time. In the work of El Karoui et al. (1998) a seriesof two-dimensional affine factor models is discussed. They find that, in theirexamples, either the continuously compounded rate is infinite or the modelcontains a nondecreasing state variable.

4.2 the long rate in factor models 85

We extend these results by showing that a term structure model with threestate variables, with one of the state variables having finite variation, is themost parsimonious model specification based on a Brownian filtration thatcan accommodate a stochastic long rate if we require that the volatility of thefactor process does not vanish over time. We thus show that the propertiesexemplified by the models in El Karoui et al. (1998) are general characteristicswhich are shared by all two-dimensional factor models. Finally, we providean explicit example of an affine factor model in which the long rate convergesalmost surely to a nondecreasing stochastic process.

The chapter is structured as follows. In Proposition 4.2.2 we prove that, upto a change of coordinates, any factor-based model in which the long-rateis not deterministic while the term structure of interest rates does not go toinfinity must have a nondecreasing state variable, where we allow the factorprocess to be time-inhomogeneous. In Proposition 4.3.1 we show that, undermild conditions, the presence of a stochastic long rate in a factor model withtwo state variables implies that the term structure is unbounded. Finally, inProposition 4.4.1 we construct a factor model with a stochastic long rate whichis finite almost surely.

4.2 the long rate in factor models

Let (Ω,F , P, (Ft)t≥0) be a filtered probability space on which is defined ad-dimensional P-Brownian motion W. The time t price p(t, T) of a zero-couponbond, paying 1 unit of currency at time T, is modelled as an Itô process for 0 ≤t ≤ T with respect to (Ft)0≤t≤T. We assume that p(T, T) = 1 and p(t, T) > 0for all 0 ≤ t ≤ T. The instantaneous forward rate is defined by r(t, x) =−∂x log p(t, t + x), while the exponential (or continuously compounded) rateis given by z(t, x) = −x−1 log p(t, t + x) for all x > 0.

As in Chapter 3, we impose as a standing assumption that no arbitrage possi-bilities exist. Following Hubalek et al. (2002), we thus assume that there existsa measure Q, locally equivalent to P, such that, for all t ≥ 0 and x ≥ 0, the quo-tient B(t)/B(t + x) is Q-integrable where B(t) := exp(

∫ t0 r(s, 0) ds), and such

that discounted bond prices B(t)−1 p(t, T) are Q-martingales for 0 ≤ t ≤ T.

Consider a time-inhomogeneous affine term structure model in which theMusiela parameterised instantaneous forward rates r : R+ ×R+ → R satisfy

r(t, x) := f (t, t + x) = h0(x) +d

∑i=1

λi(x)Yi(t) , (4.2.1)


where h0 : R+ → R and λi : R+ → R, i ∈ 1, . . . , d, are continuously differen-tiable F0-measurable functions. The stochastic factor Y is defined by

Y(t) =∫ t

0D(s, Y(s)) ds +

∫ t

0V(s, Y(s))dW(s) , (4.2.2)

in which D : R+ ×Rd×1 → Rd×1 and V : R+ ×Rd×1 → Rd×d are continuousfunctions such that (4.2.2) admits a strong solution. Notice that the assumptionY(0) = 0 is not restrictive. If Y(0) = y0 6= 0 then we may apply our analysis toZ(t) := Y(t)− y0 satisfying dZ(t) = D(t, Z(t) + y0) dt+V(t, Z(t) + y0)dW(t)and Z(0) = 0, and change h0 accordingly. We will refer to the components ofY as the state variables. The number of driving factors may be smaller thanthe number of state variables, for instance if V has rows consisting only ofzeroes. The Musiela parameterised exponential rates satisfy

z(t, x) = H(x) +d

∑i=1

Λi(x)Yi(t) , (4.2.3)

in which H(x) = 1x∫ x

0 h0(s) ds and Λ(x) = 1x∫ x

0 λ(s) ds.

For x = 0 we let Λ(x) and H(x) be defined by their righthand side limitsh0(0) and λ(0). This setup includes the model of El Karoui et al. (1998), butwe do not follow their requirements that D and V must be affine or time-homogeneous functions of the factor process Y. Indeed, to obtain the setupin El Karoui et al. (1998), we can set A(T − t) =

∫ T−t0 λ(s) ds, b(T − t) =∫ T−t

0 h0(s) ds and T = t + x, then − log p(t, T) = A(T − t)Y(t) + b(T − t).

The long instantaneous forward rate f∞ and the exponential long rate z∞ aredefined by

f∞(t) = limx→∞

f (t, t + x) and z∞(t) = limx→∞

z(t, x) ,

provided that the corresponding limits exist. We say that the limit “exists” ifthe lim sup and the lim inf are random variables that agree almost surely. No-tice that we do not require the limit to be finite. In general, the long rate inmodels of the form (4.2.1) is not necessarily a measurable function. An alterna-tive definition of the long-term interest rate, which guarantees measurability,is discussed in Goldammer and Schmock (2012) and Brody and Hughston(2016). In this paper however, we use the following characterisation of theexistence, as a random variable, of the long instantaneous forward rate.

Proposition 4.2.1. Assume that the elements of 1, Y1(t0), . . . , Yd(t0) are almostsurely linearly independent for at least one t0 ∈]0, ∞[. Then the long instantaneousforward rate exists for every t ≥ 0 if and only if limx→∞ λ(x) and limx→∞ h0(x)exist.


Proof. (If) Suppose that limx→∞ λ(x) = ζ and limx→∞ h0(x) exist. From Eq. (4.2.1)we obtain that, for all t ≥ 0, the long rate satisfies f∞(t) = f∞(0)+∑d

i=1 ζi Yi(t) .We thus find that the long instantaneous forward rate is a linear combinationof the state variables and hence is a random variable for every t ≥ 0.

(Only if) Let t0 be such that the elements of 1, Y1(t0), . . . , Yd(t0) are almostsurely linearly independent. By assumption there is a set U ⊂ Ω with P(U) =0 such that limx→∞ f (t0, t0 + x, ω) exists point-wise for every ω ∈ Uc. Considerthe set V = (Y1(t0, ω), . . . , Yd(t0, ω)) : ω ∈ Uc and suppose that dim(V) <d, that is, there exists some β ∈ Rd+1, β 6= 0 such that ∑d

j=1 β j vj = β0 wheneverv ∈ V. Then

β0 =d

∑j=1

β jYj(t0, ω) , for all ω ∈ Uc,

which contradicts the assumption that the elements of 1, Y1(t0), . . . , Yd(t0)are almost surely linearly independent. Hence there must be ω1, . . . , ωd ∈ Uc

such that yi := (Y1(t0, ωi), . . . , Yd(t0, ωi))′, i ∈ 1, . . . , d, span Rd. Write Υ =

(y1, . . . , yd)′, then Υ is a d × d matrix of full rank, therefore Υ−1 exists. By

Eq. (4.2.1) we have that limx→∞ h0(x) exists (take t = 0) and limx→∞ Υλ(x)exists (take t = t0). It follows that

limx→∞

λ(x) = limx→∞

Υ−1(Υλ(x)) = Υ−1 limx→∞

Υλ(x) exists .

The next proposition shows that in factor models with a nondeterministiclong rate, a linear combination of the state variables driving the instantaneousforward rate is nondecreasing.

Proposition 4.2.2. Suppose that the conditions of Proposition 4.2.1 hold. If the longinstantaneous forward rate exists for all t ≥ 0 and is nondeterministic for t > 0, thenthe term structure model in Eqs. (4.2.1)–(4.2.2) has an equivalent representation inwhich the nondecreasing long instantaneous forward rate coincides with one of thestate variables, plus a constant.

Proof. From Eqs. (4.2.1) and (4.2.2), we obtain

f (t, t + x)− f (0, x) =d

∑i=1

λi(x)

[∫ t

0Di(s, Y(s)) ds

+∫ t

0

d

∑j=1

Vij(s, Y(s))dWj(s)

]. (4.2.4)


By Proposition 4.2.1, we have that ζ = limx→∞ λ(x) and limx→∞ h0(x) exist.We thus find that the long rate satisfies, for all t ≥ 0,

f∞(t)− f∞(0) =d

∑i=1

ζi

[∫ t

0Di(s, Y(s)) ds

+∫ t

0

d

∑j=1

Vij(s, Y(s))dWj(s)

]. (4.2.5)

Theorem 3.3.7 implies, together with Lemma 3.3.8, that the long instantaneousforward rate is a nondecreasing process; therefore it has finite first-order vari-ation and thus cannot have a nonzero diffusion term. It follows that ζV = 0almost surely for every t ≥ 0 and

f∞(t) = f∞(0) +d

∑i=1

ζi

∫ t

0Di(s, Y(s)) ds .

If ζ = 0 then by Eq. (4.2.5) the long rate is constant which is precluded by theassumptions of the proposition. Hence ζ 6= 0 and it follows that V is singular.If the factor process Y is one-dimensional, that is, if d = 1, then either V = 0or ζ = 0 must hold. In both cases the long rate is deterministic.

Assume therefore that d ≥ 2. Because ζ 6= 0 there must be an i∗ ∈ 1, . . . , dsuch that ζi∗ 6= 0. Define the (invertible) d× d matrix M by

M =

1

1. . .

ζ1 ζ2 · · · ζi∗ · · · · · · ζd. . .

1

1

. (4.2.6)

The matrix M has zero entries outside the main diagonal and the i∗-th row.Define

Y(t) := MY(t) and λ(x) := λ(x)M−1 . (4.2.7)

The process Y(t) satisfies Y(0) = 0 and

dY(t) = D(t, Y(t)) dt + V(t, Y(t))dW(t) , (4.2.8)


in which D(t, y) = MD(t, M−1y) and V(t, y) = MV(t, M−1y) . The factormodel defined through h0, λ and Y is equivalent to the one defined by h0, λand Y. Indeed,

r(t, x) = h0(x) +d

∑i=1

λi(x)Yi(t) = h0(x) +d

∑i=1

λi(x)M−1MYi(t)

= h0(x) +d

∑i=1

λi(x)Yi(t) .

The i∗-th state variable satisfies Yi∗(t) = ∑di=1 ζiYi(t) = f∞(t)− f∞(0). We have

thus established that the model (4.2.1) has a representation in which, up to aconstant shift, the non-decreasing long instantaneous forward rate coincideswith one of the state variables.

The existence of the exponential long rate can be characterised in terms of theexistence of limx→∞ Λ(x) and limx→∞ H(x), as the following result shows.

Proposition 4.2.3. Assume that the elements of 1, Y1(t0), . . . , Yd(t0) are almostsurely linearly independent for at least one t0 ∈ [0, ∞[. Then the exponential long rateexists for every t ≥ 0 if and only if limx→∞ Λ(x) and limx→∞ H(x) exist.

Proof. The proof is similar to the proof of Proposition 4.2.1 and we omit thedetails.

We can also formulate a variation of Proposition 4.2.2 in which the conditionshave been slightly weakened.

Proposition 4.2.4. Suppose that the conditions of Proposition 4.2.3 hold. If the expo-nential long rate exists for all t ≥ 0 and is non-deterministic for t > 0, then the termstructure model in Eqs. (4.2.1)–(4.2.3) has an equivalent representation in which thenon-decreasing exponential long rate coincides with one of the state variables, plus aconstant.

Proof. As in the proof of Proposition 4.2.2 one can show that limx→∞ Λ(x) =:ζ 6= 0. Using the transformation

Y(t) = MY(t) and Λ(t) = Λ(t)M−1 , (4.2.9)

where M is defined as in Eq. (4.2.6), a representation can be established inwhich, up to a constant shift, the exponential long rate coincides with one ofthe state variables. The rest of the proof is similar to the proof for Proposi-tion 4.2.2.


