Journal of Mathematical Finance, 2012, 2, 141-158 http://dx.doi.org/10.4236/jmf.2012.22016 Published Online May 2012 (http://www.SciRP.org/journal/jmf)

Interest Rate Models

Alex Paseka1,2, Theodoro Koulis1,2, Aerambamoorthy Thavaneswaran1,2 1Department of Accounting and Finance, University of Manitoba, Winnipeg, Canada

2Department of Statistics, University of Manitoba, Winnipeg, Canada Email: thava@temple.edu

Received December 15, 2011; revised January 14, 2012; accepted January 25, 2012

ABSTRACT

In this paper, we review recent developments in modeling term structures of market yields on default-free bonds. Our discussion is restricted to continuous-time dynamic term structure models (DTSMs). We derive joint conditional mo- ment generating functions (CMGFs) of state variables for DTSMs in which state variables follow multivariate affine diffusions and jump-diffusion processes with random intensity. As an illustration of the pricing methods, we provide special cases of the general formulations as examples. The examples span a wide cross-section of models from early one-factor models of Vasicek to more recent interest rate models with stochastic volatility, random intensity jump-dif- fusions and quadratic-Gaussian DTSMs. We also derive the European call option price on a zero-coupon bond for linear quadratic term structure models. Keywords: Affine Process; Dynamic Term Structure Models; Jump-Diffusions; Quadratic-Gaussian DTSMs

1. Introduction

In this review, we summarize available continuous time technology for pricing default-free term structures. Our main goal is to provide a unified approach to the pricing exercise based on well-developed application of stochas-tic calculus to finance problems. In our discussions, we always start from state variable processes given under a risk-neutral measure. We make no attempt to systemati-cally present the transition between the actual measure and the risk-neutral measure. Thus, we leave out the question of reconciling time series properties of the un-derlying state variables (such as short rate), which are defined under the actual measure, and the properties of the market yields (or, equivalently, bond prices), which are derived under the risk-neutral measure.

We begin with affine diffusion models of interest rates. The menu of available models in this class is truly vast.1 The most redeeming features of this class of models are the availability of closed form solutions for derivatives on the short rate and the ability of models to reproduce a variety of term structure patterns. Models in this class have closed form solutions for bond and bond option prices, which makes them particularly attractive for em-pirical work. Some models in this class, especially two- or three-factor models, are flexible enough to have a sat-

isfactory fit to observed term structures. The last property is essential for pricing options on bonds. Extended ver-sions of Vasicek and CIR (e.g., [2,5,14]) allow for time- varying coefficients in state variable dynamics. This ex-tra feature allows the model to fit the initial term struc-ture.

Discontinuous movements in interest rates are caused by central banks and unexpected news announcements. In order to implement a specific policy, monetary au-thorities often use specific rate-setting rules which result in entire yield curve shifts in line with movements in the benchmark rate. Empirical evidence suggests that interest rates exhibit substantial skewness and kurtosis, and hence jump-diffusion interest rate models are more appropriate2. In deriving the term structure of spot rates for the case of discontinuous interest rates, we stay within the affine jump-diffusion class of models. The bond prices are still available in near-closed form up to the solutions of Ric-cati ordinary differential equations for the coefficients of the conditional moment generating function of the state variables.

Market spot rates are mostly positive, and a class of quadratic interest rate models is designed to restrict spot rates to be positive without ruining the bond price tracta-bility. Square-Gaussian models, in which the short rate is

2The number of models in this class is vast. Examples include [15-23], [19-21,24-26] provide evidence that jumps are essential in modeling interest rate distribution. [27] provide a general approach to interest rate derivative pricing for exponential affine jump-diffusion models of interest rates.

1Examples include the original one-factor model [1], extended Vasicek ([2,3]), one-factor Cox-Ingersoll-Ross models ([2,4-7]), two-factor Cox-Ingersoll-Ross models ([2,8,9]), three-factor models ([10-12]). For a more detailed list of models we refer the reader to [13].

