introduction to convex optimization in nancial marketsmath.ut.ee/emsdk/kevadkool2013/tartu.pdf ·...

Introduction to convex optimization in financial

markets

Teemu Pennanen1

May 16, 2013

1Department of Mathematics, King’s College London, [email protected]

Contents

1 Single-period models 91.1 Asset-liability management . . . . . . . . . . . . . . . . . . . . . 91.2 Reserving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Pricing of contingent claims . . . . . . . . . . . . . . . . . . . . . 181.4 Transaction costs and illiquidity . . . . . . . . . . . . . . . . . . . 20

2 Dynamic models 252.1 Market model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Asset-liability management . . . . . . . . . . . . . . . . . . . . . 282.3 Reserving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4 Swap rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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4 CONTENTS

Preface

Financial risks can be managed by trading in financial markets. By appropriatetrading, individuals and financial institutions may modify their net cash-flowsto better conform to their risk preferences. For example, a home owner may beable to achieve a more attractive risk profile for his future cash-flows by buyinga home insurance. Insurers, on the other hand, invest insurance premiums in fi-nancial markets in order to optimize their net cash-flow structure resulting fromdelivering insurance claims and collecting investment returns. The same prin-ciple is behind the classical Black–Scholes–Merton option pricing framework,where the seller of an option invests the premium in financial markets accordingto an investment strategy whose return matches the option payout.

In complete markets, prices of contingent claims are uniquely determinedby replication arguments but in incomplete markets, subjective factors suchas market expectations and risk preferences come in. This is pronounced ininsurance contracts, credit derivatives, weather derivatives, energy contract,etc. whose cash-flows are often largely unrelated to liquidly traded securities.Moreover, in incomplete markets, there are two distinct notions often referredto as the “price”:

1. The least amount of cash that would allow for covering a claim at anacceptable level of risk.

2. The least amount of cash one could sell a claim for without worseningone’s risk-return profile.

The first notion corresponds to the original formulation of Black and Scholes [4]who studied claims whose payoffs could be replicated exactly by an appropri-ate investment strategy. It is also used in various accounting standards andsupervisory frameworks for defining “book values” of financial liabilities. Inbanking, it is sometimes called the “economic capital” while in insurance, theterms “technical provisions” and “reserving” are often used; see e.g. Article 76of the Solvency II Directive 2009/138/EC of the European Parliament. Sucha book value is not necessarily a price at which an agent would be willing tosell or buy a claim. Offered prices are better described by the second notionwhich describes the indifference principle. For example, a seller of a financialproduct charges (at least) a price that allows him to sell the product withoutworsening the risk-return profile of his existing financial position. Indifference

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6 CONTENTS

pricing has been widely applied in various areas of finance and insurance; seee.g. Buhlmann [5], Hodges and Neuberger [21] or Bielecki et al. [3] for a smallsample. We refer the reader to Carmona [6] for further references on the topic.

These notes give an introduction to asset-liability management, reservingand indifference pricing in terms of basic optimization theory. Our aim is togive a unified treatment of the above concepts and to study their relationsand basic properties with minimal mathematical sophistication. Mathematicaltechniques are introduced only when they become necessary for the develop-ment of the theory. Working with discrete-time models allows us to avoid manyof the technicalities associated with continuous time models. This leaves roomfor practical considerations that are sometimes neglected in more mathemati-cal treatments. In particular, we extend the classical theory of mathematicalfinance by allowing for nonlinear illiquidity effects and portfolio constraints.Such features are significant in practice but they invalidate much of the classi-cal theory. We deviate from the classical theory also in that we do not insiston the existence of a numeraire. This means that one can no longer postponepayments by shorting the numeraire so the payment schedule of a claim be-comes an important issue. This is essential in practice where much of tradingconsists of exchanging sequences of cash-flows. Examples include various swapand insurance contracts where both claims and premiums involve payments atseveral points in time.

Traditionally, the main tools in mathematical finance have come from stochas-tics but convex analysis is turning out to be equally useful. Convexity is oftenindispensable in mathematical and numerical analysis of financial optimizationproblems. For example, general characterizations of the no-arbitrage propertyof a perfectly liquid market model in terms of martingale measures is largelybased on separation theorems for convex sets; see Follmer and Schied [17] andDelbaen and Schachermayer [12] for comprehensive study of the classical lin-ear model of financial markets. Techniques of convex analysis allow also forextending the classical theory of mathematical finance to more realistic modelswith portfolio constraints and illiquidity effects; see for example Dermody andRockafellar [13, 14], Cvitanic and Karatzas [8, 9], Jouini and Kallal [24, 23],Kabanov [25], Follmer and Schied [17, Chapter 9], Evstigneev, Schurger andTaksar [16], Schachermayer [44], Cetin and Rogers [7], Kabanov and Safar-ian [26] and Pennanen [34, 35]. Moreover, the optimization perspective bringsin variational and computational techniques that have been successful in moretraditional fields of applied mathematics such as partial differential equationsand operations research.

These notes take a variational approach to financial economics and risk man-agement. Basic principles in contingent claim valuation are derived by purelyalgebraic arguments using elementary convex analysis. The first part of thenotes introduces the main concepts in a simple one-period model. The secondpart extends the theory to a multiperiod setting with nonlinear illiquidity ef-fects. This builds on the market model introduced in Pennanen [34], whereconvex analysis and the theory of normal integrands were taken as the maintools of analysis. The model provides a flexible framework for modeling port-

CONTENTS 7

folio constraints and transaction costs as well as certain illiquidity effects thatarise in modern securities markets. The third part develops the duality theorythat is behind e.g. martingale representations of contingent claim prices in theclassical linear model. Again, generalizations are given to market models withconstraints and illiquidity effects.

8 CONTENTS

Chapter 1

Single-period models

Consider a financial market with a finite set J of traded assets. In this section,we consider a simplified setting where assets are traded only at two dates, thepresent time t = 0 and some time t = 1 in the future. Moreover, we assume thatthe market is perfectly liquid so that the unit prices of the traded assets do notdepend on our actions. The unit price of asset j ∈ J at time t will be denotedby sjt . We assume that the vector s0 = (sj0)j∈J of current prices is known to

us before we trade at time t = 0. The price vector s1 = (sj1)j∈J will remainuncertain until time t = 1. We model s1 as a random vector on a probabilityspace (Ω,F , P ). That is, each sj1 is a real-valued F-measurable function on Ω.The linear space of equivalence classes of real-valued random variables will bedenoted by L0 := L0(Ω,F , P ). Note that L0(Ω,F , Q) = L0(Ω,F , P ) for anymeasure Q that is equivalent to P .

Buying a portfolio x = (xj)j∈J of assets at time t = 0 costs s0 · x unitsof cash. If we hold on to the portfolio, it will be worth s1 · x at time t = 1.Clearly, s1 ·x is random. Here xj denotes the number of units of asset j ∈ J wehold. It requires investing hj = sj0x

j units of cash in asset j at time t = 0. Thevalue of our portfolio at time t = 1 can be expressed r · h, where r = (rj)j∈J is

the random vector with components rj = sj1/sj0. While in practice, it is more

common to describe investments in terms of their “market values”, formulationsin terms of units are sometimes more convenient in mathematical analysis.

1.1 Asset-liability management

Consider the problem

minimize V(c− s1 · x) over x ∈ RJ ,subject to s0 · x ≤ w, x ∈ D,

(ALM)

where w ∈ R is a given initial wealth, D ⊆ RJ describes portfolio constraints,c ∈ L0 is a random amount of cash (a claim) the agent must deliver at time

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10 CHAPTER 1. SINGLE-PERIOD MODELS

t = 1 and the function V : L0 → R measures the “risk/disutility/regret” fromthe random net expenditure c− s1 ·x at time t = 1. Throughout these notes, Rdenotes the extended real line R ∪ +∞,−∞.

Problem (ALM) may be viewed as an asset-liability management problemwhere one looks for a portfolio x ∈ D whose proceeds fit the liabilities c optimallyas measured by V. We allow c to take arbitrary real values so it may describecosts as well as income. The constraint x ∈ D may be used to model varioustrading restrictions that may be present in a given situation. Short sellingconstraints correspond to D = RJ+; see Exercise 3 for more examples.

Example 1.1 (Utility maximization) The classical utility maximization prob-lem corresponds to c = 0 and

V(d) := E[−u(−d)] ∀d ∈ L0,

where E denotes the expectation and u is a concave nondecreasing function onR. Throughout these notes, we define the expectation of an extended real-valuedrandom variable as +∞ unless its positive part has finite expectation. Theexpectation is then well-defined for any random variable. This convention ofdefining extended real-valued integrals is not arbitrary but specifically suited forstudying minimization problems.

Example 1.2 (Markovitz model) When c = 0 and V(c) = Ec+λσ2(c) for apositive scalar λ, we recover the classical Markowitz model. The mean-variancecriterion is, however, irrational in that it penalizes both the upside as well as thedownside, i.e. it is not nondecreasing. Allowing for throwing away of money,corresponds to

V(c) = infd∈L0Ed+ λσ2(d) | d ≥ c

which is nondecreasing.