4.3 the two-dimensional case

In the following proposition we consider two-dimensional factor models thatadmit a non-deterministic long rate. We show that the term structure is even-tually unbounded if the factor process is asymptotically non-deterministic andthe forward curve flattens for large expiration dates. The second condition re-flects the fact that little market information is available about the far future,hence the instantaneous rate for lending or borrowing should not differ muchbetween long-dated maturities. We say that a model is asymptotically deter-ministic almost surely if

P(

limt→∞〈Y〉t < ∞

)= 1 . (4.3.1)

Notice that t 7→ 〈Y〉t is point-wise nondecreasing hence, for any a ≥ 0 andj ∈ 1, 2,

ω ∈ Ω | limt→∞

⟨Yj⟩

t ∈ [0, a]=⋂

k∈N

ω ∈ Ω |

⟨Yj⟩

k ∈ [0, a]

,

and it follows that limt→∞ 〈Y〉t is a random variable. The forward curve is saidto flatten if

limx→∞

∂xh0(x) = 0 and limx→∞

∂xλi(x) = 0 for i = 1, 2. (4.3.2)

Consider the factor model in Eq. (4.2.1) with d = 2 and suppose that theconditions of Proposition 4.2.2 hold. We then have the following result.

Proposition 4.3.1. If the factor process Y(t) with d = 2 is not asymptotically de-terministic almost surely and the forward curve flattens for long maturities, thenlimt→∞ f∞(t) = ∞ almost surely. Moreover, the term structure x 7→ r(t, x) is un-bounded over time, that is, there exists a random variable X : Ω → [0, ∞[ such thatlimt→∞ r(t, X) = ∞.

Proof. From the proof of Proposition 4.2.2, we have that limx→∞ λ(x) = ζ 6= 0.Assume, without loss of generality, that ζ2 6= 0 and apply the coordinatetransformation defined in Eqs. (4.2.7) and (4.2.8) with

M =

(1 0

ζ1 ζ2

),

4.3 the two-dimensional case 91

so that Y1(t) = Y1(t), Y2(t) = ζ1Y1(t) + ζ2Y2(t) and λ1(x) = λ1(x)− ζ1ζ2

λ2(x)

while λ2(x) = 1ζ2

λ2(x). The factor process (4.2.2) can thus be written as

Y1(t) =∫ t

0D1(s, Y(s)) ds

+∫ t

0V11(s, Y(s))dW1(s) +

∫ t

0V12(s, Y(s))dW2(s) ,

Y2(t) =∫ t

0D2(s, Y(s)) ds .

(4.3.3)

As before, we use Theorem 3.3.7 and Lemma 3.3.8 to conclude that the secondrow of the volatility matrix must be zero, i.e. V21 = V22 = 0.

Because Q is locally equivalent to P, there exists, for every t ≥ 0, a martin-gale L such that dP/dQ = L(s) on F (s) for 0 ≤ s ≤ t. Hence, dL(s) =ϕ(s)L(s)dW(s), 0 ≤ s ≤ t, for a certain process ϕ called the market priceof risk. The HJM drift condition in Musiela form, see for example Björk andSvensson (2001, Prop. 4.1), requires that for all Y(t)

2

∑i=1

λi(x)Di(t, Y(t)) = ∂xr(t, x) +2

∑j=1

σj(t, x, Y(t))∫ x

0σj(t, u, Y(t)) du

−2

∑j=1

ϕj(t)σj(t, x, Y(t)) , (4.3.4)

in which σj(t, x, y) = ∑2i=1 λi(x)Vij(t, y) for y ∈ Rd×1 and j ∈ 1, 2. The

second term on the righthand side in Eq. (4.3.4) can be written as

2

∑j=1

σj(t, x, Y(t))∫ x

0σj(t, u, Y(t)) du

=2

∑j=1

(2

∑i=1

λi(x)Vij(t, Y(t))

) ∫ x

0

(2

∑k=1

λk(u)Vkj(t, Y(t))

)du

=2

∑i=1

2

∑k=1

(Hik(t, Y(t)) λi(x)

∫ x

0λk(u) du

), (4.3.5)

where Hik(t, Y(t)) := ∑2j=1 Vij(t, Y(t))Vkj(t, Y(t)) is the (i, k)-th element of the

symmetric matrix H = VV′. Integrating the drift condition Eq. (4.3.4) on the


interval [0, t] and using V21 = V22 = 0 (hence H12 = H21 = H22 = 0), we findthat

2

∑i=1

λi(x)∫ t

0Di(s, Y(s)) ds = ∂xh0(x)t +

2

∑i=1

∂xλi(x)∫ t

0Yi(s) ds

+L(x)∫ t

0H11(s, Y(s)) ds− λ1(x)

∫ t

0

2

∑j=1

V1j(s, Y(s))ϕj(s) ds , (4.3.6)

in which L(x) = λ1(x)∫ x

0 λ1(u) du. Due to the coordinate transformation inEq. (4.2.7), we have

limx→∞

λ1(x) = 0 and limx→∞

λ2(x) = 1 .

Moreover, the flattening of the forward curve is preserved under the transfor-mation in Eq. (4.2.7), that is

limx→∞

∂xλ(x) = 0 .

The functions ∂xh0(x), ∂xλ(x) and λ(x) are continuous with finite limits forx → ∞; hence they are bounded functions of x. By assumption, the func-tions D and V, and hence also D and V, are continuous. Because the Itô in-tegrals in Eq. (4.3.3) have versions with continuous sample paths, it followsthat the integrals over Y(t), D(t, Y(t)), H11(t, Y(t)) and V(t, Y(t)) appearingin Eq. (4.3.6) are finite almost surely for all t ∈ [0, ∞[. By taking the limitx → ∞ in Eq. (4.3.6), we thus obtain that

f∞(t)− f∞(0) =∫ t

0D2(s, Y(s)) ds = lim

x→∞L(x)

∫ t

0H11(s, Y(s)) ds . (4.3.7)

The lefthand side exists for every t ≥ 0 by the assumptions of Proposition 4.2.2.Therefore, limx→∞ L(x) exists. Notice that

∫ t0 H11(s, Y(s)) ds = 0 would imply

that H11(t, Y(t)) = 0 for any t > 0, contradicting our assumption that the longrate is non-deterministic. Likewise, limx→∞ L(x) 6= 0 because we assumed thatthe long rate is not constant.

If limx→∞ L(x) = ∞ then f∞(t) = ∞ for all t > 0 by Eq. (4.3.7). It thus remainsto prove limt→∞ f∞(t) = ∞ for the case limx→∞ L(x) < ∞. The quadraticcovariation of Yi(t) and Yk(t), with i, k ∈ 1, 2, satisfies

〈Yi, Yk〉t =∫ t

0

[Vi1(s, Y(s))Vk1(s, Y(s)) + Vi2(s, Y(s))Vk2(s, Y(s))

]ds

=∫ t

0Hik(s, Y(s)) ds .

4.3 the two-dimensional case 93

Because H12 = H21 = H22 = 0, the quadratic variation of the factor processcan be expressed as

〈Y〉t =∫ t

0diag H(s, Y(s)) ds = diag

∫ t

0M−1H(s, Y(s))(M′)−1 ds

=(

1, ζ21ζ−2

2

) ∫ t

0H11(s, Y(s)) ds , (4.3.8)

where we use “diag” to denote the main diagonal of a matrix. Due to theassumption that limt→∞ 〈Y〉t = ∞ almost surely, we conclude from Eqs. (4.3.7)and (4.3.8) that limt→∞ f∞(t) = ∞ almost surely.

We next proceed to prove that the term structure x 7→ r(t, x) is unbounded intime. Suppose that λ1 ≡ 0, then L(x) = λ1(x)

∫ x0 λ1(u) du = 0 for all x ≥ 0

which is not possible due to our assumption that the long rate is non-constant.We may thus assume that there exists a 0 ≤ x0 < ∞ such that λ1(x0) 6= 0.Consider the term structure at x = x0, i.e.

r(t, x0) = h0(x0) + Z(t) , Z(t) := λ1(x0)Y1(t) + λ2(x0)Y2(t) .

As Z has a version with continuous sample paths, the subset U of Ω for whicht 7→ Z(t) is bounded on [0, ∞[ can be expressed as

U =⋃

n∈N

⋂m∈Q+

Z(m) ≤ n ,

hence U is measurable. For ω ∈ Uc we have that r(·, x0) is unbounded on [0, ∞[.Because limx→∞ λ1(x) = 0 and limx→∞ λ2(x) = 1 it follows that λ1 is not pro-portional to λ2 and consequently there exists an 0 ≤ x1 < ∞ such that

λ1(x0)λ2(x1) 6= λ2(x0)λ1(x1) . (4.3.9)

The term structure in x = x1 satisfies

r(t, x1) = h0(x1) + a1Z(t) + a2Y2(t) ,

in which

a1 = λ1(x1)/λ1(x0) and a2 = −a1λ2(x0) + λ2(x1) .

Note that a2 6= 0 due to Eq. (4.3.9). Because Y2(t) = f∞(t)− f∞(0) it followsthat r(·, x1) is unbounded on [0, ∞[ for ω ∈ U. Set X := x0 + (x1 − x0)1U thenlimt→∞ r(t, X) = ∞ almost surely.

We thus find that, in all factor models with two state variables satisfying theconditions of Propositions 4.2.2 and 4.3.1, the long instantaneous forward rate


tends to infinity over time, and there exists at least one finite expiration datefor which the corresponding instantaneous forward rate is unbounded. Wehave shown in the proof that the finite expiration date for which this happensmay depend on the stochastic scenario.

Similar to Proposition 4.3.1, we can formulate the following result on the be-haviour of the long exponential yield in factor models with two state variables.We will say that the yield curve flattens if

limx→∞

∂x H(x) = 0 and limx→∞

∂xΛi(x) = 0, for i = 1, 2 .

Consider the factor model in Eq. (4.2.3) with d = 2 and assume that the condi-tions of Proposition 4.2.4 hold. Then we have the following proposition.

Proposition 4.3.2. If the factor process Y(t) with d = 2 is not asymptoticallydeterministic almost surely and the yield curve flattens for long maturities, thenlimt→∞ z∞(t) = ∞ almost surely. Moreover, the term structure x 7→ z(t, x) is un-bounded over time, that is, there exists a random variable X : Ω → [0, ∞[ such thatlimt→∞ z(t, X) = ∞.

Proof. From the proof of Proposition 4.2.4 we have limx→∞ Λ(x) = ζ 6= 0.Assume, without loss of generality, that ζ2 6= 0 and apply the coordinate trans-formation defined in Eqs. (4.2.8) and (4.2.9). Then Λ1(x) = Λ1(x)− ζ1

ζ2Λ2(x)

and Λ2(x) = 1ζ2

Λ2(x), while the transformed factor process Y then satisfiesEq. (4.3.3). Integrating the drift condition in Eq. (4.3.6) on the interval [0, x]and multiplying both sides by 1

x we obtain, for x > 0,

2

∑i=1

Λi(x)∫ t

0Di(s, Y(s)) ds =

1x

∫ x

0L(u) du

∫ t

0H11(s, Y(s)) ds

+h0(x)− h0(0)

xt +

2

∑i=1

λi(x)− λi(0)x

∫ t

0Yi(s) ds

− Λ1(x)∫ t

0

2

∑j=1

V1j(s, Y(s))ϕj(s) ds , (4.3.10)

in which

1x

∫ x

0L(u) du =

1x

∫ x

0λ1(u)

∫ u

0λ1(v) dv du =

12

xΛ21(x) .

Observe that, for i ∈ 1, 2,

λi(x)x

=1x

Λi(x) + ∂xΛi(x) . (4.3.11)

4.4 the higher-dimensional cases 95

Due to the coordinate transformation in Eq. (4.2.9) we have limx→∞ Λ1(x) = 0while limx→∞ Λ2(x) = 1. Moreover, the flattening of the yield curve is pre-served under the transformation, that is, limx→∞ ∂xΛ(x) = 0. Hence, therighthand side in Eq. (4.3.11) vanishes in the limit as x → ∞ for i ∈ 1, 2.Likewise, one can show that limx→∞ h0(x)/x = 0. Taking the limit x → ∞ inEq. (4.3.10) we thus obtain

z∞(t)− z∞(0) =∫ t

0D2(s, Y(s)) ds

=12

[lim

x→∞xΛ2

1(x)] ∫ t

0H11(s, Y(s)) ds .