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A. PASEKA ET AL. 142

a sum of squares of Gaussian state variables, have been investigated by [28-30]. The models of Quadratic-Gaus- sian class were studied by [31-35]. The mathematics of these models is similar to that of affine jump diffusions and also allows closed form solutions for bond prices in the exponential quadratic form.

Previous jump-diffusion TSMs treat short rate inten-sity in one of the following restrictive ways:

1) A deterministic function of the short rate ([26]); 2) Constant except for FOMC meeting days and mac-

roeconomic news announcements, in which case it is a linear function of short rate, interest rate volatility, and Fed rate target; or

3) A function of the spread between the Fed funds rate and the target.

Thus, we also consider a recent extension of a class of Linear-Quadratic term structure models (LQTSMs) to a class of jump-diffusion TSMs in which short rate jump intensity follows its own stochastic process (see [36-38] for details). We show how this model class is designed to have a closed form conditional moment generating func-tion and bond prices. Despite strong restrictions on state variable processes that are necessary to obtain closed form bond prices, [36] shows that the inclusion of ran-dom intensity improves the model fit to both the dynam-ics of the term structure and that of the volatility term structure. The main mechanism of improved fit is the short term kurtosis of the short rate.

The rest of this paper is organized as follows. In Sec-tion 2, we present a general technique of solving for joint CMGF of an affine state vector and its integral. Special cases include CMGF of an affine vector and bond prices within affine model class. Section 3 covers affine jump- diffusion DTSMs. Finally, we close with a review of quadratic Gaussian models and a more general class of linear quadratic class of DTSMs with jumps of random intensity. We also derive the price of a European call option using a generalized transform.

2. Affine Term Structure Models

In this section we provide general results for affine mod-els. As an illustration, we also give detailed account of some popular special cases.

Definition. Let be a vector of state variables, and let be the short rate. A model is affine if the state vector solves the following diffusion SDE:

Y nr

d d Σ d ,Y Y t V W

Y

(2.1)

0 1 ,r (2.2)

where Σ constant matrix),

0

n n , 1n , n n ( 1

1n n n -matrix such diag i iV Y

, , V is a diagonal that V , a ctor of

Wiener processes. We assume that an equiva-

lent martingale measure (EMM) exists, and all processes and expectations (unless stated otherwise) are understood as those under the EMM, e.g., the process in (2.1) is written under the EMM. The SDE (2.1) implies that the joint conditional moment generating function (CMGF) of the state vector is exponential affine:

nd W 1n is a ve

(2.3)

where

standard

0 1, ,tT u u Yu Ytu t E e e

T t . following ecial case of [39] with no

ju 2.1. Assume that a state vector Y satisfies

(2

The lemma is a spmps. Lemma.1). Consider the following boundary value problem:

1, 0

2

Y

YY

t

tr V f V r t Y f

fY f

(2.4)

, u Yf T Y e (2.5)

This problem has a stochas

T

tic solution of the form3

T

0 1

, exp , d

t

t st

Y

f t Y E r s Y s u Y

e

(2.6)

with the following initial conditions: and 0 0 0 1 0 u .

pplProof. A ying Ito’s lemma to function

, exp , dw

w w st

Z f w Y r s Y s

, we have4

0

0

, exp , d

, Σ d

1 Σ , d

2

T

T T st

T

t Y s

T

Y

YY

Z f T Y r s Y s

f t Y f V W

fY f

s

tr V f V r s Y f s

.

After taking the conditional expectation of both sides an

T

d noting that the first of the two integrals on the right hand side is zero due to (4), we have

T

, exp , d ,

exp , d

t t s Tt

T

t st

f t Y E r s Y s f T Y

E r s Y s u Y

3[27,39-42] used the extended transform to price derivatives in closed-form using inverse Fourier transform. 4This expression holds even if the terminal time, T, is random. In this case, the formula is known as Dynkin’s formula.