Example 1.3 (Risk measures) A function R : L∞ → R is said to be a con-vex risk measure if

1. R is convex

2. R is nondecreasing: R(c1) ≤ R(c2) whenever c1 ≤ c2 almost surely

3. R(c+ α) = R(c) + α for every constant α ∈ R.

A risk measure associates a monetary value to a random claim. Receiving R(c)units of cash at time t = 1 compensates for delivering c in the sense that the riskof the net cash-flow c−R(c) is zero (by the third property). Given a convex riskmeasure, one can extend it to a monotone convex function on L0 by defining

V(c) = infd∈L∞

R(d) | c ≤ d;

see Exercise 8.

1.1. ASSET-LIABILITY MANAGEMENT 11

Problem (ALM) depends on the agent’s

1. views as described by the probability measure P ,

2. risk preferences as described the function V,

3. financial position as described by (w, c).

All these factors are subjective. Agents with differences in views, preferences orfinancial positions may have very different investment strategies.

We will assume throughout that D is a convex set containing the origin andthat V is nondecreasing, normalized and convex. That V is nondecreasing meansthat V(c1) ≤ V(c2) whenever c1 ≤ c2 almost surely, i.e. one always prefers morecash. Normalization means that V(0) = 0. It is posed mainly for notationalconvenience. As long as 0 ∈ domV, normalization can be achieved by adding aconstant to the objective1. Convexity corresponds to the fundamental diversifi-cation principle risk management; the risk associated with a convex combinationof two claims c1 and c2 should be no higher than the convex combination of theindividual risks.

The convexity of D and V imply that (ALM) is a convex optimization prob-lem, i.e. a problem of minimizing a convex function over a convex set. Indeed,the objective is the composition of V with the linear function x 7→ s1 · x fromRJ to L0 so it is convex as a function of x; see Exercise 1.

In pricing and hedging of contingent claims as well as in reserving for finan-cial liabilities one is concerned with how the quality of the risk profile dependson the financial position (w, c) ∈ R× L0. We will denote the optimum value of(ALM) by

ϕ(w, c) := infV(c− s1 · x) |x ∈ D, s0 · x ≤ w.The following simple fact turns out to have important consequences in financialrisk management.

Lemma 1.4 The value function ϕ of (ALM) is convex and ϕ(0, 0) ≤ 0. It isnonincreasing in w and nondecreasing in c.

Proof. Exercise.

The greatest among all convex nondecreasing normalized functions on L0 is2

V = δL0−, (1.1)

where L0− := c ∈ L0(Ω,F , P ) | c ≤ 0 P -almost surely. This corresponds to a

completely risk averse agent who deems even the smallest possibility of a lossunacceptable. In the most risk averse case,

ϕ(w, c) = δD(w, c),

1Given an extended real-valued function f , the set dom f = x | f(x) < +∞ is called thedomain of f

2Here and in what follows δC denotes the indicator function of a set C. That is, δC(x) = 0if x ∈ C and δC(x) = +∞ otherwise.


where

D = (w, c) ∈ R× L0 | ∃x ∈ D : s0 · x ≤ w, c ≤ s1 · x P -a.s.. (1.2)

We have (w, c) ∈ D if and only if an investment of w at time t = 0 allows us toguarantee payment of c at time t = 1 without any risk of ending with negativewealth. Note that the value function can be written in terms of D as

ϕ(w, c) = infdV(c− d) | (w, d) ∈ D.

Exercises

An extended real-valued function f on a linear space X is convex if its epigraphepi f := (x, α) | f(x) ≤ α is convex in X × R. It is not hard to check that afunction f is convex iff

f(α1x1 + α2x2) ≤ α1f(x1) + α2f(x2)

whenever x1, x2 ∈ dom f and α1, α2 > 0 are such that α1 + α2 = 1. A functionf is concave if −f is convex. The lower level sets levα f := x | f(x) ≤ α of aconvex function are convex sets but the converse does not hold in general. Anextended real-valued function is said to be proper if it never takes the value −∞and it is not identically equal to +∞.

A set K ⊆ X is said to be a cone if αx ∈ K for every x ∈ K and α ≥ 0.Given a set C ⊆ X, the cone

C∞ = y |x+ αy ∈ C ∀x ∈ C ∀α > 0

is called the recession cone of C. The recession cone gives the directions inwhich C is “unbounded”.

1. Let X and U be linear spaces and show that

(a) a set C ⊂ X is convex if and only if δC is convex function.

(b) if f1, f2 are convex functions on X, then f(x) = f1(x) + f2(x) is aconvex function on X. Here and in what follows, we define ∞ +(−∞) = (−∞) +∞ =∞.

(c) if h is a convex function on U and A : X → U is linear, then f(x) =h(Ax) is a convex function on X.

(d) if fii∈I are convex functions on X, then f(x) = supi∈I fi(x) is aconvex function on X.

(e) if f is a convex function on X × U , then

ϕ(u) = infx∈X

f(x, u)

is convex on U .


2. Write problem (ALM) in terms of the decision variables h ∈ RJ whereh = (hj)j∈J gives the amounts hj of money invested in asset class j ∈ J .Is the resulting problem convex?

3. Specify the constraint set D in order to model the following constraintson the portfolio.

(a) At most 50% of initial wealth is invested in a given asset k ∈ J .

(b) The variance of the portfolio value at time t = 1 is at most σ asviewed by a supervisor who believes that investment returns followa multivariate normal distribution with expectation m ∈ RJ andvariance Σ.

(c) The δ% Value at Risk of the portfolio return is less than q as viewedby the same supervisor as in part (b) above.

4. Prove Lemma 1.4.

5. Show that

(a) the recession cone of a convex set is a convex cone.

(b) if C is a convex cone, then C∞ = C.

6. Show that if R is a convex risk measure in the sense of Example 1.3, thenA = lev0R is a convex set with L∞− ⊂ A∞ and we have

R(c) = infα ∈ R | c− α ∈ A.

Conversely, if A ⊂ L∞ is any convex set with L∞− ⊂ A∞, then R(c) =infα ∈ R | c− α ∈ A is a convex risk measure.

7. Assume that R : L∞ → R satisfies properties 1 and 2 of Example 1.3 butinstead of 3 it only satisfies R(α) = α for every α ∈ R. Show that R is arisk measure.

8. If R is a convex risk measure, then the function

V(c) = infd∈L∞

R(d) | d ≥ c

is a convex nondecreasing function on L0.

9. Let v : R× Ω→ R be an B(R)⊗F-measurable function such that v(·, ω)is nondecreasing and convex for every ω ∈ Ω. Show that

V(c) = Ev(c) :=

∫Ω

v(c(ω), ω)dP (ω)

is a nondecreasing convex function on L0. Again, the integral is definedas +∞ unless the positive part of the integrand is integrable.


Further reading

Portfolio optimization has had a central role in mathematical finance ever sincethe pioneering work of Harry Markovitz [31]. Sharpe et al. [45] gives an excellentnon-technical overview of optimal investment and financial markets. Much ofthe theory has been developed in a continuous-time setting, most famously byMerton [32]; see Rogers [43] for a recent account.

Classical references on the analysis of convex functions and sets include thelecture notes of Moreau [33] and the book of Rockafellar [38]. Although [38] isconcerned with finite-dimensional spaces, many of the facts found there extendto more general vector spaces; see e.g. [39, 15].

Convex nondecreasing functions V used in the formulation of the asset-liability management problem (ALM) can be seen as extensions of risk mea-sures introduced in Artzner, Delbaen, Eber and Heath [1]; see e.g. Follmer andSchied [17] or Rockafellar [41] for general discussions on risk measures and de-scription of risk preferences. Unlike a risk measure, as defined in the abovereferences, the values of V need not represent monetary values of uncertaincash-flows. Such values will be studied in Sections 1.2 and 1.3 below.

1.2 Reserving

In financial reporting and supervision of financial institutions, one is often inter-ested in determining the least amount of capital that would suffice for “covering”a financial liability. In the single-period model of Section 1.1, where financialliabilities are described by random cash-flows c ∈ L0(Ω,F , P ) at time t = 1,such an amount can be defined as

π0(c) = infw ∈ R |ϕ(w, c) ≤ 0.

Indeed, π0(c) gives the least amount of initial capital w needed to construct ahedging strategy x ∈ D whose value at time t = 1 covers a sufficient part of theliability c in the sense that the residual risk V(c − s1 · c) is no higher than therisk of doing nothing at all (recall that V(0) = 0). We will call π0(c) the reservefor c ∈M.

If the optimum in (ALM) is attained for every w and c, we have

π0(c) = infw | ∃x ∈ D, s0 · x ≤ w, V(c− s1 · x) ≤ 0

so that π0(c) equals the optimum value of the convex optimization problem

minimize s0 · x over w ∈ R, x ∈ D,subject to V(c− s1 · x) ≤ 0.

(1.3)

If we ignore the existence of financial markets and assume that all wealth (bothpositive and negative amounts) is invested “under the mattress”, we get

π0(c) = infw | V(c− w) ≤ 0 (1.4)

1.2. RESERVING 15

which is a risk measure in the sense of Example 1.3 (exercise). This corre-sponds to classical premium principles in actuarial mathematics. By an ap-propriate choice of a trading strategy, one may be able to lower the reserve.The same principle is behind the famous Black–Scholes–Merton option pricingmodel where the price of an option is defined as the least amount of initial cap-ital that allows for the construction of a trading strategy whose terminal valueequals the payout of the option. In practice, however, exact replication is oftenimpossible so one has to define preferences concerning the uncertain amount ofwealth remaining after the delivery of the claim.