The rest of the proof is similar to the proof of Proposition 4.3.1.

4.4 the higher-dimensional cases

In this section we show that the result of Proposition 4.3.1 does not necessarilyhold for d > 2 by constructing an explicit example of an affine factor modelwith d = 3 in which the long rate is finite and non-deterministic, while the fac-tor process is not asymptotically deterministic. We will use Ai and Bi to denotethe Airy functions of the first and second kind respectively, see Abramowitzand Stegun (1964, Chapter 10). The Airy functions are linearly independentsolutions v to the Stokes equation xv− ∂xxv = 0.

Let µ1, κ1, κ2, σ1 and σ2 be constants satisfying 2µ1 > σ21 and σ2 > 0. We define

processes Y1(t) and Y2(t) by

Y1(t) =∫ t

0[µ1 − κ1Y1(s)] ds +

∫ t

0σ1

√Y1(s)dW1(s) , (4.4.1)

and

Y2(t) = 1−∫ t

0κ2Y2(s) ds +

∫ t

0σ2

√Y2(s)dW2(s) . (4.4.2)

A strong solution exists for the square-root processes Y1(t) and Y2(t) due to theYamada-Watanabe theorem, see Karatzas and Shreve (1991, Theorem 5.2.13). Athird state variable Y3(t), which we define by

Y3(t) =∫ t

0Y2(s) ds , (4.4.3)

is nondecreasing and finite almost surely, because Y2(t) is positive and meanreverting to zero in finite time with probability one. Observe that the pro-cess Y(t) defined by Eqs. (4.4.1)–(4.4.3) is not asymptotically deterministicas limt→∞ 〈Y1〉t = ∞.


Consider the factor model

r(t, x) = h0(x) + λ1(x)Y1(t) + λ2(x)Y2(t) + Y3(t) , (4.4.4)

with Y(t) as in Eqs. (4.4.1)–(4.4.3) and

λ1(x) = λ1(0)[

cosh(

12

γx)+

κ1

γsinh

(12

γx)]−2

, (4.4.5)

where γ = (κ21 + 2λ1(0)σ2

1 )12 and

λ2(x) =2ψ

φ +12

ψx−(

C1∂1Ai(φ + 12 ψx)− C2∂1Bi(φ + 1

2 ψx)C1 Ai(φ + 1

2 ψx)− C2 Bi(φ + 12 ψx)

)2 , (4.4.6)

in which

φ = (κ22 + 2λ2(0)σ2

2 )/ψ2, ψ = (2σ2)2/3,

while

C1 = κ Bi(φ)− ψ∂1Bi(φ), C2 = κ Ai(φ)− ψ∂1Ai(φ), (4.4.7)

and

h0(x) = h0(0) + µ1

∫ x

0λ1(s) ds . (4.4.8)

Suppose that λ1(0) > 0 and λ2(0) > 0. The next proposition shows that thelong rate in this model is finite and non-deterministic.

Proposition 4.4.1. The factor model in Eqs. (4.4.4)–(4.4.8) is arbitrage-free. More-over, the long instantaneous forward rate in this model is finite almost surely andsatisfies

f∞(t) = h0(0) +2µ1

γ + κ1+∫ t

0Y2(s) ds . (4.4.9)

Before proving Proposition 4.4.1, we need the following auxiliary result.

Lemma 4.4.2.

limx→∞

∂1 Bi(u(x))Bi2(u(x))

= 0 , limx→∞

[u(x)−

(∂1 Bi(u(x))

Bi(u(x))

)2]= 0 , (4.4.10)

in which u(x) := φ + 12 ψx, with ψ > 0.


Proof. By l’Hôpital and the identity ∂xx Bi(x) = x Bi(x) we have

limx→∞

(Bi(x)

∂1 Bi(x)

)2

= limx→∞

2 Bi(x)∂1 Bi(x)2∂1 Bi(x)∂11 Bi(x)

= limx→∞

1x= 0 . (4.4.11)

Another application of l’Hôpital yields the following expression for the rela-tive asymptotic behaviour of ∂1 Bi(x) and Bi(x)

limx→∞

(√x Bi(x)

∂1 Bi(x)

)2

= limx→∞

2x Bi(x)∂1 Bi(x) + Bi2(x)2x Bi(x)∂1 Bi(x)

= limx→∞

(1 +

Bi(x)2x∂1 Bi(x)

)= 1 . (4.4.12)

Because limx→∞ ∂1 Bi(x) = ∞, it follows from Eq. (4.4.12) and l’Hôpital that

limx→∞

∂1 Bi(x)Bi2(x)

= limx→∞

∂1 Bi(x)√x Bi(x)

limx→∞

√x

Bi(x)

= 1 · limx→∞

12√

x ∂1 Bi(x)= 0 .

This proves the first part of Eq. (4.4.10).

By the chain rule and the identity ∂11 Bi(x) = x Bi(x) we have

∂x∂1 Bi(u(x)) = 2ψ−1∂xx Bi(u(x)) =12

ψu(x)Bi(u(x)) . (4.4.13)

From Eqs. (4.4.11) and (4.4.13) together with l’Hôpital we thus find

limx→∞

[u(x)−

(∂1 Bi(u(x))

Bi(u(x))

)2 ]= lim

x→∞

u(x)Bi2(u(x))− (∂1 Bi(u(x)))2

Bi2(u(x))

= limx→∞

[ 1ψ Bi(u(x))∂1 Bi(u(x))

( 12

ψ Bi2(u(x))

+ ψu(x)Bi(u(x))∂1 Bi(u(x)) − 2∂1 Bi(u(x))∂x∂1 Bi(u(x)))]

= limx→∞

Bi(u(x))2∂1 Bi(u(x))

= 0 . (4.4.14)

This establishes the second part of Eq. (4.4.10).

We can now prove Proposition 4.4.1.


Proof of Proposition 4.4.1.

Let r(t, x) = h0(x) + ∑3i=1 λi(x)Yi(t) with Y as defined in Eqs. (4.4.1)–(4.4.3).

By integrating the drift condition Eq. (4.3.4) on the interval [0, x] with ϕ = 0we find that, to exclude arbitrage, the functions A(x) =

∫ x0 λ(s) ds and b(x) =∫ x

0 h0(s) ds must satisfy∂x A1(x)∂x A2(x)∂x A3(x)

=

∂x A1(0)

∂x A2(0)

∂x A3(0)

+

−κ1 0 0

0 −κ2 1

0 0 0

A1(x)

A2(x)A3(x)

− 12

σ21 A2

1(x)σ2

2 A22(x)0

,

(4.4.15)

and

∂xb(x) = ∂xb(0) + µ1A1(x) . (4.4.16)

The first and third ODE admit closed-form solutions:

A1(x) = 2λ1(0)[

κ1 + γ coth(

12

γx)]−1

and A3(x) = x .

Differentiation yields Eq. (4.4.5) and λ3(x) = 1. The solution to the secondODE can be expressed in terms of solutions of the Stokes equation. Indeed, ifwe apply the transformations

A2(x) =2∂xw(x)σ2

2 w(x), w(x) = e−

12 κ2xz(x) , z(x) = v(φ+

12

ψx) , (4.4.17)

then v(x) satisfies the Stokes equation 0 = xv(x) − ∂xxv(x). The solution tothis equation is v(x) = C3 Ai(x) + C4 Bi(x) and the values of C3 and C4 aredetermined by the boundary condition A2(0) = 0. It follows that

A2(x) =− κ2

σ22+

4ψ2

C1∂1 Ai(φ + 12 ψx)− C2∂1 Bi(φ + 1

2 ψx)C1 Ai(φ + 1

2 ψx)− C2 Bi(φ + 12 ψx)

,

with C1 and C2 as in Eq. (4.4.7). Differentiation of this expression yields Eq. (4.4.6).Using Lemma 4.4.2 and Eqs. (4.4.5)–(4.4.6) we find

limx→∞

λ1(x) = limx→∞

λ2(x) = 0 , (4.4.18)

and

limx→∞

h0(x) = h0(0) +2µ1

γ + κ1. (4.4.19)


Hence, f∞(t) = f∞(0) + Y3(t) and Eq. (4.4.9) follows. The long rate is finitebecause Y2(t) hits zero in finite time with probability one.

The requirement in Proposition 4.3.1 which states that the factor process mustbe asymptotically non-deterministic cannot be omitted. Indeed, observe fromProposition 4.4.1 that the factor model defined by r(t, x) = h0(0)+λ2(x)Y2(t)+Y3(t) satisfies all other requirements, but it has a finite and non-deterministiclong rate.

In this chapter, we have shown how to construct a parsimonious factor modelof the term structure which has the property that the long rate of interestis non-deterministic. Up to a coordinate transformation, the dynamics of thelong rate in any such model must coincide with the dynamics of one of thestate variables. This state variable thus has finite first-order variation as aconsequence of the Dybvig-Ingersoll-Ross theorem. To exclude unrealistic be-haviour, such as the long rate tending to infinity over time or the forwardrates eventually becoming deterministic, at least three state variables mustbe included in the model. Finally, we have proposed a tractable term structuremodel in which the long rate follows an integrated Cox-Ingersoll-Ross process.An open question is whether similar results can be derived using the more gen-eral definitions of the long rate appearing in Goldammer and Schmock (2012)and Brody and Hughston (2016).

5 T E R M S T R U C T U R E E X T R A P O L AT I O N A N DA S Y M P T O T I C F O RWA R D R AT E S

This chapter is based on J. de Kort and M.H. Vellekoop, Term structure extrap-olation and asymptotic forward rates, Insurance: Mathematics and Economics, 67

(2016): 107–119.

In this chapter, we investigate inter- and extrapolation methods for term struc-tures under constraints in order to generate market-consistent estimates whichdescribe the asymptotic behavior of forward rates. Our starting point is themethod proposed by Smith and Wilson, which is used by the European insur-ance supervisor EIOPA. We use the characterization of the Smith-Wilson classof interpolating functions as the solution to a functional optimization problemto extend their approach in such a way that forward rates will converge toa value which is an outcome of the optimization process instead of a valuewhich is given a priori. Precise conditions are stated which guarantee that theoptimization problems involved are well-posed on appropriately chosen func-tion spaces. As a result, a well-defined optimal asymptotic forward rate can bederived directly from prices and cashflows of traded instruments. This allowspractitioners to use raw market data to extract information about long termforward rates, as we will show in a study which analyzes historical EURIBORswap data.

5.1 introduction

Every financial institution that uses a discount curve to estimate the currentvalue of future cashflows has to decide how to construct such a curve frominformation that can be found in the fixed income markets. In such markets,cashflows are defined for a limited number of payment dates so curve con-struction will require certain interpolation methods. If the financial institutionneeds to assess the value of cashflows with maturities that exceed the ma-turities of observable market instruments, or when the market informationfor these longer maturities is known to be unreliable, there is also a need forextrapolation of the curve. This last problem arises naturally for many insur-ance companies and pension funds, since by the very nature of their busi-

101

102 term structure extrapolation and asymptotic forward rates

ness their liabilities stretch out much further into the future than those of theaverage financial institution. At the same time, regulatory frameworks suchas the Solvency II proposals for European insurers stress the importance ofmarket-consistent pricing of liabilities. This raises the question how one cansystematically incorporate market information in a discount curve, while atthe same time recognizing that a subjective choice has to be made for maturi-ties in between or beyond those that can be directly related to available marketinstruments.