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A. PASEKA ET AL. 143

In order to prove the exponential affine form of the solution (2.6) to (2.4), it suffices to use it as a guess and find the equations determining coefficients1n 0 and 1 . Substitution of ,f t Y 0 1e tY

n that mu into

st hold for (2.4 uces the following any value of the state vector:

) prod equatio

1 1

0 1 1

1 1

Σ Σ

1

2

t t

tr VV

Y Y

V V

0 1 0r Y

The left hand side of the above equation is an affine function in . Since the equation must hold for arbitrary vector , both coefficients of the affine function must

requirement gives two Riccati ODEs for coefficients

Y

0

Ybe zero. This

and 1 :

0 1 1 0 1 0

10

2t H (2.7)

1 1 1 1 1 1

10,

2t H (2.8)

where 0H and 1H can be inferred from the following expressi

It follows from (2.5) that the initial values are

io r an affine diffusion is just a special case of the above result for

on:

0 1

Σ diag Σ Σ diag Σ

i iVV Y

H H Y

0 0 0 and 0 u . The express

1n for MGF fo

0 :5

(2.9)

Es:

0 1, .tT u u Yu Ytu t E e e

The coefficients satisfy the following Riccati OD

0 1 1 0 1

10,

2tu u u H u

0, 1 1 1 1 1

1

2tu u u H u

0 10 0, 0 .u u u

Zero coupon bond prices are a special case f the Lemma. Assuming again that the short rate process is affine in the state variables as given in (2.2), the price of a zero-coupon default-free bond with maturity and face value of $1 is given by

o

T

, exp , dt t t st

D t Y E r s Y s

(2.10)

where the expectation is under a EMM. The expectation in (2.10) is a special case of (2.6) with 1

T

and 0u . Given these restrictions, the bond price has exponential affine form

0 1, tYt tD t Y e

and satisfies the following PDE:

,

1

t YD Y D r t Y D

(2.11)

Σ 02 YYtr V D V

Bond price coefficients 0 and 1 (a sp

solve the ons ecial case of

(2.7)-(2.8)): following system of Riccati equati

0 1 1 0 1 0

10

2t H (2.12)

1 1 1 1 1 1

1( ) ( ) 0

2t H (2.13)

0 10 0 0 (2.14)

The spot rate is given by

0 1

1ln ,

t t tr D t Y

tY

(2.15)

Taking limits as 0 , we obtain the short rate hich is also given in (2.2)

tr (w )

0 1 0 1 t0 0t tr Y Y

2.1. One-Factor Interest-Rate Models

In edetails. Under the EMM,

the evolution of this process is described by a di usion Gauss-Markov process of the form:

this section we provide closed form expressions of th default-free bond prices with

ff

d d d ,r t t t t W t (2.16)

where , 0W t t is a standard Brownian motion and t , t are suitable processes adapted to

,0t W u u t . The usual assumption is that t and t are simply functions of the short rate r t and time t , i.e., t r t , t and ,t r t t .

rem 2.2. Con

Theo sider the Hull and White model of the f rm o

d d ,r t t t W t (2.17)

e

dr t t t 5In (2.9), the expectation is usually computed under the actual measure. In practice, the actual moments of the state variables are used instead of the risk neutral ones.

t , t and -random t wher are non func-tions of t such that

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A. PASEKA ET AL. 144

Hence the default-free time bond price is given by 2 d .

T

t t t t (2.18) 0

The price at time of a zero-coupon deaying at time T is

, (2.19)

where

t fault free bond p $1

, expB t T A , ,t T r t C t T

2

d d

T T

T u

t t

1, d d

2 t s

g u

,

A t T s u sg s

g us s u

g s

0

, d and

exp d .

T

t

t

g uC t T u g t

g t

s s

　　　　

Note: The Merton [43] model, Vasicek [1] model, and the Ho-Lee model [44] are special cases of (2.17).