In order to clarify the connections between reserving and risk-neutral valu-ation (as e.g. in the Black–Scholes–Merton model), consider a completely riskaverse agent whose risk preferences are described by (1.1). In this case, π0(c)becomes the superhedging cost

πsup(c) := infw | ∃x ∈ D : s0 · x ≤ w, c ≤ s1 · x P -a.s..

While πsup(c) gives the least amount of initial capital needed for riskless hedgingof c, the subhedging cost

πinf(c) := supw | ∃x ∈ D : s0 · x+ w ≤ 0, c+ s1 · x ≥ 0 P -a.s.

gives the greatest amount of initial debt one could risklessly cover when receivingc units of cash at time t = 1. Clearly, πinf(c) = −πsup(−c). The super- andsubhedging costs depend on the probability measure P only through its nullsets. Indeed, πsup(c) does not change if P is replaced by any measure equivalentto P . The super- and subhedging costs are often called “arbitrage bounds”for the price of c. If it was possible to sell the claim c for more than πsup(c),one could make riskless profit by selling the claim and following a superhedgingstrategy. Similar opportunity would exist if we could buy the claim for less thanπinf(c).

The reserve π0(c) defines an extended real-valued function on L0. The fol-lowing theorem summarizes some of its properties. We will say that a claimc ∈ L0 is replicable if there is an x ∈ D such that −x ∈ D and s1 · x = c almostsurely.

Theorem 1.5 The function π0 is convex, nondecreasing and π0(0) ≤ 0. Wealways have π0(c) ≤ πsup(c). If π0(0) = 0, then

πinf(c) ≤ π0(c) ≤ πsup(c)

with equalities throughout for replicable c.

Proof. The first claim is left as an exercise. The function π0 is nondecreasingsimply because V is. Since 0 ∈ D and V(0) = 0, the zero portfolio is feasiblewhen c = 0 and thus, π0(0) ≤ 0. Since δL0

−is the greatest among all nonde-

creasing convex functions V on L0 with V(0) = 0, we have ϕ ≤ δD, where D is


the set defined in (1.2). Thus,

π0(c) = infw |ϕ(w, c) ≤ 0≤ infw | δD(w, c) ≤ 0= infw |x ∈ D, s0 · x ≤ w, c ≤ s1 · x = πsup(c).

Since πinf(c) = −πsup(−c), we also have πinf(c) ≤ −π0(−c). By convexity of π0,

π0(0) = π0(1

2c+

1

2(−c)) ≤ 1

2π0(c) +

1

2π0(−c)

so if π0(0) = 0 we get −π0(−c) ≤ π0(c), which completes the proof of the“sandwich” inequalities. Assume now that c is replicable with s1 · x = c almostsurely for some x ∈ D with −x ∈ D. Such an x superhedges c while −xsubhedges c. Thus πsup(c) ≤ s0 · x ≤ πinf(c), which completes the proof.

The function π0 behaves much like a risk measure in Example 1.3. However,while convexity and monotonicity of V imply the convexity and monotonicity ofπ0, the same does not hold for the translation property. While a risk measuregives the amount of cash required at time t = 1, the reserve gives the requiredamount of cash at time t = 0. However, as long as long the market model allowsfor long positions in cash, we have π0(c+α) ≤ π0(c)+α for all α ≥ 0 (exercise).The full translation property holds if arbitrary long and short positions in cashare allowed. While this is a standard assumption in financial mathematics (seeExample 2.5 below), it rarely holds in practice.

The condition π0(0) = 0 means that zero liabilities require no reserves.It holds, in particular, if D ⊆ RJ+ and s0 ∈ RJ+ (exercise). In general, π0

depends on the the probability measure P and the function V (views and riskpreferences) but the last part of Theorem 1.5 says that for replicable claims,the π0 is independent of such subjective factors (since the superhedging cost issuch).

We have assumed implicitly that one can solve problem (ALM) exactly.This is rarely the case in practice, so one has to resort to numerical techniquesthat can only generate approximate solutions and optimum values. When (1.3)cannot be solved exactly, the reserve should be defined as the least value afinancial institution can achieve in (1.3). It then depends also on the tradingexpertise of a financial institution. Institutions that are good at hedging theirliabilities can operate at lower capital.

Exercises

1. Prove the first claim of Theorem 1.5.

2. (a) Show that the function R(c) = infw | V(c−w) ≤ 0 is a convex riskmeasure in the sense of Example 1.3.

(b) Let V(c) := E[eγc − 1] and show that

R(c) =1

γlnEeγc.

1.2. RESERVING 17

This function is known as the entropic risk measure.

(c) Show that if c1 and c2 are independent random variables, then theentropic risk measure satisfies

R(c1 + c2) = R(c1) +R(c2).

3. Show that the condition π0(0) = 0 in Theorem 1.5 holds if

(a) D ⊆ RJ+ and s0 ∈ RJ+.

(b) domV ⊆ L0− and the market satisfies the weak no-arbitrage condition:

there is no x ∈ D such that

s0 · x < 0 and s1 · x ≥ 0 P -a.s..

4. It is often reasonable to assume that one can invest arbitrary positiveamounts of wealth in cash (money under mattress). This can be modeledby setting one of the components, say 0 ∈ J , of the price vectors st equalto one and by allowing arbitrary positive positions in this special asset.That is, we could set st = (1, st) and D = (x0, x) |x0 ≥ 0, x ∈ D, wherest denotes the prices of the “risky” assets J = J \ 0.

(a) Show that if the market model allows for long positions in cash, then

π0(c+ α) ≤ π0(c) + α ∀α ≥ 0.

Hint: prove first that ϕ(w, c+α) ≤ ϕ(w−α, c) for any c ∈ L0, w ∈ Rand α ≥ 0.

(b) Show that if the market model allows for arbitrary positions in cash,i.e. D = (x0, x) |x0 ∈ R, x ∈ D, then

π0(c+ α) = π0(c) + α ∀α ≥ 0.

Further reading

Our formulation of the reserve π0 corresponds to the usual definition of an “op-tion price” as the least amount of initial cash needed for hedging the option.The most famous example is the classical Black–Scholes–Merton model whereessentially all claims are replicable (“attainable” in the terminology of [17]) byappropriate trading; see [4]. When replication is not possible, the terminalwealth is necessarily uncertain and one has to specify what kind of terminalpositions are acceptable. The completely risk averse attitude leads to super-hedging. Comprehensive treatments of superhedging in incomplete markets canbe found in [17, 12, 26]. More reasonable risk preferences are studied in a liquidmultiperiod model in Follmer and Schied [17, Chapter 8]. Illiquid generaliza-tions can be found e.g. in Davis, Panas and Zariphopoulou [11] and Section 2.3below. Expressions of the form (1.4) are often used in actuarial mathematics aspremium principles for insurance claims; see e.g. Buhlmann [5].


Our formulation of (1.3) is also motivated by modern financial supervisorystandards, such as the Solvency II Directive 2009/138/EC of the European Par-liament, which promotes market consistent accounting principles that recognizethe risks in both assets and liabilities. The interplay between reserves and assetmanagement was recently studied in Artzner, Delbaen and Koch-Medona [2].An implementation of a dynamic version of (1.3) (see Section 2.3 below) forpension insurance liabilities is described in Hilli, Koivu and Pennanen [20].

1.3 Pricing of contingent claims

Consider now the problem of valuing a contingent claim c ∈ L0(Ω,F , P ) fromthe point of view of an agent whose current financial position is given by aninitial wealth w ∈ R and liabilities c ∈ L0(Ω,F , P ). This time we are notlooking for the amount of capital that allows the agent only to survive with agiven liability but a price at which he could sell a claim c without worsening therisk-return profile associated with his current financial position. It is intuitivelyclear that such a price should depend on the agent’s financial position. Thefinancial position matters also when buying a claim. For instance, the value ofa European option to an agent probably depends on whether the agent ownsthe underlying stock or not. Similarly, a wheat farmer is more likely to buya futures contract on wheat than somebody who’s income does not depend onwheat price. In fact, most financial instruments exist because of differences infinancial positions of different agents. Appropriately designed financial contractsallow all counterparties to reduce their financial risks.

The least cash-payment at time t = 0 that would compensate an agent fordelivering a claim c at time t = 1 can be expressed in the model of Section 1.1as

π(w, c; c) = infw |ϕ(w + w, c+ c) ≤ ϕ(w, c), (1.5)

where again, ϕ(w, c) denotes the optimum value of (ALM). Any price w >π(w, c; c) would allow the agent to optimize his portfolio after the trade so thatthe risk with his new financial position (w + w, c+ c) is no greater than it wasbefore the trade. The value given by (1.5) is often referred to as an indifferenceprice of c. While π gives the least price an agent would be willing to sell a claimfor, the buying indifference price for c is given by

πb(w, c; c) = supw |ϕ(w − w, c− c) ≤ ϕ(w, c).

Clearly, πb(w, c; c) = −π(w, c;−c).When there are no constraints, i.e. when D = RJ , we can bound the indif-

ference prices between the super- and the subhedging costs much as we did forreserving in Section 1.2.