Yields and forward rates provide natural coordinates to characterize discountcurves and several parametrized functional forms have therefore been pro-posed which describe the yield curve or the forward curve rather than thediscount curve itself. Well-known examples include the approaches of Nelsonand Siegel (1987) and Svensson (1994) which use given functions with threeand four free parameters respectively to fit the entire yield curve. Since it is im-possible to recover the exact prices of all market instruments using only a fewparameters, some optimality criterion must be defined to decide which valuesof the parameters are chosen. In practice one often minimizes a weighted sumover the squares of the pricing errors for a given set of fixed income instru-ments.

If one insists that all observed market prices must be fitted exactly and allowsthe introduction of a larger number of parameters to achieve this, we face astandard interpolation problem. An important criterion to decide which in-terpolation methods should be applied, is the desired degree of smoothnessof the resulting interpolating function. It is clear that piecewise linear func-tions can perfectly interpolate a given set of yields for different maturities, butthis leads to a yield curve which will not be differentiable, and the associatedforward curve will be discontinuous.

To achieve smoother discount-, yield- and forward curves, standard spline-based interpolation methods have been proposed and certain modificationsbased on the particular application at hand have been suggested. An overviewof the many possible methods is given in a paper by Hagan and West (2006).Smoothness is important, but there are other desirable properties of interpo-lating functions which are analyzed in that study, such as preserving mono-tonicity or convexity in the given dataset. The authors also investigate whetherchanges in the input data which are restricted to the vicinity of a particularmaturity may influence the values of the interpolated discount curve at maturi-ties far away from it. Strong non-local sensitivities in an interpolation methodmake it less suitable for risk management purposes since it complicates the

5.1 introduction 103

use of fixed income market instruments to mitigate the interest rate risk in aportfolio of liabilities.

Similar considerations play a role when deciding on a method to extrapolatethe discount curve beyond the last maturities for which reliable market infor-mation is available. Compared to the interpolation problem, the extrapolationproblem for term structures has received considerably less attention in the lit-erature until recently. The new Solvency II regulatory framework for Europeaninsurers EIOPA (2010) has changed this, since the prescribed method to con-struct the term structure from market data incorporates the assumption thatforward rates must converge to an a priori specified value which was chosento be 4.2%. The class of interpolating functions that is used is the one proposedin earlier work by Smith and Wilson (2000). The Smith-Wilson functions areexponential tension splines with a given tension parameter. They were intro-duced by Schweikert (1966) and first proposed as interpolating functions forterm structures by Barzanti and Corradi (1998a,b). Andersen (2007) character-ized the corresponding interpolating curves in terms of a more convenientbasis of functions and showed how non-local sensitivities are controlled bythe tension parameter.

Exponential tension spline functions form the solution to an explicit optimiza-tion problem for interpolation under the constraint that a set of given datapoints must be fitted exactly. The optimization criterion aims at making boththe first order derivatives and the second order derivatives of the interpolatingfunction small, in the sense that one minimizes the integral over all maturitiesof their squared values. Criteria based on second-order derivatives have oftenbeen used as a basis for term structure interpolation, see for example Adamsand Van Deventer (1994), Adams (2001) and Lim and Xiao (2002). Smith andWilson then added the constraint that the asymptotic forward rate is given apriori.

In this chapter we use similar criteria to extend the interpolation problem tothe case where the limit of forward rates is assumed to exist, but not givena certain value on beforehand. This not only makes it possible to constructdiscount curves without such a restrictive assumption but it also providesinformation on the behavior of forward rates at high maturities in historicaldata, without making specific model assumptions. This provides a way to es-tablish whether the values that are imposed in regulatory frameworks such asthe one by EIOPA are consistent with what is found in market data at timesof high market liquidity. There is already some empirical evidence that longterm rates are in fact highly uncertain. This is for example the conclusion of arecent paper by Balter et al. (2014) where a Bayesian approach is used to de-


scribe parameter uncertainty in a Vasicek short rate model. At the end of thischapter some further empirical evidence will be presented, which suggeststhat long term forward rates have been rather stable before the financial crisisof 2008 but that they have decreased considerably after the crisis.

The structure of the chapter is as follows. First we define the constrained termstructure interpolation problem and the solution proposed by Smith & Wilsonin the next two sections. We analyse its properties in section 5.4 and use thisto solve a related but unconstrained optimization problem in section 5.5. Insection 5.6 another method is presented which directly describes the interpo-lation problem in terms of forward or yield curves instead of discount curves.This has the advantage of generating a very intuitive and simple formula forthe asymptotic forward rate in terms of market yields at different maturities.In section 5.7 we show some numerical results based on market data and wedraw conclusions and provide some proofs in the last two sections.

5.2 term structure interpolation

Term structures can be represented in many equivalent ways. As in earlierchapters, we use the notation p(t, T) for the amount to be paid at a time t ≥ 0to receive a certain single unit of currency at a later time T ≥ t, i.e. the zero-coupon bond price at time t for maturity T. The continuous-time yield y(t, T)and instantaneous forward rate f (t, T) are then defined by

f (t, T) = − ∂

∂Tln p(t, T), y(t, T) = − ln p(t, T)

T − t,

so

p(t, T) = e−(T−t)y(t,T) = e−∫ T

t f (t,u)du.

Since we do not have a continuum of bond prices available we must use inter-polation methods and it is necessary to use extrapolation methods for timesbeyond the maximal bond maturity that is available in the market.

To do so, we will assume that at the current time t = 0 we know the marketprices mi of certain fixed income instruments which pay given amounts cij atgiven times uj ≥ 0. Here i ∈ I = 1, 2, ..., nI refers to a certain instrumentand j ∈ J = 1, 2, .., nJ to all possible times where payments may occur. Ifasset i does not pay anything at time uj the corresponding value cij is simplyset to zero. Note that we only consider the static problem in this chapter, whichdeals with the term structure at a fixed time t that we take to be time zero

5.2 term structure interpolation 105

without loss of generality. This term structure may then be used as an inputfor dynamic models for interest rates: see for example the books by Cairns(2004) or Brigo and Mercurio (2007) for details on such models. We will alsoassume that prices must be fitted exactly. Alternatively one may use a leastsquares approach, maximum likelihood methods or a Bayesian framework, asin Fisher et al. (1995), Cairns (1998), Sack (2000) and Cairns and Pritchard(2001). We will not require that the interpolating curve for forwards is locallymonotone or convex when the inputs for the forward curve have the sameproperty, see Hagan and West (2006) for a method based on given forwardvalues (instead of fixed income prices) which satisfies this requirement.

An interpolating curve p(0, t) for the observed fixed income instruments mustthus satisfy

(∀i ∈ I) mi = ∑j∈J

cij p(0, uj). (5.2.1)

To decide how the curve is defined in between the time points uj, a specificcriterion is needed since there are many possible ways to do this. Often acriterion is chosen which regulates the smoothness of the interpolating curve.One could for example require the first and second order derivatives of aninterpolating curve g to stay small in a quadratic sense. This means that onetries to minimize

Lα[g] :=∫ ∞

0[g′′(s)2 + α2g′(s)2] ds, (5.2.2)

over all functions g in a given function space. Notice that a parameter α > 0is needed for the trade-off between the possibly conflicting requirements thatboth slope and curvature remain small.

We thus want to define a discount curve p(0, t) which is smooth and is market-consistent in the sense that it matches quoted market prices of certain fixedincome instruments by satisfying (5.2.1). The European insurance regulatoralso requires that forward rates converge to an a priori given value f∞ whenthe maturity goes to infinity. In the formulation of Smith and Wilson (2000)this is implemented by defining, instead of p(0, t) = g(t),

p(0, t) = (1 + g(t))e− f∞t. (5.2.3)

We thus look for a sufficiently smooth function g which minimizes Lα[g] underthe constraint that g(0) = 0 while satisfying

mi = ∑j∈J

cij(1 + g(uj))e− f∞uj , (5.2.4)


for a given nI × nJ matrix C of cashflows and an nI × 1 vector m of marketprices. We will always assume that the rows of C are linearly independentvectors, so there are no superfluous instruments in our set.

The solution to this problem can be written in terms of certain basis functionsW(t, uj) which depend on time t and the maturities uj. We will derive someproperties of the functions W in the next section, before proving that theyprovide a solution to the constrained optimization problem defined above.

5.3 a class of interpolating functions

The functions SW introduced by Smith and Wilson (2000) are scaled versionsof functions W that we will find more convenient to work with: SW(t, u) =e− f∞(t+u)W(t, u) in which

W(t, u) = α min(t, u)− 1/2e−α|t−u| + 1/2e−α(t+u), (5.3.1)

for (t, u) ∈ R+ ×R+ and with α > 0 a given constant. These functions W canbe characterized as follows. The proof (and other proofs in the sequel whichdo not follow directly after a proposition) can be found in Section 5.9.

Proposition 5.3.1. For a given α > 0 consider the ordinary differential equation

∂2t w(t, u) = α2w(t, u)− α3 min(t, u), (5.3.2)

for (t, u) ∈ R+ × R+. For every u > 0 the function W(t, u) is the only twicecontinuously differentiable solution to this equation which is zero for t = 0 and has afinite limit for t→ ∞.

We notice that W(t, u) = W(u, t) and (5.3.2) shows that the function t →W(t, u) is twice differentiable in the point t = u but not three times differ-entiable. The functions W are called exponential tension splines and they wereintroduced in 1966 by Schweikert (1966). Their properties have been shown tohave certain advantages which are useful for modeling yield curves and forother applications, see for example Lim and Xiao (2002) , Andersen (2007) andAndersen and Piterbarg (2010).

In EIOPA documentation about the Smith-Wilson interpolation method, seeEIOPA (2010), it is mentioned that the function W(t, u) is related to the covari-ance function of an integrated Ornstein-Uhlenbeck process. However, a paperby Andersson and Lindholm (2013) shows that W does not exactly correspond

5.3 a class of interpolating functions 107

to that covariance function unless some further restrictive assumptions are im-posed1. In the following propositions, we use their probabilistic interpretationof the functions W, to derive some properties that we will need in later proofs.We also show how functions related to W characterize the best linear predictorfor the yield on a bank account and the discount factor in a Hull-White shortrate model.

Proposition 5.3.2. Let Zt be a standard Brownian Motion and Lt be an Ornstein-Uhlenbeck process Lt =

∫ t0 e−α(t−s)dVs with V a standard Brownian Motion indepen-

dent of Z and α > 0 a given constant. Then W can be characterized as the covariancefunction between two Gaussian processes: W(t, u) = α cov(Zt + Lt, Zu − Lu).

Proposition 5.3.3. If It is an integrated Ornstein-Uhlenbeck process, so It =∫ t

0 Luduwith Lt =

∫ t0 e−α(t−s)dVs, then

E[ It | Iu1 , Iu2 , ...Iun ] = ∑j

ej(t)Iuj ,

where the vector e(t) = H−1b(t) can be defined in terms of the matrix Hij :=H(ui, uj) and the vector bj(t) = H(t, uj) if we choose

H(t, u) = W(t, u)− (1− e−αt)(1− e−αu) ,

provided the matrix H is invertible.

This shows how H can be used to define the best linear predictor for the yield1t ln Bt of a bank account in a Hull-White model2, since in that model ln Btcorresponds to an integrated Ornstein-Uhlenbeck process with mean rever-sion parameter α if we take B0 = 0. The best linear predictor of a stochasticprocess Xt in terms of Xu1 , . . . , Xun by definition equals ∑n

i=1 wiXui if the coef-ficients (w1, .., wn) solve

min(w1,..,wn)

E[(Xt −n

∑j=1

wjXuj)2] . (5.3.3)

From the proposition we find as a direct corollary that

E[ln Bt | ln Bu1 , ln Bu2 , ..., ln Bun ]

= E[ln Bt] +n

∑j=1

(n

∑i=1

(H−1)ijH(t, ui)

)(ln Buj −E[ln Buj ]).