Proof. Under these assumptions (2.18) the Equation (2.17) has a unique (strong) solution

0

0 g s g s

Since r r t is the Markov process, it foll

d d .t

s

s sr t g t r s W

ows th

t Tat

B , E exp d

E exp ,

T

t

t T r s s r t

I t T r t

　　　

where

, dT

t

I t T r s s is conditionally normal. The

conditional mean, variance, and the Laplace transform are given as:

d ,

T u g us s u

E ,

d dT

I t T r t

g ur t u

g t g s

t t t

2

var , d d ,T T

t s

g uI t T r t s u s

g s

and

E exp ,

1exp Var , E , .

2

I t T r t

I t T r t I t T r t

t

B , exp , , .t T A t T r t C t T

Note 1. The Vasicek model is a special case with con-stant coefficients: t r t and t ( 0 ), such that t, ,t t

2 that . It

follows from Theorem

1, d 1

TT tt y

t C t T e e y e (2.20)

2

2 2

2 2

1d d d

1, d ,

T T Tv y v y

t v

T T

e e y e e y v

C v T v C v

.21)

2 2 2 2

2

,

2

d2

, 2 ,.

4

v

t t

A t T

T v

C t T T t C t

(2

Note 2. For the extended Vasicek model,

T

t t , t , it follows from Theorem 2.2 that

1, d 1

TT tt y

t

C t T e e y e

(2.22)

2

2 2

2 2

22

,

1d d d

1, d , d

2

, , .

2 2

T T Tv y v y

t t

A t T

v

v T v v C v T v

C t T T t C t T

Theorem 2.3. Consider the Cox, Ingersoll and Ross (CIR) model, the interest rate process

2t v v

T T

e v e y e e y

C

(2.23)

r t is given by

d dr t r t t r t W t d ,

where r t n bon

is given. The price at time of a coupo d paying $1 at time T is

where

t zero-

, ,, ,r t C t T A t TB t T e

sinh, ,

1c sh sinh( )

T tC t T

T t T t

o2

1

,

T t

A t

2

2

2log ,

1cosh sinh

2

T

e

T t T t

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A. PASEKA ET AL. 145

2 212

2 and .

Proof. See the Appendix.

2.2. Multi-Factor TSMs

There is substantial evidence that bond yields have tim - varying conditional volatilities (see, e.g., [45,46]). Even though general single-factor DTSMs do have built-in time varying bond yield volatilities (except for basic

sian affine and l rmal models), however, they can not match the variation observed in practice. For ex- am nd to understate the vola- tility of long-term yields, overstate the correlation be-

aturities, require mean re-

e

Gaus ogno

ple, single-factor models te

tween yields of different mversion properties that cannot simultaneously match cross section and time series of bond yields. There is also evi- dence that linear drift term in single factor models is misspecified. Multifactor models are designed to address some of these issues.

Here, we give an example of a three-factor affine fam- ily model presented in [11]. The short rate is a CIR proc- ess with long run mean (central tendency) following Ornstein-Uhlenbeck process and stochastic volatility fol- lowing a CIR process all under an EMM. The state vec-tor , ,

TY r V is defined as follows:

d d rr r t V W

d dt dW

d d dV VV a b V t V W

where d d dr VE W W t , and all other correlations are zero.

According to (2.11), the price of a zero coupon bond in th sfies the following PDE: is model sati

2 212

2

t r V

rr V VV V rV

D r D D a b V D

V D D V D V D rD

0

Since the model is in the affine category in the following f

, we look for asolution orm:

0 1, , , tY tD t r V e

where vector 1 has three components 11 , 12 , and 13

th the bo. The Riccati Equations (2

g wi undary conditions (2.14).12) and imply (2.13) alon

2 20 12 13 12

10

2ab (2.24)

11 11 1 0 (2.25)

12 11 12 0

2 2 213 11 13 11 13

1 10

2 2 V Va (2.27)

0 0 1 0 0 (2.28)