Theorem 1.6 The function π(w, c; ·) is convex, nondecreasing and π(w, c; 0) ≤0. If there are no portfolio constraints, then π(w, c; c) ≤ πsup(c). If in addition,π(w, c; 0) = 0, then

πinf(c) ≤ πb(w, c; c) ≤ π(w, c; c) ≤ πsup(c)

1.3. PRICING OF CONTINGENT CLAIMS 19

with equalities throughout when c is replicable.

Proof. The first claim is left as an exercise. For any w > πsup(c), there is anx′ ∈ RJ such that s0 · x′ ≤ w and c ≤ s1 · x′ almost surely. When there are noconstraints, the change of variables x→ x− x′ in (ALM) gives

ϕ(w, c) = ϕ(w + s0 · x′, c+ s1 · x′).

The growth properties of ϕ in Lemma 1.4 then give

ϕ(w, c) ≥ ϕ(w + w, c+ c)

and thus, π(w, c; c) ≤ w. Since w > πsup(c) was arbitrary, we must haveπ(w, c; c) ≤ πsup(c).

By convexity of π(w, c; ·),

π(w, c; 0) ≤ 1

2π(w, c; c) +

1

2π(w, c;−c)

so if π(w, c; 0) = 0, we get −π(w, c;−c) ≤ π(w, c; c). The “sandwich” inequal-ities now come from the identities πb(w, c; c) = −π(w, c;−c) and πinf(c) =−πsup(−c). The last claim follows from the last claim of Theorem 1.5.

The condition π(w, c; 0) = 0 means that one cannot lower the initial wealthwithout affecting the optimum value of (ALM). It holds, in particular, if ϕ(w, c)is strictly decreasing in the initial endowment w. In general, selling and buyingprices depend on an agent’s financial position (w, c), future views as describedby P and risk preferences V, all of which are subjective factors. The inequalityπb(c; c) ≤ π(c; c) just means that two agents with identical characteristics haveno incentive to trade with each other. By the second part of Theorem 1.6, pricesof replicable claims are given by the superhedging cost πsup which is independentof such subjective factors. Moreover, the convexity of πsup and the concavity ofπinf imply that, on the space of replicable claims, prices are linear in c.

Combining Theorems 1.5 and 1.6, we see that in unconstrained market mod-els, reserves and indifference prices of replicable claims coincide if π0(0) = 0 andπ(w, c; 0) = 0. Thus, in complete market models (models where all claims canbe replicated) such as the classical Black–Scholes model, there is no need todistinguish between reserves and selling or buying prices of contingent claims.In practice, however, reserves maybe very be different from offered buying andselling prices.

The pricing principle (1.5) assumes that one can solve (ALM) to optimality.This assumption rarely holds in practice. However, (1.5) still makes sense if oneredefines ϕ as the least value one can achieve in (ALM). Besides the financialposition (w, c), future views P and risk preferences V, offered prices thus dependon the agents’ trading expertise in optimizing their portfolios.


Exercises

1. Let X be a linear space and let C be a convex subset of X×R. Show that

f(x) = infα | (x, α) ∈ C

is a convex function on X.


Further reading

Indifference pricing goes back, at least, to Hodges and Neuberger [21] who stud-ied the pricing of contingent claims that cannot be exactly replicated due totransaction costs; see Carmona [6] for further references. Indifference pricing is apopular approach for pricing contingent claims that cannot be exactly replicatedby trading in financial markets. Examples include various insurance contractsand credit derivatives; see Buhlmann [5] and Bielecki et al. [3], respectively.We will extend indifference pricing to general swap contracts in a multiperiodsetting in Section 2.4 below.

1.4 Transaction costs and illiquidity

The market model considered so far describes liquid markets where unit pricesof securities do not depend on whether one is buying or selling nor on thequantity of the traded amount. In real securities markets based on auctionmechanisms, however, the unit price for buying is usually strictly higher thanthe unit price for selling. The best available price for buying is known as the ask-price and the best available price for selling is known as the the bid-price. Theirdifference is called the bid-ask spread. Proportional transaction costs can alsobe described in terms of the bid-ask spread. Indeed, transaction costs definedas a fixed proportion δ of the traded amount effectively increase the ask-priceby a multiple of (1 + δ) and decrease the bid-price by a multiple of (1 − δ).Moreover, the numbers of securities that are available at the bid- and ask-pricesare limited. As traded quantities increase, the prices move against us. This isoften referred to a illiquidity.

Many securities are traded in limit order markets, where market participantssubmit buying or selling offers defined by limits on quantity and unit price.When buying, the quantity available at the lowest submitted selling price isfinite. When buying more, one gets the second lowest price and so on. Themarginal price c(x) of buying is thus a piecewise constant nondecreasing functionof the number x of units bought. Thus, the total cost

C(x) =

∫ x

0

c(z)dz

of buying x units is a piecewise linear convex function on R+; see Exercise 1.Analogously, the marginal price for selling securities is a nonincreasing piecewise

1.4. TRANSACTION COSTS AND ILLIQUIDITY 21

constant function r of the number of units sold. Thus, the total revenue R(x)of selling x units is a piecewise linear concave function on R+.

Given the cost and revenue functions C and R, the market is “cleared” bysolving the convex optimization problem

minimize C(x)−R(x) over x ∈ R+.

Optimal solutions are characterized by the market clearing condition

∂C(x) + ∂[−R](x) 3 0,

where the set-valued mappings ∂C : R ⇒ R and ∂[−R] : R ⇒ R are obtainedfrom the marginal price functions by closing the gaps in the graphs of c and−r by vertical lines. In the terminology of convex analysis, ∂C and ∂[−R]are the subdifferentials of the convex functions C and −R, respectively; seeExercise 2. When multiple solutions exist, the greatest solution x is implementedby matching x units of the most generous selling and buying offers. The pricesof the remaining selling offers are then all strictly higher than the prices of theremaining buying offers and no more trades are possible before new offers arrive.

The above describes what happens in an opening auction that takes placeat the beginning of a trading day before continuous trading begins. The offersremaining after market clearing are recorded in the so called limit order book.It gives the marginal prices for buying or selling a given quantity at the bestavailable prices. Interpreting negative purchases as sales, the marginal pricescan be incorporated into a single function x 7→ s(x) giving the marginal pricefor buying positive or negative quantities of the asset. Figure 1.4 presents anexample of a marginal price curve s taken from Copenhagen stock exchange.Since the highest buying price is lower than the lowest selling price, the marginalprice curve s is a nondecreasing piecewise constant function on R. Consequently,

S(x) =

∫ x

0

s(z)dz

defines a piecewise linear convex function on R; see Exercise 1. It gives the costof buying x shares. Again, negative x is interpreted as sales and a negativecost as revenue. Note that the perfectly liquid market model studied earliercorresponds to marginal prices s(x) being independent of the traded quantityx in which case the cost function S is linear. Sections 1.1–1.3 could be readilyextended to allow for illiquidity effects. We will do this in the next section in adynamic setting.

What we have explained above, applies to so called market orders. In amarket order, a trader buys a desired number x of securities at best availableprices. Besides market orders, market participants may also submit limit orderswhich do not lead to an immediate transaction if the offered price does notmatch available prices in the book. Such orders are incorporated into the limitorder book. While market orders reduce liquidity by truncating the limit orderbook, limit orders increase liquidity.


236

237

238

239

240

241

242

-100000 -50000 0 50000

QUANTITY

PR

ICE

Figure 1.1: Marginal price curve for shares of the Danish telecom companyTDC A/S observed in Copenhagen Stock Exchange on January 12, 2005 at13:58:19.43. The horizontal axis gives the cumulative depth of the book mea-sured in the number of shares. Negative order quantity corresponds to a sale.The prices are in Danish krone. The data was provided by OMX market re-search.

Exercises

1. Let φ : R→ R be nondecreasing and a ∈ R such that φ(a) is finite. Thenthe function

f(x) =

∫ x

a

φ(z)dz

is convex.

2. Given an extended real-valued function f on RJ , a vector v ∈ RJ is saidto be a subgradient of f at x ∈ RJ if

f(x) ≥ f(x) + v · (x− x) ∀x ∈ RJ .

The set of subgradients of f at x is known as the subdifferential of f at xand it is denoted by ∂f(x). Let f be a convex function on R. Show that∂f is a nondecreasing curve.

3. Let f be as in Exercise 1 above and show that φ(x) ∈ ∂f(x) wheneverx ∈ R is such that φ(x) ∈ R. Thus, since f does not change if we changeφ at countably many points, we may conclude that both left and rightlimits of φ (which exist since φ is nondecreasing) are subgradients of f .

More on subdifferentials: Being the intersection of closed half-spaces, ∂f(x)is a closed convex set. It can be shown that if f is differentiable at x, then ∂f(x)

1.4. TRANSACTION COSTS AND ILLIQUIDITY 23

reduces to a single point, the gradient of f at x. Clearly, an x minimizes f ifand only if 0 ∈ ∂f(x). This generalizes the classical Fermat’s rule which saysthat the gradient of a differentiable function vanishes at a minimizing point.

Further reading

A general, nontechnical account of various trading protocols used in securitiesmarkets can be found in Harris [18]. Limit order markets have attracted a lotof attention in mathematical finance in recent years. The multivariate char-acteristics of limit order books provide many challenges in statistical modelingof financial markets. A simple parametric approach is presented in Malo andPennanen [30].