1 See Lagerås and Lindholm (2016) for a related discussion.2 See Hull and White (1990).


The conditional expectation on the left hand side is defined to be the bestpredictor and since it is equal to the projection on the right hand side, whichis linear in ln Buj , the best linear predictor is actually the best predictor in thiscase.

The following corollary shows that the best linear predictor of the stochasticdiscount factor B−1

t also has a representation in terms of the function H andthus in terms of the exponential tension spline W.

Corollary 5.3.4. In a Hull-White model with mean reversion parameter α, the bestlinear predictor of the stochastic discount factor B−1

t in terms of B−1u1

, . . . , B−1un is given

by

n

∑i=1

(n

∑j=1

(H−1)ij H(t, uj)

)B−1

ui, (5.3.4)

provided that the matrix Hij = H(ui, uj), 1 ≤ i, j ≤ n , defined by

ln H(t, u) = E[ln B−1

t

]+E

[ln B−1

u

]+

12

α−3(H(t, t) + H(u, u) + 2H(t, u))

,

is invertible.

Proposition 5.3.2 characterizes W as a covariance between two different pro-cesses. It is possible to characterize it as an (auto)covariance process of a singleGaussian process as well on any finite interval [0, T], as the following resultshows.

Proposition 5.3.5. For a given T > 0 and α > 0 let (zn)n∈N be the countablyinfinite number of solutions to the equation

z3 tan(zαT)− (1 + z2)32 tanh((1 + z2)

32 αT)− 2z2 = 1, (5.3.5)

and define

φn(t) =ψn(t)√∫ T

0 ψ2n(u)du

,

and

ψn(t) = znsin(αtzn)

cos(αTzn)+√

1 + z2n

sinh(αt√

1 + z2n)

cosh(αT√

1 + z2n)

.

5.4 the constrained optimization problem 109

Then the Gaussian process

Xt =1√α

∞

∑n=0

φn(t)εn

zn√

1 + z2n

,

with (εn)n∈N independent standard Gaussian stochastic variables, has the functionW as covariance function on its domain [0, T], so E(XtXu) = W(t, u) for all u andt in [0, T].

This result may seem rather superfluous at first but we can use it to showthat the matrix with entries Wij = W(ui, uj) is invertible for any sequence ofmaturities 0 ≤ u1 < u2 < ... < unJ . This is in fact a result that is tacitlyassumed in the Smith-Wilson interpolation procedure.

To prove this, let a be any vector in Rn. Then aTWa equals the variance of theprocess ∑j ajXuj which is easily seen to be strictly positive when ‖a‖ 6= 0. Thisshows that the matrix W is invertible for any finite value of nJ and any finiteset of distinct values uj with j ∈ J := 1, 2, .., nJ .

5.4 the constrained optimization problem

Now that some elementary properties of the functions W have been estab-lished, we show how these functions can be used to solve the constrainedoptimization problem that was proposed to find a discount curve based oninterpolation and extrapolation of market data.

We define the following subset of the space L2(R+) of square-integrable func-tions on the positive reals:

E = g ∈ L2(R+) : limt→∞

g(t) = 0,

and a subset of the space of two times differentiable functions on the samedomain

Fa = g ∈ C2(R+) : g(0) = a, g′′(0) = 0, g′ ∈ E , g′′ ∈ E,

for any a ∈ R. The space F0 turns out to be the appropriate space to findfunctions g which define a discount curve p(0, t) = (1 + g(t)) exp(− f∞t) thatfits given market data, has an asymptotic3 forward rate f∞, and is as smoothas possible in the sense that it minimizes the weighted integrals over first andsecond order derivatives as defined by Lα[g] in (5.2.2).

3 The European insurance supervisor EIOPA uses the term ultimate forward rate. We call it anasymptotic forward rate because in our case its value will usually not be reached for any finitematurity.


Problem 5.4.1. (Smoothest discount curve for a given asymptotic forwardrate)Find

ming∈F0

Lα[g] (5.4.1)

subject to

∑j∈J

cijg(uj) = mi, for all i ∈ I , (5.4.2)

with cij = cije− f∞uj and mi = mi −∑j∈J cij, to ease the notation of (5.2.4).

The solution to this problem has been given by Smith and Wilson (2000). Weuse a slightly different notation and give a different proof (in Section 5.9) sincethis will allow us to prove certain extensions in later sections of this chapter.

Theorem 5.4.2. If a solution to Problem 5.4.1 exists it must take the form

g(t) = ∑i∈I

ζi ∑j∈J

cijW(t, uj) , (5.4.3)

with (ζi)i∈I a vector of weights and the functions W as defined in (5.3.1).

This result shows that the method proposed by Smith and Wilson has theinterpretation of the best possible interpolation under a criterion based onthe operator Lα, since the functions SW that they propose are simply scaledversions of the exponential tension splines W. We prefer to work with thefunctions W since they can be defined without referring to the asymptoticforward rate f∞, a property that we will need in later sections.

The weights (ζi)i∈I in (5.4.3) can be calculated using

ml = ∑k∈J

clkg(uk) = ∑k∈J

∑j∈J

∑i∈I

clkW(uk, uj)cijζi.

If we define matrices W , C and a vector Z as follows:

Wkj = W(uk, uj), Cij = cij, Zi = ζi ,

then this equation reads m = CWCTZ so we find the parameters using4

Z = (CWCT)−1m,

4 The inverse of CWCT exists by our requirement that the rows of C, and thus of C, are lin-early independent. Indeed, since W is non-singular and symmetric we can write W = LΛLT

for a lower triangular matrix L and a diagonal matrix Λ with strictly positive diagonal ele-ments: Λii > 0. A vector x satisfying CWCTx = 0 would then imply that 0 = xTCWCTx =yTΛy = ∑i y2

i Λii for y = LTCTx 6= 0 which is impossible since both LT and CT have linearlyindependent columns.

5.5 the unconstrained optimization problem 111

and finally, using (5.4.3),

p(0, t) = (1 + ∑j∈J

ηjW(t, uj) )e− f∞t , (5.4.4)

with η = CTZ = CT(CWCT)−1m. Note that for square and invertible matricesC, such as the case where only distinct zero coupon bond prices are given, thisreduces to η = W−1C−1m.

We will show numerical examples for this interpolation method using histor-ical swap date in a later section. But first we formulate a different version ofthe optimization problem, where we no longer impose to which fixed valuethe forward rates must converge.

5.5 the unconstrained optimization problem

The Smith-Wilson procedure defines a method to obtain an extrapolated dis-count curve based on given market data. The asymptotic forward rate f∞ isgiven a priori so it is an input parameter for the methodology. Smoothness,as measured by the functional Lα, is used as a criterion for the interpola-tion between maturities for which data are available. But the requirement ofa given asymptotic forward rate means that the discount curve may becomeless smooth around the last maturity for which liquid market data are avail-able. It would be more consistent to take the asymptotic forward rate as a freeparameter, which is chosen in such a way that we get the smoothest possiblediscount curve for all relevant maturities. Mathematically this means that wepropose to optimize the functional Lα for a given value of α over all possiblefunctions g in a certain class but also over all possible values of f∞. This willnot just give a smoother end result for the discount curve, but also providesan objective estimation method for the asymptotic forward rate in terms ofmarket quotes for bond and swap data.

Problem 5.5.1. (Smoothest discount curve with converging forward rates)Find the minimizer for f∞ in

minf∞

ming∈H( f∞)

Lα[g] (5.5.1)

in which

H( f∞) = g ∈ C2(R+) | g ∈ F0, ∑j∈J

cijg(uj) = mi, for all i ∈ I ,

with the cij and mi as defined in (5.4.2).


It turns out that the asymptotic forward rate can be characterized explicitly asthe solution of a matrix equation. We define the matrices

Wij = W(ui, uj), D fij = e− f uj 1i=j, Uij = uj1i=j ,

and e a vector full of ones in RnJ .

Theorem 5.5.2. For an nI × nJ cashflow matrix C and price vector m ∈ RnI theoptimized asymptotic forward rate f = f∞ of Problem 5.5.1 solves

(m− CD f e)T(CD f W D f CT)−1CD f U

×(

e + W D f CT(CD f W D f CT)−1(m− CD f e))

= 0 .

If the cashflow matrix C is invertible this simplifies to

∑j∈J

∑k∈J

(ujπjef uj) [W−1]jk (πke f uk − 1) = 0 .

with π = C−1m.

This equation can be solved rather easily using numerical algorithms sinceonly the matrix D f contains the parameter to vary over. Calculating the op-timized asymptotic forward rate therefore typically takes only a fraction of asecond.

5.6 an alternative method

The Smith-Wilson interpolation method is a transparent method which is easyto implement since it can be directly applied to bond and swap data. But it hassome important disadvantages. The parameter α, which influences the speedof convergence to the asymptotic forward rate, needs to be defined a priori andfor some values of α the resulting zero coupon bond prices may become nega-tive. It is also possible that the discount function t → p(0, t) is not decreasingfor certain values of t, which corresponds to the possibility of a negative for-ward rate. Moreover, there is the question of smoothness of forward rates. TheHull-White model that we mentioned earlier is a relatively simple and popularMarkovian model for stochastic interest rates, which allows a perfect fit to agiven initial term structure. In that model, one assumes that short rates followa diffusion process driven by a Brownian Motion process with mean reversion

5.6 an alternative method 113

at a certain speed κ > 0 towards a time-varying level θ(t). The function θ canbe used to give a perfect calibration for any initial term structure p(0, t) byusing the corresponding initial forward curve f (0, t) and its derivative withrespect to maturity:

θ(t) =1κ

∂

∂tf (0, t) + f (0, t) +

σ2

2κ2 (1− e−2κt) .

If our initial term structure is modeled using the Smith-Wilson method, thenθ(t) depends on ∂

∂t f (0, t) = − ∂2

∂t2 (ln p(0, t)) which involves a combination ofsecond order derivatives of the functions W(t, uj). As shown in proposition5.3.1, these second order derivative functions are continuous but not differen-tiable which leads to a lack of smoothness in Hull-White calibration results inany of the maturity points uj.

To overcome some of these problems, we now propose two alternative ex-trapolation methods. Where the Smith-Wilson approach takes the first andsecond order derivatives of the (transformed) discount curve as a criterion forsmoothness, we will instead focus on the smoothness of the forward curveor the yield curve itself. More precisely, we will minimize the weighted in-tegrals over first and second order derivatives of the forward or yield curveunder the conditions that forward rates converge and that the market pricesof a set of given fixed income instruments is fitted perfectly. Since p(0, t) =

exp(−∫ t

0 f (0, s)ds) = exp(−ty(0, t)) this will also guarantee that p, and henceall coupon bond prices, will always be strictly positive. If we apply the crite-rion to the yield curve the order of differentiability of the discount curve doesnot change but if we apply the criterion to the forward curve, it will lead to adiscount curve with a higher order of differentiability.

We formalize the new optimization problem as follows.

Problem 5.6.1. (Smoothest forward or yield curve with converging forwardrates)Find

ming∈H f

Lα[g], ming∈Hy

Lα[g], (5.6.1)

in which

H f =

g ∈ C2(R+)

∣∣∣ g ∈ Fa, ∑nJj=1 cij e−

∫ uj0 g(s) ds = mi , for i ∈ I

,

Hy =

g ∈ C2(R+)∣∣∣ g ∈ Fa, ∑

nJj=1 cij e−ujg(uj) = mi , for i ∈ I

.


Figure 5.6.1: Scaled functions −W(t, u)/(αu) and W(t, u) for α = α = 1/10 and−W(−t, u)/(αu) and W(−t, u) for α = α = 1/2. W in red (bottom) andW in blue (top).

Note that g(0) = a for functions g in H f or Hy so the initial short rate r0 :=f (0, 0) equals a given constant a. This implies that we assume that the shortrate is given. If this is not the case, we may determine its value by includingit in the optimization procedure. We will show later in this section how to dothis.