Solutions to Equations (2.25) and (2.26) are available in closed form:

11

1 e

(2.26)

12

1 1.

e e

Explicit solutions for

e

13 and 0 are not available. However, numtain.

p-Diffusion DTSMs

There is a growing body of empirical research showing that discontinuous behavior of interest rates is essential in correctly describing their distribution and fitting terms structures to the data. e.g., [24-26] all find that introduc-ing jumps substantially improves the fit of e condi-tional distribution of short-term interest rates compared to odels.6

le mathematics involved in these models. In- the mean re-interest rates

nclude jumps. The C

erical solutions are easy to ob-

3. Affine Jum

th

that of nested diffusion m The relative scarcity of empirical work on models of

discontinuous interest rates is most likely related to far less tactabterest rates are mean reverting. In general,version feature combined with jumps in the makes the moments of jump-diffusion distribution de-pendant on the timing of the jumps. However, in jump- diffusion models it turns out that the density of equity returns does not depend on the jump times. In order to derive the expression for the CMGF of the state variables and bond prices for the case of affine jump diffusion we generalize the Lemma 2.1 in order to i

MGF for a jump-diffusion process turns out to be the product of CMGFs of continuous and jump parts of the processes.

Definition. Let Y be a vector of n state variables and r is the short rate. A model is affine if the state vector solves the following diffusion SDE:

d d Σ d dY Y t V W J N (3.1)

where n n , 1n , Σ n n constant matrix,

0 , 11

n , V is a diagonal n n -matrix such that diag i iVV Y , 1nW is a vector of stand-

6Jump-diffusion models of interest rates are numerous. e.g., [15] gen-eralise the equilibrium model of CIR by including jumps in the dy-namics of underlying state variables. [21,22], and [18] augment the Vcl

asicek model with jumps in the short rate. [39] consider a general ass of affine jump-diffusion models.

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A. PASEKA ET AL. 146

ard Wiener processes, q is a Poisson counter with state-dependent intensity, a positive affine function of Y , 0 1Y Y , and J

u Ees

r

pun

is a u Je

are

ent statxp

derst

jump size with ti We assuncorrelat

mp e varia

me-homo- me that jump size geneous MGF

p arrivpa f thlated ac

xists,

J .al ime pr

ross diffe

o

and jum t u ed with the diffu- sion rt o ocess. In general, ju sizes can be corre bles.

As before, (2.2) is the e ression for the short rate. We again assume that an equivalent martingale measure (EMM) e and all rocesses and expectations (unless stated otherwise) are od as those under the EMM. The SDE (3.1) implies that the joint conditional moment generating function (CMGF) of the state vector is expo-nential affine. To see why this is the case, we extend the lemma of section 2 to include jumps.

Lemma 3.1. Assume that a state vector Y satisfies (3.1). Then, the joint CMGF of process Y and its inte-gral is given by the following expression

0 1

, exp , d

t

T

t s Tt

Y

f t Y E r s Y s u Y

e

(3.2)

where T t . This CMGF solves the llowing boun- dary value problem:

fo

( , 0Y f t t f Af r

,

3.3)

u Yf T Y e (3.4)

where

2

,

ΣY YY

,

1

t

Af Y f tr V f V

J f t

(

of stficients

Y

r nctio ef

E

genen

f t Y

rato f. Co

Y

is the infinitesimal the fu

3.5)

ate variable vector Y applied to 0 and 1 satisfy the followin

g ODEs

:

0 1 1 0 1

0

1

2

1 0

H

0 1

t

J

(3.6)

1 1 1

1

2t 1 1

11 0

H

1 1J

(

3.7)

0 1,0 0 0 u

Proof. The proof is somewhat similar to the proof of Lemma 2.1 To find equations for the coefficients

(3.8)

0

1

0 1 1

1 1

1Σ

2

1 ,

and 1 (3.2) in

, we insert exponential affine solutionsion to (3.3). The resulting expression

expres-

0f

t t

Jt

Y Y

tr V V

Y E e r t Y

Model

The valuation of fixed income securities requires transi-tion from actual to EMM measure. In general, this task can be accomplished by specifying a stochastic discount factor (SDF) for the economy. In continuous time, the SDF,

must hold for all values of the state variables, which re-sults in ODEs (3.6)-(3.8).