Another generalization of the classical linear model of financial markets wasproposed in Kabanov [25]; see also Kabanov and Safarian [26]. In Kabanov’smodel, all assets are treated symmetrically, much like in currency markets, andbudget constraints are described in terms of “solvency cones”, which can beinterpreted as the negatives of the sets of portfolios that are freely available inthe market. In the model of limit order markets, such a set can be expressedas x ∈ RJ |S(x) ≤ 0. In Kabanov’s original model the sets were polyhedralcones. Extensions to more general convex sets were studied in Pennanen andPenner [37].

An extensive treatment of differential properties of convex functions onfinite-dimensional spaces can be found in Rockafellar [38, Part V].

Chapter 2

Dynamic models

Consider now a dynamic setting where uncertainty is still modeled by a proba-bility space (Ω,F , P ) but now one may trade at multiple points t = 0, . . . , T intime. The information available at time t is described by a σ-algebra Ft ⊆ F inthe sense that, at time t, we do not know which scenario ω will eventually realizebut only which element of Ft it belongs to. We will assume that the sequence(Ft)Tt=0 is a filtration, i.e. Ft ⊆ Ft+1. This just means that the informationincreases over time.

A portfolio xt chosen at time t may depend on the information observed sofar but not on information that will only be observed in the future. This meansthat xt is an Ft-measurable function from Ω to RJ , or in other words, the tradingstrategy x = (xt)

Tt=0 is adapted1 to the filtration (Ft)Tt=0. The linear space of

adapted trading strategies will be denoted by N . Unless FT has only a finitenumber of elements with positive probability, N is an infinite-dimensional space.We will assume that F0 = Ω, ∅ so that x0 is independent of ω. Implementinga portfolio process x ∈ N requires buying a portfolio ∆xt := xt − xt−1 at timet. Again, negative purchases are interpreted as sales.

2.1 Market model

The cost of buying a portfolio of assets depends on the financial market. We willdescribe the financial market by an (Ft)Tt=0-adapted sequence (St)

Tt=0 of random

lower semicontinuous2 (lsc) convex functions on RJ such that St(0) = 0 almostsurely for t = 0, . . . , T . More precisely, we will assume that St is a real-valuedB(RJ)⊗Ft-measurable function on RJ ×Ω such that St(·, ω) are lsc and convexand vanish at the origin. Such a sequence is called a convex cost process.

1Some authors describe trading strategies by “predictable” processes ξ = (ξt)Tt=0 with ξtdenoting the portfolio that was chosen at time t− 1. This is only a notational difference withξt = xt−1.

2A function f : RJ → R is lower semicontinuous if the lower level set x ∈ RJ | f(x) ≤ αis closed for every α ∈ R.

25

26 CHAPTER 2. DYNAMIC MODELS

The value of St(x, ω) is interpreted as the cost we would have to pay for aportfolio x ∈ RJ at time t in state ω; see Section 1.4. Implementing a tradingstrategy x ∈ N requires investing St(∆xt) units of cash at time t. The measur-ability condition implies that ω 7→ St(∆xt(ω), ω) is Ft-measurable for each t,i.e. the cost of a portfolio is known at the time of purchase. Indeed, the cost isthe composition of the measurable functions ω 7→ (∆xt(ω), ω) and St.

Example 2.1 If st is an RJ -valued Ft-measurable price vector for each t =0, . . . , T , then the functions

St(x, ω) = st(ω) · x

define a linear cost process. This corresponds to a frictionless market, whereunit prices are independent of purchases.

Example 2.2 If st and st are RJ -valued Ft-measurable price vectors with st ≤st, then the functions

St(x, ω) =∑j∈J

Sjt (xj , ω),

where

Sjt (xj , ω) =

sjt (ω)xj if xj ≥ 0,

sjt (ω)xj if xj ≤ 0

define a sublinear (i.e. convex and positively homogeneous) cost process. Thiscorresponds to a market with transaction costs or bid-ask spreads, where unlim-ited amounts of all assets can be bought or sold for prices st and st, respectively.When s = s, we recover Example 2.1.

Example 2.3 If Zt is an Ft-measurable set-valued mapping3 from Ω to RJ ,then the functions

St(x, ω) = sups∈Zt(ω)

s · x

define a sublinear cost process. When Zt = [st, st] we recover Example 2.2.

Proof. Sublinearity is left as an exercise. By [42, Example 14.51], St(x, ω) isan Ft-measurable normal integrand4. By [42, Corollary 14.34], this implies thatSt is B(RJ)⊗Ft-measurable.

We will model portfolio constraints by requiring that the portfolio xt(ω)chosen at time t and state ω belongs to a given set Dt(ω). We will assumethat ω 7→ Dt(ω) is Ft-measurable closed convex-valued and that 0 ∈ Dt almostsurely. The measurability condition simply means that the set Dt of feasible

3A set-valued mapping ω 7→ C(ω) is Ft-measurable if ω ∈ Ω |C(ω) ∩ U 6= ∅ ∈ Ft forevery open set U .

4A function f : RJ × Ω → R is an Ft-measurable normal integrand if f(·, ω) is lowersemicontinuous for every ω ∈ Ω and if ω 7→ epi(·, ω) is an Ft-measurable set-valued mapping.

2.1. MARKET MODEL 27

portfolios is known to us at time t when we choose xt. The condition 0 ∈ Dt

means that we are always allowed not to hold any of the traded assets.

Besides short selling constraints which correspond to Dt(ω) ≡ RJ+, the port-folio constraints can be used to model situations where one encounters differentinterest rates for lending and borrowing; see Exercise 4.

Exercises

1. Show that the functions St(·, ω) in Example 2.3 are sublinear.

2. Show that

(a) a function f is sublinear if and only if

f(α1x1 + α2x2) ≤ α1f(x1) + α2f(x2)

for every xi ∈ dom f and αi > 0.

(b) a positively homogeneous function f is convex if and only if

f(x1 + x2) ≤ f(x1) + f(x2)

for every xi ∈ dom f .

3. The mark-to-market value of a portfolio x ∈ RJ is given by s · x, wheres ∈ RJ is a vector of market prices. At time t, market prices are given byan Ft-measurable vector st with st(ω) ∈ ∂St(0, ω), i.e.

St(z, ω) ≥ St(0, ω) + st(ω) · z ∀z ∈ RJ .

Assume that st are componentwise strictly positive and express the tradingcost St(∆xt) in terms of the mark-to-market values hjt = sjtx

jt of the

investments xt.

4. Specify a market model (S,D) that describes a situation where the inter-est rates for lending and borrowing unlimited amounts of cash over eachperiod (t, t+ 1] are given by r+

t and r−t , respectively.

Further reading

Classical references on financial market models with a proportional transactioncosts include Hodges and Neuberger [21], Jouini and Kallal [24] and Cvitanicand Karatzas [9]. Models with nonlinear cost functions have been studied inCetin and Rogers [7] and Pennanen [34]. For a mathematical study of generalmeasurable set-valued mappings and normal integrands, we refer the reader toRockafellar [40] and Rockafellar and Wets [42, Chapter 14].


2.2 Asset-liability management

Consider an agent whose financial position is described by an (Ft)Tt=0-adaptedsequence c = (ct)

Tt=0 of cash-flows in the sense that the agent has to deliver

a random amount ct of cash at time t. Allowing c to take both positive andnegative values, endowments and liabilities can be modeled in a unified manner.In particular, −c0 may be interpreted as an initial endowment while the subse-quent payments ct, t = 1, . . . , T may be interpreted as the cash-flows associatedwith financial liabilities. A typical example would be an insurance portfolio thatmay obligate an insurer to claim payments over several points in time. Simplerclaims such as European options correspond to processes c with ct = 0 for allbut one t. We will denote the space of (Ft)Tt=0-adapted sequence of cash-flowsby

M := (ct)Tt=0 | ct ∈ L0(Ω,Ft, P ).

Problem (ALM) can be generalized to the dynamic setting as follows

minimize

T∑t=0

Vt(St(∆xt) + ct) over x ∈ ND, (ALM-d)

where Vt are nondecreasing, convex functions on L0(Ω,Ft, P ) and

ND := x ∈ N |xt ∈ Dt, t = 0, . . . , T − 1, xT = 0.

We always define x−1 = 0 so the elements of ND describe trading strategies thatstart and end at liquidated positions. We allow Vt to be extended real-valuedbut assume that they are normalized so that Vt(0) = 0.

Problem (ALM) can be interpreted as an asset-liability management problemwhere one looks for a trading strategy x ∈ ND whose proceeds fit the givenliabilities c ∈ M as well as possible. The functions Vt measure the disutilitycaused by the net expenditure St(∆xt)+ct of updating the portfolio and payingout the claim ct at time t. When Vt = δL0

−for t < T , we can write (ALM-d)

with explicit constraints as

minimize VT (ST (∆xT ) + cT ) over x ∈ ND,subject to St(∆xt) + ct ≤ 0, t = 0, . . . , T − 1.

(2.1)

The constraint means that the claim ct has to be completely financed by tradingin financial markets. Recall that both the claim and the trading cost may benegative. When T = 1 and St(x, ω) = st(ω) · x, we recover the single-periodproblem (ALM) from Section 1.1.