This problem can be solved explicitly in terms of the earlier defined functionsW and a different class of interpolating and extrapolating functions:

W(t, u) = 1 − e−αt cosh(αu)− 11/2α2u2

+ 1t≤u

(cosh(α(u− t))− 1− 1/2α2(u− t)2

1/2α2u2

). (5.6.2)

These functions can be written as affine combinations of integrals of Smith-Wilson functions and are therefore smoother than the Smith-Wilson functionsthemselves, see Figure 5.6.1.

5.6 an alternative method 115

Theorem 5.6.2. If solutions for Problem 5.6.1 exist, they must be of the form

g f (t) = g(0) + ∑i∈I

ζfi ∑

j∈Jπ

fj cij u2

j W(t, uj) ,

gy(t) = g(0) + ∑i∈I

ζyi ∑

j∈Jπ

yj cij ujW(t, uj) ,

(5.6.3)

with the functions W as defined in (5.6.2) and where the (ζ fi )i∈I and (π

fj )j∈J solve,

for all i ∈ I ,

mi = ∑j∈J

cijπfj , (5.6.4)

and, for k ∈ J ,

−(ln πfk )/uk = g(0) + ∑

i∈Iζ

fi ∑

j∈Jπ

fj ciju2

j1uk

∫ uk

0W(s, uj)ds , (5.6.5)

while the (ζyi )i∈I and (π

yj )j∈J should solve, for all i ∈ I ,

mi = ∑j∈J

cijπyj (5.6.6)

and, for k ∈ J ,

−(ln πyk )/uk = g(0) + ∑

i∈Iζ

yi ∑

j∈Jπ

yj cijujW(uk, uj). (5.6.7)

For every u > 0 we have that W(0, u) = 0 and limt→∞ W(t, u) = 1 and thefunctions t → W(t, u) become linear for very small positive values of t inthe sense that ∂2

1W(0, u) = 0 . Moreover, if α and α go to infinity, the scaledinterpolating functions t → W(t, u)/(αu) and t → W(t, u) will converge to1 for t > u. In the limit, the interpolating functions g(t), gy(t) and g f (t) arethus all constant after the last maturity point un which is considered in theinterpolation5.

5 Before EIOPA introduced the ultimate forward rate methodology some central banks, suchas the Dutch Central Bank, used an extrapolation method where the forward rate for the lastliquid maturity was taken to be the value for all forward rates beyond that maturity.


We can use this to derive expressions for the asymptotic forward rate underboth methods. Since limt→∞ W(t, u) = 1 and limt→∞ W(t, u) = αu for all u > 0,this follows from (5.6.3):

f f∞ = lim

t→∞g f (t) = g(0) + ∑

i∈Iζ

fi ∑

j∈Jπ

fj ciju2

j , (5.6.8)

f y∞ = lim

t→∞gy(t) = g(0) + ∑

i∈Iζ

yi ∑

j∈Jπ

yj cijαu2

j . (5.6.9)

We will now show that we can write these two estimated asymptotic forwardrates in terms of a linear combination of yields at different maturities. Let

yk = y(uk) = − ln p(0, uk)/uk,

denote the yield for maturity uk when 1 ≤ k ≤ n. We define u0 := 0 in orderto write y0 = f (0, 0) = y(u0) for the short rate.

Corollary 5.6.3. The optimized asymptotic forward rates can be written as a linearcombination of the yields:

f∞ =n

∑k=0

vkyk.

The weights vk, k = 1 . . . n equal vk = ∑nj=1[G

−1]jk and v0 = 1−∑nk=1 vk, where

the matrices G = G f and G = Gy for the forward and yield curve based methods are:

G fkj =

1uk

∫ uk

0W(s, uj)ds , Gy

kj =1

αujW(uk, uj) .

If the matrix with cashflows Cij is invertible, we can observe the yields directlysince π = C−1m and yk = −(ln πk)/uk. In that case f f

∞ and f y∞ can be calcu-

lated as a linear combination of directly observable quantities in the markets,with weights that are fixed and can be determined beforehand for a given setof maturities (ui)i∈I .

If the short rate is not given a priori it can be chosen as a free parameter inthe optimization problems. An explicit formula for this value of the short ratewhich gives the smoothest curve for small maturities can be derived when thecashflow matrix is invertible.

5.7 estimates of asymptotic forward rates 117

Corollary 5.6.4. If nI = nJ = n, (cij)i∈I , j∈J is an invertible matrix and theinitial short rate is not known, then the previous formula holds if we substitute for theunknown short rate the optimized value

y0 =

n

∑j=1

1uj

n

∑k=1

G−1jk

y(uj) + y(uk)

2n

∑j=1

1uj

n

∑k=1

G−1jk

,

where G equals G f or Gy as defined above.

Figure 5.7.1: Inter- and extrapolation of Datastream swap data up to maturity 20 onMarch 29, 2013. Blue lines show yield curves, red lines forward curves.Dashed lines correspond to the EIOPA forward rate limit of 4.2% whilesolid curves are based on forward rates with a limit that is not specifieda priori.


5.7 estimates of asymptotic forward rates

In this section we present the results of an empirical study of the extrapolationmethods that we introduced earlier. Input data consisted of historical EURI-BOR swap rates on all trading days in the last ten years that were obtainedfrom Datastream. All rates were middle rates and quotes were given with anaccuracy that varied from 0.50 basis point (in the beginning of the dataset) upto 0.01 basis point (for the most recent data). Data for swaps with maturities1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 20, 30, 40 and 50 years were downloaded and wealways used all swap rates up to and including maturity 20 but sometimesincluded maturity 30 as well and sometimes included all rates, as will be indi-cated below.

An example of inter- and extrapolated yield and forward curves is shown inFigure 5.7.1 which shows results for the market data of March 29, 2013. Weused all swap data up to and including maturity 20 years and took α = 1/10,the value used in the original proposal of EIOPA.

The solid lines represent the smoothest extension of the discount curve (top),yield curve (bottom left) and forward curve (bottom right) as measured by ouroperator L, whereas the dashed lines correspond to extrapolation to the ulti-mate forward rate of 4.2% specified by EIOPA. We see that the restriction thatthe forward rate must converge to that asymptotic value causes the forwardcurve to show a kink around maturity 20, i.e. at the last maturity for whichmarket data was included in the construction of the curves. The solid lines inthe three figures show that in all the three methods proposed in this chapter,where such a restriction is no longer imposed, the kink disappears. As a resultwe find asymptotic values for forward rates that are much lower than 4.2%.Moreover, the asymptotic values and the ensuing term structures seem to bevery consistent across the three methods.

Figure 5.7.2 can be used to examine this consistency between the three pro-posed methods, as well as the influence of the maturities that are included toconstruct the curves. It shows monthly updates for the estimated asymptoticforward rate f∞ generated by the methods which smooth the discount curve(top graph), the yield curve (middle graph) and the forward curve (bottomgraph). Estimates are displayed for the cases where instruments are interpo-lated up to 20 years (red line), 30 years (blue line) and 50 years (purple line).The three graphs show that the first two methods provide estimates which aremore stable over time than the last method. Indeed, the third method onlygives good results when the last swap with maturity 50 is included; if thisnot the case then the estimates are too volatile to be of practical use. We also

5.7 estimates of asymptotic forward rates 119

Figure 5.7.2: Inter- and extrapolation based on Datastream swap data up to maturityT = 20, 30, 50 when applying our objective functional L to the discountcurve (top figure), yield curve (middle figure) and forward curve (bottomcurve).


Figure 5.7.3: The same quantities are shown as in the previous figure but now on adaily basis. The forward curve method in the bottom graph gave veryvolatile results after 2009 when the last included maturity was taken tobe 20 or 30, so these results have been omitted.

5.8 conclusions 121

note that before the financial crisis of 2008, asymptotic rates were close to theassumed value of 4.2% but they have dropped substantially since.

After 2012 the first two methods show values which do not differ much whenmarket data beyond maturity 20 are included. Between 2009 and 2012 thereis a more marked effect but the difference between results up to and beyondmaturity 30 are smaller than 25 basis points at all times for the second method,which is based on yield curve smoothening. We conclude that extrapolationmethods which implement an explicit trade-off between first and second orderderivatives of the yield curve give stable estimates when data up to maturity20 are used. But inclusion of the data for maturity 30 seems preferable sincethis does not lead to more volatile estimates, and swap rates of maturities upto 30 or up to 50 years give roughly the same estimates.

The effects discussed above are even more pronounced in Figure 5.7.3, inwhich the same estimates of the asymptotic forward rate are plotted but nowfor a daily update frequency. For the last method only results for maturity 50are shown since the estimates are too volatile when earlier maturities are takenas the last liquid point. This suggests that extrapolation of yields is possiblebeyond maturity 20 but that market-implied forward rates are not smoothenough to allow extrapolation if information from the far end of the curve isnot used. We notice again that if extrapolation is applied to the discount curveor the yield curve then the choice of the last used maturity only mildly affectsthe asymptotic estimates that are found after 2011. This supports the use ofsuch extrapolation methods to estimate long-term yields.

5.8 conclusions

We have investigated inter- and extrapolation techniques that can be used tocreate discount curves from observed market data under different assump-tions on asymptotic forward rates. We have shown that Smith-Wilson extrap-olation, which requires the a priori specification of the asymptotic forwardvalue, can be extended in such a way that the value of this limit is implied bymarket data. Moreover, we show how the smoothness criterion of the originalproblem can also be applied to the forward or yield curve instead of the dis-count curve. This leads to a market-consistent asymptotic forward rate whichcan be written as a weighted combination of yields at earlier maturities, withweights that can be calculated beforehand. This provides an intuitive char-acterization of asymptotic forward rates in terms of well understood marketinformation.


On days where reliable market information is available for high maturitiesthese quotes can be easily incorporated into the extrapolation method, whileat times of reduced liquidity for the highest maturities the same method canbe used with a restricted set of maturities. We find that since 2012 the use ofmarket data up to maturities 20, 30 or 50 gives similar results when extrapo-lation is based on yield curves. Before 2012 it seems best to use all availablematurities up to 50 since the inclusion of later maturities does not lead to anincrease in the volatility of the estimates. This suggests that the market infor-mation at higher maturities is reliable enough to be of use in the estimationprocess.

In this chapter we always assumed that market prices must be fitted exactly.If one allows some mispricing of market instruments in return for smoothercurves, a different optimization problem must be solved. One could keep thesame optimization functional Lα, but one would have to impose, for example,that the weighted sum of pricing errors does not exceed a certain thresholdvalue. Taking that value equal to zero will give back our old solutions butby varying it we can implement a tradeoff between smoothness and pricingaccuracy. Such problems have been studied for the case where there are noconstraints on the asymptotic behavior of forward rates and when forwardrates are observed directly (see for example Andersen and Piterbarg (2010)).Finding the solution under our additional requirements seems a challengingopen problem.

5.9 proofs

Proof of Proposition 5.3.1.Assume that W(t, u) solves this differential equation and satisfies the condi-tions stated for a certain u > 0. We consider the differential equations fort > u and t < u separately. In both cases we have a linear equation which iseasily seen to have αu and αt as possible solutions respectively while the homo-geneous equation is solved by linear combinations of eαt and e−αt. Therefore

t < u : W(t, u) = αt + c1eαt + c2e−αt,

t > u : W(t, u) = αu + c3eαt + c4e−αt.

A finite limit for t → ∞ is only possible if c3 = 0, and the solution being zeroat t = 0 implies that c1 = −c2 so we can actually reduce the solution to

t < u : W(t, u) = αt + c1(eαt − e−αt),

t > u : W(t, u) = αu + c4e−αt.

5.9 proofs 123

Matching value and derivative at t = u gives c1eαu = (c1 + c4)e−αu and α(1 +c1eαu + c1e−αu) = −αc4e−αu. This gives c1 = 1/2e−αu and c4 = 1/2(eαu − e−αu)which proves the result.