Example 2.4 (Single-period models) When T = 1, problem (2.1) becomes

minimize V1(S1(∆x1) + c1) over x ∈ ND,subject to S0(∆x0) + c0 ≤ 0,


where ND = (x0, x1) ∈ N |x0 ∈ D0, xT = 0. Recalling that x−1 = 0 bydefinition and that x0 is deterministic since we have assumed F0 = Ω, ∅, theproblem can be written as

minimize V1(S1(−x0) + c1) over x0 ∈ D0,

subject to S0(x0) + c0 ≤ 0.

In the linear case where St(x, ω) = st(ω) · x, this becomes

minimize V1(c1 − s1(ω) · x0) over x0 ∈ D0,

subject to s0(ω) · x0 ≤ −c0,

which is the single-period problem (ALM) with initial wealth w = −c0.

With the traditional model of liquid markets, problem (2.1) can be writtenin terms of stochastic integrals.

Example 2.5 (Numeraire and stochastic integration) Assume that thereis a perfectly liquid asset (numeraire), say 0 ∈ J , such that

St(x, ω) = s0t (ω)x0 + St(x, ω),

Dt(ω) = R× Dt(ω),

where x = (x0, x), s0 is a strictly positive adapted process and S and D arethe cost process and the constraints for the remaining risky assets J = J \ 0.Expressing all costs in terms of the numeraire, we may assume s0 ≡ 1. Wecan then use the budget constraint to substitute out the numeraire from problem(2.1). Indeed, defining

x0t = x0

t−1 − St(∆xt)− ct t = 0, . . . , T − 1,

the budget constraint holds as an equality for t = 0, . . . , T − 1 and

x0T−1 = −

T−1∑t=0

St(∆xt)−T−1∑t=0

ct.

Then also

ST (∆xT ) + cT = ∆x0T + ST (∆xT ) + cT = x0

T +

T∑t=0

St(∆xt) +

T∑t=0

ct.

Recalling that xT = 0 for x ∈ ND, problem (2.1) can thus be written as

minimize VT

(T∑t=0

St(∆xt) +

T∑t=0

ct

)over x ∈ ND.

This shows that, in the presence of a numeraire, the timing of the paymentsbecomes irrelevant; only their sum matters. Furthermore, in the linear case


St(x, ω) = st(ω) · x, the cumulated trading costs can be written as the stochasticintegral

T∑t=0

St(∆xt) =

T∑t=0

st ·∆xt = −T−1∑t=0

xt ·∆st+1.

We then recover a discrete-time version of the classical problem of optimal in-vestment in perfectly liquid markets.

We will denote the optimal value of problem (ALM-d) by ϕ(c). That is,

ϕ(c) := infx∈ND

T∑t=0

Vt(St(∆xt) + ct).

This is an extended real-valued function on the space M of adapted sequencesof claims. In the completely risk averse case where Vt = δL0

−for all t = 0, . . . , T ,

we get ϕ = δC , where

C = c ∈M|∃x ∈ ND : St(∆xt) + ct ≤ 0 P -a.s. t = 0, . . . , T, (2.2)

the set of claim processes that can be superhedged without a cost. On the otherhand, since the functions Vt are nondecreasing, we can always write the valuefunction ϕ in terms of C as

ϕ(c) = infx∈ND

T∑t=0

Vt(St(∆xt) + ct)

= infx∈ND,d∈M

T∑t=0

Vt(dt)

∣∣∣∣∣ dt = St(∆xt) + ct

= infx∈ND,d∈M

T∑t=0

Vt(dt)

∣∣∣∣∣ dt ≥ St(∆xt) + ct

= infd∈M

V(d) | c− d ∈ C ,

where V is the convex function on M defined by

V(c) :=

T∑t=0

Vt(ct).

Above, the variable d may be interpreted as investments that the agent makesto his portfolio over time. Alternatively, a simple change of variables gives

ϕ(c) = infd∈CV(c− d), (2.3)

where d now represents the part of liabilities that can be costlessly covered bytrading in financial markets. If we ignore the financial market so that C =M−,we simply have ϕ = V.


While the set C consists of the claims that can be superhedged without acost, its recession cone

C∞ = c ∈M| c+ αc ∈ C ∀c ∈ C, ∀α > 0

consists of the claims that can be superhedged without a cost at unlimitedamounts. The following lemma summarizes some key properties of the valuefunction ϕ and the set C.

Lemma 2.6 The value function ϕ is convex, ϕ(0) ≤ 0 and

ϕ(c+ c) ≤ ϕ(c) ∀c ∈M, c ∈ C∞.

In particular, the set C is convex and C∞ ⊆ C. If the trading costs St aresublinear and portfolio constraints Dt are conical, then C is a cone and C∞ = C.

Proof. Convexity of C as well as the fact that C is a cone when St(·, ω) aresublinear and Dt(ω) are conical is proved in Exercise 3 below. The convexity ofϕ now follows from expression (2.3); see Exercise 4. If c ∈ C∞, then c+c−d ∈ Cwhenever c− d ∈ C so that

infd∈M

V(d) | c+ c− d ∈ C ≤ infd∈M

V(d) | c− d ∈ C ,

which means that ϕ(c + c) ≤ ϕ(c). The second claim follows by applying thefirst part of the lemma to the case where Vt = δL0

−for all t and c = 0. When C

is a convex cone, we have C∞ = C when C, by Exercise 5 in Section 1.1.

Note that since M− ⊆ C∞, the growth property in Lemma 2.6 implies thatϕ(c1) ≤ ϕ(c2) whenever c1 ≤ c2 almost surely. This simply means that theagent is always better off with more money.

We will say that a claim c ∈M is redundant if

c ∈ C∞ ∩ (−C∞),

i.e. if c + αc ∈ C for every c ∈ C and every α ∈ R. In particular, arbitrarypositive as well as negative multiples of redundant claims can be superhedgedwithout a cost. By Lemma 2.6, the optimal value of (ALM-d) is invariant withrespect to redundant claims. Redundancy generalizes the classical notion of“attainability” in the classical liquid market model as illustrated by the followingexample. Redundant claims will play an important role in relating indifferenceswap rates to risk-neutral prices in Section 2.4 below.

A market model (S,D) is said to satisfy the no-arbitrage condition if

C ∩M+ = 0, (NA)

Violation of this condition would mean that there exist nonzero nonnegativeclaims that could be costlessly superhedged by trading in financial markets.The possibility of doing so is known as an arbitrage opportunity.


Example 2.7 When St(x) = st · x and Dt = RJ , a claim c ∈ M is redundantif there is an x ∈ ND such that st ·∆xt + ct = 0. The converse holds under theno-arbitrage condition.

Proof. By Lemma 2.6, C is now a cone and C∞ = C. Thus, a claim c ∈ M isredundant iff c ∈ C and −c ∈ C, which means that there exist x1, x2 ∈ ND suchthat

st ·∆x1t + ct ≤ 0 and st ·∆x2

t − ct ≤ 0 ∀t. (2.4)

If there is an x ∈ ND such that st ·∆xt + ct = 0, then (2.4) holds with x1 = xand x2 = −x. On the other hand, (2.4) implies st · ∆(x1

t + x2t ) ≤ 0, where

equality must hold under (NA). Combining this equality with (2.4), we get

ct ≤ −st ·∆x1t = st ·∆x2

t ≤ ct

so both x1 and −x2 satisfy the condition st ·∆xt + ct = 0.

Example 2.8 (Numeraire and stochastic integration) In the classical lin-ear model with St(x) = x0 + st · x and Dt = RJ (see Example 2.5), we have

C = c ∈M|∃x ∈ ND :

T∑t=0

ct ≤T−1∑t=0

xt ·∆st+1

and a claim c ∈ C is redundant if there exists an x ∈ ND such that

T∑t=0

ct =

T−1∑t=0

xt ·∆st+1 (2.5)

The converse holds under the (NA) condition which, in this classical model, canbe stated as

x ∈ ND :

T−1∑t=0

xt ·∆st+1 ≥ 0 =⇒T−1∑t=0

xt ·∆st+1 = 0.

Proof. The expression for C follows from Example 2.5 by recalling that thevalue function ϕ becomes the indicator of C when Vt = δL0

−for every t =

0, . . . , T . By Lemma 2.6, C is now a cone and C∞ = C. Thus, a claim c ∈M isredundant iff c ∈ C and −c ∈ C, which means that there exist x1, x2 ∈ ND suchthat

T∑t=0

ct ≤T−1∑t=0

x1t ·∆st+1 and −

T∑t=0

ct ≤T−1∑t=0

x2t ·∆st+1. (2.6)

If there is an x ∈ ND such that (2.5) holds, then (2.6) hold with x1 = x andx2 = −x. On the other hand, (2.6) implies

T−1∑t=0

(x1t + x2

t ) ·∆st+1 ≥ 0.

where equality must hold under (NA). Combining this equality with (2.6), wesee that both x1 and −x2 satisfy (2.5).


Exercises

1. Consider an optimal consumption problem where an agent is endowedwith an adapted sequence e ∈M of cash-flows. At each time t = 0, . . . , Tthe agent may consume part of the endowment and invest the remainingpart in financial markets described by (S,D). Assume that the agent’spreferences are described by a sequence of disutility functions Vt and showthat this problem can be written as (ALM-d).

2. Assume that s is a componentwise strictly positive (Ft)Tt=0-adapted “mar-ket price” process as in Exercise 3 of Section 2.1. Write the asset-liabilitymanagement problem (ALM-d) in terms of the mark-to-market valueshjt = sjtx

jt .