Proof of Proposition 5.3.2.By independence α cov(Zt + Lt, Zu − Lu) = α cov(Zt, Zu)− α cov(Lt, Lu) so theresult follows immediately from

cov(Lt, Lu) =∫ min(t,u)

0e−α(t−s)e−α(u−s)ds = 1/2αe−α(t+u)(e2α min(t,u) − 1)

= 1/2αe−α(max(t,u)−min(t,u)) − 1/2αe−α(t+u)

= 1/2αe−α|t−u| − 1/2αe−α(t+u).

Proof of Proposition 5.3.3.Since I is Gaussian with mean zero, the conditional expectation E[ It | Iu1 , ..., Iun ]equals ∑j ej(t)Iuj with e(t) = Σ−1b(t) for Σij = EIui Iuj and bj(t) = EIt Iuj . But

since dL = dV − αLdt = dV − αdI, we have It = α−1(Vt − Lt) = α−1∫ t

0 (1−e−α(t−s))dVs so we find

EIt Iu = α−2∫ min(t,u)

0(1− e−α(t−s))(1− e−α(u−s))ds

= α−2[ min(t, u) + cov(Lt, Lu)− (e−αt + e−αu)∫ min(t,u)

0eαsds ]

= α−3[α min(t, u) + 1/2e−α|t−u| − 1/2e−α(t+u)

+ (e−αt + e−αu)(1− eα min(t,u))]

= α−3[α min(t, u)− 1/2e−α|t−u| + 1/2e−α(t+u)

+ (−e−α(t+u) + e−αt + e−αu − 1) ]

= α−3(W(t, u)− (1− e−αt)(1− e−αu)).

This gives the result by defining Hij = α3Σij.

Proof of Corollary 5.3.4.Differentiating Eq. (5.3.3) with Xt = B−1

t with respect to wi we obtain, for1 ≤ i ≤ n,

0 = −E[2(B−1t −

n

∑j=1

wjB−1uj

)B−1ui

] .


Rearranging terms yieldsE[B−1

u1B−1

u1

]. . . E

[B−1

u1B−1

un

]...

...

E[B−1

un B−1u1

]. . . E

[B−1

un B−1un

]

w1...

wn

=

E[

B−1t B−1

u1

]...

E[

B−1t B−1

un

] . (5.9.1)

The random variables ln Bui = Iui , 1 ≤ i ≤ n, have a multivariate Gaussianlaw with a moment generating function that can be expressed in terms ofits mean and covariance matrix. Using the results obtained in the proof ofproposition 5.3.3 we find that, for all 1 ≤ i, j ≤ n,

lnE[

B−1ui

B−1uj

]= lnE

[e− ln Bui−ln Buj

]= lnE

[e−Iui−Iuj

]= −E [Iui ]−E

[Iuj

]+

12

Var(Iui) +12

Var(Iuj) + Cov(Iui , Iuj)

= −E [Iui ]−E[

Iuj

]+

12α3

(H(ui, ui) + H(uj, uj) + 2H(ui, uj)

).

Hence Eq. (5.9.1) can be written as Hw = H(t, u) and from the assumed in-vertibility of H we conclude that wi = ∑j∈J (H−1)ijH(t, uj).

Proof of Proposition 5.3.5.We first solve the eigenvalue problem λφ(t) =

∫ T0 W(t, u)φ(u)du by writing

λφ(t) =∫ t

0φ(u)[αu− 1/2e−α(t−u) + 1/2e−α(t+u)]du

+∫ T

tφ(u)[αt− 1/2eα(t−u) + 1/2e−α(t+u)]du

=∫ t

0φ(u)[αu− e−αt sinh(αu)]du +

∫ T

tφ(u)[αt− e−αu sinh(αt)]du,

so in particular φ(0) = 0 and

λφ(T) =∫ T

0φ(u)[αu− e−αT sinh(αu)]du .

Differentiating gives

λφ′(t) = α∫ t

0φ(u)e−αt sinh(αu)du + α

∫ T

tφ(u)[ 1− e−αu cosh(αt) ]du,

5.9 proofs 125

and in particular

λφ′(T) = αe−αT∫ T

0φ(u) sinh(αu)du .

Differentiating twice gives

λφ′′(t) = αφ(t)e−αt sinh(αt)− αφ(t)(1− e−αt cosh(αt))

−α2∫ t

0φ(u)e−αt sinh(αu)du

+α2∫ T

tφ(u)[ −e−αu sinh(αt) ]du

= −α2∫ t

0φ(u)e−αt sinh(αu)du + α2

∫ T

tφ(u)[ −e−αu sinh(αt) ]du

= α2( λφ(t)− α∫ t

0uφ(u)du− αt

∫ T

tφ(u)du )

which shows that φ′′(0) = α2λφ(0) = 0 and

λφ′′(T) = α2( λφ(T)− α∫ T

0uφ(u)du )

= −α2∫ T

0φ(u)e−αT sinh(αu)du = −αλφ′(T),

so φ′′(T) = −αφ′(T). Differentiating once more we find

λφ′′′(t)− λα2φ′(t) = −α3∫ T

tφ(u)du,

λφ′′′′(t)− λα2φ′′(t) = α3φ(t),

and φ′′′(T) = α2φ′(T). Substitution of φ(t) = exp(mt) in the last equationgives λ(m2)2 − λα2m2 − α3 = 0 so m2 = 1/2α(α±

√α2 + 4α/λ). Define z2 =

1/2(√

1 + 4/(αλ)− 1) then m2 = −z2α2 or m2 = (1 + z2)α2 and αz2(1 + z2) =1/λ. Solutions must therefore take the form

φ(t) = c1 sin(tzα) + c2 cos(tzα) + c3 sinh(tα√

1 + z2) + c4 cosh(tα√

1 + z2) .

Since φ(0) = φ′′(0) = 0 we must have c2 = c4 = 0. The boundary conditionφ′′′(T)− α2φ′(T) = 0 shows that

0 = c1(−z3α3 − α2αz) cos(Tzα)

+ c3((1 + z2)3/2α3 − α2α

√1 + z2) cosh(T

√1 + z2α),

0 = −c1z(z2 + 1) cos(Tzα) + c3z2√

1 + z2 cosh(T√

1 + z2α),

0 = −c1

√1 + z2 cos(Tzα) + c3z cosh(T

√1 + z2α) .


This shows that φ(t) must be of the form

φ(t)/c = zsin(tzα)

cos(Tzα)+√

1 + z2 sinh(tα√

1 + z2)

cosh(Tα√

1 + z2),

for some c which is not zero. The last boundary condition φ′′(T) = −αφ′(T)then gives the equation for the zn. We see that there are countably infinitemany solutions since for all k ∈N there is at least one solution on the interval](k− 1/2)π/(αT), (k + 1/2)π/(αT)[. This is because the left hand side of (5.3.5)is continuous in z on that interval and goes to −∞ for z ↓ (k− 1/2)π/(αT) andto +∞ for z ↑ (k + 1/2)π/(αT). The φn are orthogonal on [0, T] since

λn

∫ T

0φn(s)φm(s)ds =

∫ T

0

∫ T

0W(s, u)φn(u)duφm(s)ds

= λm

∫ T

0φn(u)φm(u)du,

and λm 6= λn. This makes φn(t) an orthonormal set of functions and ourdefinition of X then gives

E(XtXu) =1α

∞

∑n=0

φn(t)φn(u)z2

n(1 + z2n)

=∞

∑n=0

λnφn(t)φn(u),

so for all m ∈N we find∫ T

0[W(t, u)−E(XtXu)]φm(u)du = λmφm(t)− λmφm(t) = 0.

Since the (φm)m∈N form an orthonormal basis of L2[0, T] by Mercer’s theorem,this proves the result.

Proof of Theorem 5.4.2.We introduce Lagrange multipliers α3ζi to get the unconstrained optimizationproblem

Lα[g] =∫ ∞

0[g′′(s)2 + α2g′(s)2]ds + α3 ∑

i∈Iζi(mi − ∑

j∈Jcijg(uj)).

Local minima must satisfy for every h in F0:

0 = 1/2∂

∂εLα[g + εh]

∣∣∣ε=0

=∫ ∞

0[g′′(s)h′′(s) + α2g′(s)h′(s)]ds − 1/2α3 ∑

i∈I∑j∈J

ζi cijh(uj)

=∫ ∞

0h′(s) [ −g′′′(s) + α2g′(s)− 1/2α3 ∑

i∈Iζi ∑

j∈Jcij1s≤uj ] ds ,

5.9 proofs 127

where we have performed a partial integration on the function h′(t)g′′(t) andused that h, g ∈ F0 which implies that h(0) = 0 and that g′′(t) vanishes fort = 0 and t → ∞. This must hold for all h ∈ F0 so we can thus conclude thatfor all s > 0

−g′′′(s) + α2g′(s)− 1/2α3 ∑i∈I

ζi ∑j∈J

cij1s≤uj = 0. (5.9.2)

This ODE has the solution (for real constants ai )

g′(t) = a1eαt + a2e−αt + 1/4α2∫ t

0(eα(s−t) − e−α(s−t)) ∑

j∈Jej1s≤uj ds,

where we have lightened notation by writing ej = ∑i∈I ζi cij. The last termequals

1/4α2 ∑j∈J

ej

∫ min(t,uj)

0(eα(s−t) − e−α(s−t))ds

= 1/4α ∑j∈J

ej(eα(min(t,uj)−t) + e−α(min(t,uj)−t))

= 1/2α ∑j∈J

ej(1t<uj + cosh(α(t− uj))1t>uj) .

Integrating g′(t) and using that g(0) = 0 thus shows that the optimal solutionmust have the form

g(t) = a3 sinh(αt) + a4(1− e−αt)

+ 1/2 ∑j∈J

ej(α min(t, uj) + sinh(α(t− uj))1t>uj) ,

for certain real constants a3 and a4. For t → ∞ the value of g(t) will in-volve a term eαt times (1/2a3 + 1/4 ∑j∈J eje

−αuj) so we must choose a3 =

−1/2 ∑j∈J eje−αuj to make this term vanish. We then get

g(t) = a4(1− e−αt) + ∑j∈J

ej(α min(t, uj) + 1/2eα(t−uj)(1t>uj − 1)

− 1/2e−α(t−uj)1t>uj + 1/2e−α(t+uj))

= a4(1− e−αt) + ∑j∈J

ej(α min(t, uj)− 1/2e−α|t−uj| + 1/2e−α(t+uj))

= a4(1− e−αt) + ∑j∈J

∑i∈I

ζi cijW(t, uj) .


From proposition 5.3.1 it follows that g′′(0) = −α2a4. Due to the requirementg′′(0) = 0 we thus have a4 = 0, the result follows.

Proof of Theorem 5.5.2.From equation (5.4.4) and the remarks below it we know that g(t) = ∑j ηjW(t, uj)

with η = CT(CWCT)−1m. We now remark that for any g ∈ F0 ⊇ H( f∞) wehave that

Lα[g] =∫ ∞

0(g′′(s)2 + α2g′(s)2)ds

=[g′(s)g′′(s)

]s=∞s=0 −

∫ ∞

0g′(s)g′′′(s)ds + α2

∫ ∞

0g′(s)2ds

=∫ ∞

0g′(s) ( −g′′′(s) + α2g′(s) )ds. (5.9.3)

But optimality conditions for g in the proof of Theorem 5.4.2, see (5.9.2), implythat −g′′′(s)2 + α2g′(s) = 1/2α3 ∑j ηj1s≤uj and g(0) = 0 so

Lα[g] = 1/2α3 ∑j

ηj

∫ uj

0g′(s)ds = 1/2α3 ∑

jηj(g(uj)− g(0))

= 1/2α3 ∑j

ηj ∑k

ηkWjk = 1/2α3 · mT(CWCT)−1CWCT(CWCT)−1m

= 1/2α3 · mT(CWCT)−1m.