3. Show that C is a convex set containing the set M− of nonpositive claimprocesses. Show also that if S is sublinear and D is conical, then C is acone.

4. Let f1 and f2 be convex functions on a linear space X. Show that theirinfimal convolution

(f1 f2)(x) := infz∈Xf1(x− z) + f2(z)

is a convex function.

5. Show that the no-arbitrage condition can be expressed as follows: if x ∈ND is such that

St(∆xt) ≤ 0 P -a.s. t = 0, . . . , T

then

St(∆xt) = 0 P -a.s. t = 0, . . . , T.

6. Let C be a convex set in a linear space and assume that C is algebraicallyclosed in the sense that α ∈ R |x+αy ∈ C is a closed interval for everyx, y ∈ X. Show that y ∈ C∞ if there exists even one x ∈ C such thatx+ αy ∈ C for every α > 0. In this case, we thus have the expression

C∞ =⋂α>0

α(C − x)

which is independent of the choice of x ∈ C.

Further reading

Much of the theory of optimal investment has been developed in a continous-time setting, most famously by Merton [32] in a complete market model. Ex-tensions to illiquid markets can be found e.g. in He and Pearson [19], Karatzas,


Lehoczky, Shreve and Xu [27] and Kramkov and Schachermayer [29]. Illiquid ex-tensions have been given in Hodges and Neuberger [21], Davis and Norman [10]and Cetin and Rogers [7]. A recent overview of optimal investment can be foundin Rogers [43].

Besides the discrete-time and illiquidity effects, our model deviates from theabove references in that it does not require the existence of a numeraire. This issignificant from the practical point of view. Indeed, in practice, much of trad-ing consists of exchanging sequences of cash-flows, exactly because one cannottransfer cash quite freely in time. Problem (ALM) can also be interpreted as anilliquid discrete-time version of the optimal consumption-investment problemwith random endowment studied in Karatzas and Zitkovic [28]; see Exercise 1.

The notion of redundancy extends the notion of attainability from the clas-sical linear market model to general nonlinear market models. Indeed, thecondition in Example 2.8 means that

∑Tt=0 ct is “attainable at price zero” as

defined e.g. in [12, Definition 2.2.1].

2.3 Reserving

Analogously to the single-period model of Section 1.2, we define the reservefor a financial liability as the least amount of initial capital that would allowan agent to pay the liability cash-flows at an acceptable risk of bankruptcy.Liabilities are now described by sequences c ∈ M of cash flows and the leastrisk achievable by trading is given by the optimum value ϕ(c) of (ALM-d).Defining p0 = (1, 0, . . . , 0),

π0(c) = infα ∈ R |ϕ(c− αp0) ≤ 0

gives the least amount of initial capital one needs in order to construct a hedgingstrategy x ∈ ND whose proceeds cover a sufficient part of the liabilities c sothat the risk associated with the residual is no higher than the risk from doingnothing at all (recall that Vt(0) = 0, by assumption). It is natural to assumethat c0 = 0 since a nonzero value of c0 would just add directly to the requiredinitial capital π0(c).

If we ignore the financial market so that ϕ = V, we simply have π0(c) =infα | V(c − αp0) ≤ 0. If Vt = δL0

−for t < T as in (2.1) and if the optimum

value of (ALM-d) is attained, the reserve for a claim c ∈M with c0 = 0 is givenby the optimum value of (exercise)

minimize S0(x0) over x ∈ ND,subject to St(∆xt) + ct ≤ 0, t = 1, . . . , T − 1,

VT (ST (−xT−1) + cT ) ≤ 0.

(2.7)

The corresponding solution x ∈ ND gives an optimal hedging strategy for theliabilities c ∈ M. The same idea is behind the classical Black–Scholes–Mertonoption pricing model where the price of an option is defined as the least amountof initial capital that allows for the construction of a trading strategy whose

2.3. RESERVING 35

terminal value equals the payout of the option. Unlike in the Black–Scholes–Merton model, however, exact replication is often impossible in practice so thereserve depends one the risk preferences concerning the uncertain amount ofwealth remaining after paying out the claims. Also, we have not assumed theexistence of a numeraire so the natural domain of definition for π0 is the spaceM of sequences of cash-flows. This is important in practice where financialliabilities often have several payout dates.

In the deterministic case, we recover the classical discounting principle fromactuarial mathematics.

Example 2.9 (Actuarial best estimate) Assume that c and S are deter-ministic with St(x, ω) = x/Pt where Pt is the price at time 0 of a zero-couponbond maturing at time t. If VT (α) > 0 for positive constants α, problem (2.7)can be written as

minimize x0 over x ∈ ND,subject to ∆xt/Pt + ct ≤ 0, t = 1, . . . , T,

where ND now consists of sequences (xt)Tt=0 of real numbers with xT = 0. This

can be solved explicitly for xt =∑Ts=t+1 Pscs and the optimum value

π0(c) =

T∑s=1

Pscs.

Such formulas are frequently encountered in actuarial mathematics where ct isusually defined as the expected claims to be paid at time t. In that context, theabove expression is known as the “best estimate” of the claims.

It should be noted, however, that the “best estimate” is appropriate only forthe valuation of deterministic cash-flows. For uncertain ones, it can result inquite unreasonable values. For example, the “best estimate” of a European calloption is usually much higher than its market or Black–Scholes value. Neverthe-less, the “best estimate” is still widely used in the insurance industry and it formsthe basis of the regulatory standard in the Solvency II Directive 2009/138/ECof the European Parliament.

When Vt = δL0−

for all t, the reserve π(c) coincides with the superhedgingcost

π0sup(c) = infα ∈ R | c− αp0 ∈ C

which gives the least amount of initial capital required for delivering c withoutany risk of loosing money. Analogously, we define the subhedging cost by

π0inf(c) = supα ∈ R |αp0 − c ∈ C.

Note that, the super- and subhedging costs depend on the probability measureP only through its null sets. The following summarizes some basic propertiesof the reserve.


Theorem 2.10 The function π0 is convex, π0(0) ≤ 0 and

π0(c+ c′) ≤ π0(c) ∀c ∈M, ∀c′ ∈ C∞.

We always have π0(c) ≤ π0sup(c). If π0(0) = 0, then

π0inf(c) ≤ π0(c) ≤ π0

sup(c)

with equalities throughout when c− αp0 ∈ C ∩ (−C) for some α ∈ R.

Proof. Defining B = c ∈M|ϕ(c) ≤ 0, we have

π(c) = infα | c− αp ∈ B.

Lemma 2.6 implies that B is a convex set with C∞ in its recession cone. Thisimplies the convexity and monotonicity of π; see Exercise 1 in Section 1.3. SinceL0− is the greatest among all nondecreasing convex functions Vt with Vt(0) = 0,

we have ϕ ≤ δC and thus,

π0(c) = infα |ϕ(c− αp0) ≤ 0≤ infα | δC(c− αp0) ≤ 0= infα | c− αp0 ∈ C = π0

sup(c).

Since π0inf(c) = −π0

sup(−c), we must also have π0inf(c) ≤ −π0(−c). By convexity,

π0(0) = π0(1

2c+

1

2(−c)) ≤ 1

2π0(c) +

1

2π0(−c)

so if π0(0) = 0 we get −π0(−c) ≤ π0(c), which completes the proof of the“sandwich” inequalities. If c−αp0 ∈ C ∩ (−C) for some α ∈ R, we get π0

sup(c) ≤α ≤ π0

inf(c), which completes the proof.

The convexity and monotonicity properties of π0 make it reminiscent ofconvex risk measures. Indeed, since M− ⊂ C∞ we have π0(c1) ≤ π0(c2) when-ever c1 ≤ c2. However, the “translation invariance property” often required ofrisk measures, is not relevant in markets without a numeraire. The conditionπ0(0) = 0 means that an agent without liabilities needs no reserves. It holds, inparticular, if V0 = δL0

−, the cost function S0 is nondecreasing and if short selling

is prohibited, i.e. if D0 ⊆ RJ+. In the completely risk averse case with Vt = δR−for all t, the condition π0(0) = 0 means that it is not possible to superhedge thezero claim when starting from a strictly negative initial wealth. In the classicallinear model, such a condition is known as the “weak no-arbitrage” condition;see Exercise 3b in Section 1.2. As the name suggests, the weak no-arbitragecondition is a weaker condition than the no-arbitrage condition (NA).

In general, π0(c) depends on the probability measure P and the disutilityfunctions Vt (views and preferences). By the last part of Theorem 2.10, however,π0(c) is independent of such subjective factors when c−αp0 ∈ C∩(−C) for someα ∈ R. Indeed, the super- and subhedging costs do not depend on P or V.

2.3. RESERVING 37

Example 2.11 (Numeraire and stochastic integration) Consider again theclassical model of liquid markets in Example 2.8. We get

π0sup(c) = infα | ∃x ∈ ND :

T∑t=0

ct ≤ α+

T−1∑t=0

xt ·∆st+1.

Since now C∞ = C, we have c−αp0 ∈ C∩(−C) if and only if c−αp0 is redundant.By Example 2.8, this holds if there is an x ∈ ND such that

T∑t=0

ct = α+

T−1∑t=0

xt ·∆st+1.

Under the no-arbitrage condition, the converse holds.