We denote CWCT by A to lighten notation. Setting the derivative with respectto the asymptotic forward rate f equal to zero gives

0 =∂

∂ f(mTA−1m) = mT(2A−1 ∂m

∂ f−A−1 ∂A

∂ fA−1m ).

Since ∂m∂ f = ∂

∂ f (m− CD f e) = −CD f Ue = −CUe and

∂A∂ f

=∂

∂ f(CD f W D f CT) = 2CD f UW(D f )TCT = 2CUWCT,

we must have that

0 = −2mTA−1CU(e + WCTA−1m ) ,

which gives the first result after substitution of C = CD f and m = m− CD f eto write everything in terms of the original cashflow matrix C and price vectorm. If C is invertible, we can write π = C−1m and reduce the equation to theform (D− f π − e)TW−1U(D− f π) = 0 which gives the second result.

5.9 proofs 129

Proof of Theorem 5.6.2.Suppose that g f = g is the solution to Problem 5.6.1 for the forward curve,then g must be a local optimum of the following functional with Lagrangianparameters ζi for the constraints:

Lα[g] = 1/2

∫ ∞

0g′′(s)2 + α2(g′(s))2ds+ 2α2 ∑

i∈Iζi

(−mi + ∑

j∈Jcije−

∫ uj0 g(s)ds

),

under the additional requirements g′′(0) = limt→∞ g′′(t) = 0 and for a giveng(0) but free limit limt→∞ g(t). This implies that for every h satisfying h′′(0) =limt→∞

h′′(t) = h(0) = 0, the perturbed solution g := g + εh must satisfy 0 =

( ddε Lα[g + εh])ε=0. Define πj = exp(−

∫ uj0 g(s)ds), then

0 =∫ ∞

0

[g′′(s)h′′(s) + α2 g′(s)h′(s)

]ds− 2α2 ∑

i∈I∑j∈J

ζi cij πj

∫ uj

0h(v)dv

= [h′(s)g′′(s)]s=∞s=0 +

∫ ∞

0h′(s)

[−g′′′(s) + α2 g′(s)

]ds

−2α2 ∑i∈I

∑j∈J

ζi cij πj

( [h(t) t

]uj0 −

∫ s=uj

s=0h′(s) s ds

)=

∫ ∞

0h′(s)

[−g′′′(s) + α2 g′(s)− 2α2 ∑

i∈Iζi ∑

j∈Jcij πj (uj − s)1s≤uj

]ds .

In the last step we used that ujh(uj) = uj∫ uj

0 h′(s)ds since h(0) = 0. Variationover h gives

−g′′′(s) + α2 g′(s) = 2α2 ∑j∈J

e fj (uj − s)1s≤uj (5.9.4)

where e fj = πj ∑i ζicij. The solution to this differential equation can be written

in terms of the functions W(·, u) which are three times differentiable in allpoints and satisfy for all u > 0

∂21W(0, u) = lim

t→∞∂2

1W(t, u) = 0, limt→∞

W(t, u) = 1, W(0, u) = 0,

while

−∂31W(t, u) + α2∂1W(t, u) = 2α2/u2(u− t)1t≤u, (5.9.5)

as may be shown by direct differentiation. The solution to the ODE (5.9.4)must thus be of the form

g(s) = a0 + a1e−αt + a2eαt + ∑j∈J

ηfj W(s, uj),


with ηfj = u2

j e fj = u2

j πj ∑i ζicij and for certain constants a0, a1 and a2 in R.The constraint that g′′(0) = lim

t→∞g′′(t) = 0 gives a1 = a2 = 0 and we have

a0 = g(0). Substituting the expression for g in the definition of (πj)j∈J thengives the result (5.6.4)-(5.6.5) for the forward curve.

For the problem of finding the best yield curve g = gy, we want to minimize

Lα[g] = 1/2

∫ ∞

0[g′′(s)2 + α2g′(s)2]ds− α3 ∑

i∈Iζi(mi − ∑

j∈Jcije−ujg(uj)),

under the constraints g′′(0) = limt→∞ g′′(t) = 0 and for a given g(0) butunspecified limit limt→∞ g(t). We set 0 = ( d

dε Lα[g + εh])ε=0 with h′′(0) =limt→∞ h′′(t) = h(0) = 0:

0 =∫ ∞

0[g′′(s)h′′(s) + α2g′(s)h′(s)]ds− α3 ∑

iζi ∑

jujcije

−ujg(uj)h(uj)

=∫ ∞

0h′(s) [ −g′′′(s) + α2g′(s)− α3 ∑

iζi ∑

jujcije

−ujg(uj)1s≤uj ] ds,

so

0 = −g′′′(s) + α2g′(s)− α3 ∑j

eyj uj1s≤uj ,

with eyj = πj ∑i ζicij and πj = e−ujg(uj). We know from our reasoning after

equation (5.9.2) that the solution of this ODE, for a given value of g(0) andunder the constraints that g′′(0) = limt→∞ g′′(t) = 0, equals

g(s) = g(0) + ∑j

ηyj W(s, uj) = g(0) + ∑

jujπj ∑

iζicijW(s, uj) ,

with ηyj = uje

yj . Therefore, if we can find (πj)j∈J and (ζi)i∈I such that

−(ln πk)/uk = g(0) + ∑i

∑j

ζicijujπjW(uk, uj)

mi = ∑j

cijπj,

then all optimality conditions are fulfilled.

5.9 proofs 131

Proof of Corollary 5.6.3.Define as before η

fj = ∑i ζ

fi cijπ

fj u2

j and ηyj = ∑i ζ

yi cijπ

yj uj. Then we have by

(5.6.4)-(5.6.9) that

yk = g(0) + ∑j

ηfj G f

kj f f∞ = g(0) + ∑

jη

fj

yk = g(0) + ∑j

ηyj Wy

kj f y∞ = g(0) + ∑

jη

yj (αuj)

= g(0) + ∑j(η

yj αuj)G

ykj

so

f f∞ = g(0) + ∑

j∑k[G f ]−1

jk (yk − g(0)) ,

and

f y∞ = g(0) + ∑

j∑k[Gy]−1

jk (yk − g(0)) ,

which gives the result after rearranging terms.

Proof of Corollary 5.6.4.We first note that for any g in H f or Hy and any β ∈ R we have by (5.9.3) that

Lβ[g] =∫ ∞

0g′(s)(−g′′′(s) + β2g′(s))ds. (5.9.6)

For g(s) = g f (s) we found g(s) = g(0) + ∑j ηfj W(s, uj) and in the proof of

Corollary 5.6.3 we established ηfj = ∑k[G f ]−1

jk (yk − g(0)) so because of (5.9.4)and (5.9.5) this gives

Lα[g] = 2α2 ∑j

ηfj u−2

j

∫ uj

0g′(s)(uj − s)ds

= 2α2 ∑j

ηfj u−2

j

∫ uj

0(g(s)− g(0))ds

= 2α2 ∑j

ηfj u−1

j (yj − g(0))

= 2α2 ∑j

u−1j ∑

k[G f ]−1

jk (yk − g(0))(yj − g(0)) .


Setting the derivative with respect to the free parameter g(0) equal to zeroand rewriting then establishes the result for the forward curve optimization.

For the result of the yield curve optimization we have g(s) = gy(s) = g(0) +∑j η

yj W(s, uj) and η

yj = 1

αuj∑k[Gy]−1

jk (yk − g(0)). Then (5.9.6) gives

Lα[g] =∫ ∞

0g′(s)α3 ∑

jη

yj 1s≤uj ds

= α3 ∑j

ηyj (g(uj)− g(0)) = α3 ∑

jη

yj (yj − g(0))

= α3 ∑j

1αuj

∑k[Gy]−1

jk (yk − g(0))(yj − g(0)),

so we find the same result but with a different matrix G.

S A M E N VAT T I N G ( S U M M A RY I N D U T C H )

Levensverzekeraars en pensioenfondsen bieden financiële producten aan waar-van de waarde afhangt van sterfte- en renteontwikkelingen op de lange termijn.Deze lange-termijn risico’s vormen het onderwerp van dit proefschrift. Indeel I wordt het effect van langlevenrisico op investeringsbeslissingen vanrationele individuen bestudeerd. Deel II behandelt modellen die kunnen bij-dragen aan een betere beschrijving van de dynamiek van de lange-termijnrente.

Hoofdstuk 2

In hoofdstuk 2 wordt de optimale consumptie en investeringsstrategie vaneen rationele investeerder gekarakteriseerd in een markt waarin onverwachteveranderingen in sterfte-risico verhandelbaar zijn middels een survival bond.We tonen aan dat, onder geschikte voorwaarden, een unieke optimale strategiebestaat wanneer de rente en de sterfte-intensiteit gemodelleerd worden doormiddel van een Cox-Ingersoll-Ross proces. Dit proces is niet-negatief en be-schrijft een stochastische fluctuatie rondom een deterministisch tijdsafhankelijkgemiddelde. Semi-analytische uitdrukkingen voor de optimale consumptie-en investeringsstrategie worden afgeleid. Condities die de existentie van eendergelijke optimale strategie garanderen, worden uitgedrukt in termen van deparameters van het model.

Hoofdstuk 3

In hoofdstuk 3 behandelen we de convergentie van de forward rente in Heath-Jarrow-Morton (HJM) modellen, wanneer de expiratiedatum naar oneindiggaat. Ten eerste vergelijken we bestaande resultaten uit de literatuur met be-trekking tot bijna zekere convergentie en convergentie in kans naar een vastelange-termijn rente. Ten tweede formuleren we nodige en voldoende voor-waarden voor de HJM model parameters die garanderen dat het forward renteproces als semimartingaal uniform convergeert in gemiddelde.

133


Hoofdstuk 4

In hoofdstuk 4 bestuderen we de implicaties van de Dybvig-Ingersoll-Rossstelling voor factor modellen voor de termijnstructuur. Deze stelling houdtin dat de lange-termijn rente niet kan dalen als functie van de tijd. Dit laatechter onverlet dat de lange-termijn rente een stochastisch verloop kan heb-ben. Reeds in El Karoui et al. (1998) is opgemerkt dat het eenvoudig is omeen arbitrage-vrij HJM model te formuleren waarbij de lange-termijn rentestijgt en stochastisch is. Hoofdstuk 4 beantwoordt een open vraag in dit arti-kel, namelijk of een dergelijk model ook bestaat binnen de klasse van factormodellen voor de termijnstructuur. De modellen in deze klasse hebben eenduidelijke interpretatie, omdat ze te schrijven zijn als een inproduct van eenvector-waardige stochastische factor met een set parametrisaties van de ren-tetermijnstructuur. We laten zien dat de constructie van een dergelijk modelmogelijk is, mits het model ten minste drie toestandsvariabelen heeft, en wegeven een expliciet voorbeeld.

Hoofdstuk 5

Prijzen van obligaties en swaps zijn in de markt beschikbaar voor een beperktaantal looptijden. Omdat verplichtingen van verzekeraars en pensioenfondsentypisch verbonden zijn aan de levens van polishouders, strekken de looptijdenvan deze verplichtingen zich veelal uit voorbij die van de langstlopende liqui-de instrumenten in de markt. Bij het waarderen van zulke verplichtingen is ex-trapolatie daarom onvermijdelijk. In hoofdstuk 5 wordt een nieuwe extrapola-tiemethode voorgesteld die gebaseerd is op een interpretatie van Smith-Wilsonextrapolatie als de oplossing van een variationeel optimalisatie probleem. Integenstelling tot de Smith-Wilson procedure, waarin de lange-termijn rente(de ultimate forward rate) bekend wordt verondersteld, maken we het niveauvan de lange-termijn rente onderdeel van het optimalisatieprobleem. Het re-sultaat is een extrapolatiemethode die de meest gladde voortzetting geeft vande in de markt geobserveerde termijnstructuur, waarbij de limiet van de geëx-trapoleerde curve een schatting geeft voor de ultimate forward rate.

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