Proof. In the classical linear model, we have, by Example 2.8, that

C = c ∈M|∃x ∈ ND :

T∑t=0

ct ≤T−1∑t=0

xt ·∆st+1

so

π0sup(c) = infα | c− αp0 ∈ C

= infα | ∃x ∈ ND :

T∑t=0

(ct − αp0t ) ≤

T−1∑t=0

xt ·∆st+1.

Since p0 = (1, 0, . . . , 0), this gives the expression for π0sup(c). The rest follow

from the characterization of redundancy in Example 2.8.

Exercises

1. Show that when T = 1 and St(x, ω) = st(ω) · x, problem (2.7) reduces toproblem (1.3) in the one-period setting.

2. Show that if Vt = δL0−

for t < T and if the optimum value of (ALM-d)

is attained for every c ∈ M, then π0(c) is given by the optimum value of(2.7) as claimed.


4. Show that, in the perfectly liquid market model of Example 2.5, the no-arbitrage condition (NA) implies the weak no-arbitrage condition. Give anexample of a market model that satisfies the weak no-arbitrage conditionbut fails the no-arbitrage condition.


Further reading

Our definition of the reserve π0 corresponds to the usual definition of an “optionprice” as the least amount of initial cash needed for hedging the option. Wehave avoided the term “price” since that is more appropriate for the notionof indifference price to be studied in the next section; see also Section 1.3.Extensive treatments of superhedging can be found e.g. in [17] and [12]. Morereasonable risk preferences are studied in a liquid multiperiod model in Follmerand Schied [17, Chapter 8]. Illiquid generalizations can be found e.g. in Davis,Panas and Zariphopoulou [11]. The presented approach has been applied to thevaluation of pension liabilities where in [20].

In the context of the classical linear market model, the condition that c −αp0 ∈ C∩(−C) for some α is often known as “attainability”; see e.g. [17] and [12].The notion of “weak no-arbitrage” condition has been studied in the contextof perfectly liquid markets in [22] under the name “no-arbitrage of the secondtype” and in fixed-income markets with illiquidity effects in [13, 14].

2.4 Swap rates

Much of trading in practice consists of exchanging sequences of cash-flows. Thisis the case in various swap and insurance contracts where both claims as well aspremiums are paid over several points in time. The timing of payments matterssince in the absence of a numeraire, one cannot transfer both positive as wellas negative amounts of cash quite freely through time. This section extendsthe classical indifference pricing principle to general swap contracts in illiquidmarkets. We show that indifference swap rates lie between arbitrage boundssuitably generalized to account for nonlinear illiquidity effects and general pre-mium processes.

Consider an agent whose current financial position is described by a sequenceof cash-flows c ∈ M (recalling that negative values of c are interpreted asincome) and a possibility to enter a swap contract where the agent agrees todeliver a sequence c ∈M of random payments in exchange for receiving anothersequence proportional to a given premium process p ∈ M. The lowest swaprate (premium rate) that would allow the agent to enter the contract withoutworsening his risk-return profile is given by

π(c, p; c) := infα ∈ R |ϕ(c+ c− αp) ≤ ϕ(c).

Similarly,πb(c, p; c) := supα ∈ R |ϕ(c+ αp− c) ≤ ϕ(c)

gives the highest swap rate the agent would accept for taking the opposite side ofthe trade. Clearly, πb(c, p; c) = −π(c, p;−c). We will call π(c, p; c) and πb(c, p; c)the indifference swap rates.

When p = (1, 0, . . . , 0) and c = (0, . . . , 0, cT ), we recover the traditionalpricing problem where one looks for an initial payment that compensates fordelivering a claim with single payment date. The value of π(c, p; c) still depends

2.4. SWAP RATES 39

on the agent’s existing liabilities c and it is, in general, different from the reserveπ0(c) studied in the previous section. In an interest rate swap (with equallyspaced payment dates), one has p = (N, . . . , N) and c = (r0N, . . . , rTN), whereN is the notional and rt the floating rate. In a credit default swap, the premiumprocess p is a constant sequence until the default of the reference entity afterwhich it is zero while the claim c is zero except at the first t after default whenit pays a fixed amount. Collateralized debt obligations also fit the above format.More examples can be found in the insurance industry.

The indifference swap rates can be bounded by the super- and subhedgingswap rates defined by

πsup(c) = infα ∈ R | c− αp ∈ C∞ and πinf(c) = supα ∈ R |αp− c ∈ C∞,

where C∞ is the recession cone of C; see Section 2.2. Clearly, πinf(c) = −πsup(−c).Note that the super- and subhedging swap rates depend on the underlyingprobability measure P only through its null sets. When C is a cone andp = (1, 0, . . . , 0), we have C∞ = C and the functions πsup and πinf coincidewith the super- and subhedging costs π0

sup and π0inf considered in the previous

section.

Theorem 2.12 Let c, p ∈ M. The function π(c, p; ·) is convex, π(c, p; 0) ≤ 0and

π(c, p; c+ c′) ≤ π(c, p; c) ∀c ∈M, ∀c′ ∈ C∞.

We always have π(c, p; c) ≤ πsup(c) and if π(c, p; 0) = 0, then

πinf(c) ≤ πb(c, p; c) ≤ π(c, p; c) ≤ πsup(c)

with equalities throughout if c− αp ∈ C∞ ∩ (−C∞) for some α ∈ R.

Proof. Defining B(c) = c ∈M|ϕ(c+ c) ≤ ϕ(c), we have

π(c, p; c) = infα | c− αp ∈ B(c).

Lemma 2.6 implies that B(c) is a convex set with C∞ in its recession cone. Thisimplies the convexity and monotonicity of π(c, p; ·); see Exercise 1 in Section 1.3.

By convexity,

π(c, p; 0) ≤ 1

2π(c, p; c) +

1

2π(c, p;−c),

so that πb(c, p; c) ≤ π(c, p; c) when π(c, p; 0) = 0. If c − αp ∈ C∞, Lemma 2.6gives ϕ(c + c − αp) ≤ ϕ(c) and thus π(c, p; c) ≤ α. Taking the infimum oversuch α, we get π(c, p; c) ≤ πsup(c). Since πinf(c) = −πsup(−c) and πb(c, p; c) =−π(c, p;−c) we also get πinf(c) ≤ πb(c, p; c), which completes the proof of theinequalities. If c − αp ∈ C∞ ∩ (−C∞) for some α, we get πsup(c) ≤ α andπinf(c) ≥ α which completes the proof.


The condition π(c, p; 0) = 0 means that one cannot deliver strictly positivemultiples of the premium p without worsening the optimal value of (ALM). Theinequality πb(c, p; c) ≤ π(c, p; c) means that two agents with identical character-istics will not trade with each other. Differences in the financial position, marketexpectations and/or risk preferences (as described by c, P and Vt, respectively)may provide incentive for trading. The second part of Theorem 2.12 says thatif c − αp is redundant for some α, then the indifference swap rates are givenby πsup which is independent of such subjective characteristics. In particular,when C is a cone and p = (1, 0, . . . , 0) (so that π0

sup = πsup and π0inf = πinf),

Theorems 2.12 and 2.10 imply that, indifference swap rates coincide with thereserve if c − αp is redundant for some α and π0(0) = 0 and π(c, p; 0) = 0.Moreover, the convexity of πsup and the concavity of πinf imply that swap ratesare linear in c on the linear subspace of claims c such that c−αp ∈ C∞∩(−C∞).

While redundant claims relate indifference prices to classical “risk neutralprices”, their practical significance should not be over-estimated.

Remark 2.13 If there is an arbitrage-free price process s such that st·x < St(x)for x 6= 0, then C∞ ∩ (−C∞) = 0. Indeed, if c ∈ C∞ ∩ (−C∞), there existx1, x2 ∈ ND such that St(∆x

1t ) + ct ≤ 0 and St(∆x

2t )− ct ≤ 0 and thus,

st ·∆(x1t + x2

t ) ≤ St(∆x1t ) + St(∆x

2t ) ≤ 0,

with the first inequality being strict unless ∆x1t = ∆x2

t = 0. If s is arbitrage-free,we must have ∆x1

t = ∆x2t = 0 and thus, c = 0.

A market model is complete if for any c ∈ M and p /∈ C∞ ∩ (−C∞) there isan α ∈ R such that c − αp0 ∈ C∞ ∩ (−C∞). In complete market models withp /∈ C∞ ∩ (−C∞), the indifference swap rate coincides with the superhedgingswap rate for every c ∈ M. However, completeness is satisfied only by somespecial instances of perfectly liquid market models such as the binomial modelor the continuous-time Black–Scholes model.

Exercises

1. Show that C∞ ∩ (−C∞) is a linear space.

2. Show that if p ∈ M is such that ϕ(c + p) > ϕ(c) for some c ∈ M, thenp /∈ C∞ ∩ (−C∞).

Further reading

Indifference pricing has been studied extensively in incomplete financial mar-kets; see the references at the end of Section 1.3. It has mostly, however, beenrestricted to models with a numeraire and to claims with a single payout date.The more general framework described here was introduced in [36]. An exten-sive treatment of recession cones in finite-dimensional spaces can be found inSections 8 and 9 of [38].

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introduction to convex optimization in nancial marketsmath.ut.ee/emsdk/kevadkool2013/tartu.pdf ·...

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