3.1. One-Factor Jump-Diffusion Vasicek

tM , places a restriction on a price of a zero- coupon bond according to the following Euler condition

tD

(see [47]):

d d d dD M M Dt t tE E E

D M M D

In a model adapted from [22], the SDF satisfies a jump-diffusion SDE:

dd d d dW J

Mr t W N t

M (3.9)

where W is the price of diffusion risk, J is the price of jump risk, d dN t

cess with intenis a demeaned (compensated)

Poisso sity n pro jump r year. They also assume that the sort rate follows O-U rocess with jumps:

s pe p

d d dr t W J N .

The zero-coupon bond price, being a fun, and the short rate, satisfies the follow

r k

ction of time, ing Ito SDE: t r ,

21d d

2

d , , d

r rr

r

D D k r D D t

D W D r J t D r t N

Define

, ,tq E D r J t D r t and substitute

the above differential into the Euler condi on to obtain the

tifollowing PDE for the zero-coupon bond price:

dtE D

21

2r rr

W r J

D k r D D q

rD D q

(3.10)

d , , dt W r JrD t E D D r J t D r t t

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A. PASEKA ET AL. 147

21

2

1 0

Q

Q

The resulting PDE describes the evolution of the b

Wr rr

J

D k r D Dk

q rD

(3.11)

ond price under EMM. Under this risk-neutral measure the short rate process satisfies

d Q d d dQ Qr k r t W J N

Q W

k

d dQWW W dt

The transformed counting process has new risk-ad- justed intensity 1Q

J at generates ex

. The PDE (3.11) be-longs to a class th ponential-afprices. Thus, we look for a solution to the bonthe following form:

fine bond d price in

, .A B rD r e

Inserting this expression into the PDE, we have

2 21

2

1 0.

Q

QJ

A B r k r B B

B r

where

BJJ B Ee

xpression is an iis the MGF of the jump size. The

above e dentity that ust hold for any values of time and short rate. This requirement leads to se

m

parate ODEs for coefficients A and B:

2 211 ,

2Q Q

JA k B B B (3.12)

1,B kB (3.13)

0 0A B 0. (3.14)

e fact that the bond has face value of $1 at maturity. Subject to (3.14), solutions to (3.12)-(3.13) are given by

The terminal condition (3.14) follows from th

1,

keB

k

2 2

0

1( ) ( ) ( ( )) 1 d .

2Q Q

J

A

k k B s B s B s s

3.2. One-Factor Cox-Ingesoll-Ross MoJumps

[4] develops a general equilibrium

economy with a representative agent pre

he square root SDE7 (in this case, we also augment it with a jump-diffusion component; see also 5]):

del with

model for a standard

with log-utility ferences, and shows that the short rate is a linear

function of the single state variable that drives the eco- nomy. The short rate therefore inherits the same dynamic properties as the state variable and follows t

[1

d d dQ Q Q Qr k r t r W J N d (3.15)

According to the first fundamental theorem of fina e, discounted prices are martingales. Thus, a zero-coupon bond price satisfies the following PDE:

nc

0,QD A D rD

e Qwher A is the infinitesimal generator of the short rate (3.15):

2

2

, , .

Q Q Qr rr

Q Qt

rA D k r D D

E D r J t D r t

The relation between risk neutral and actual measure parameters is based on the CIR pricing kernel:

d ,tdM

d d dW Jr t r W N M

d d d ,QWW W r t

where 1QJ , Q

Wk k and

Q

W

k

k

. Substituting a solution of the form

A Be r

functions

into the above PDE, just like in the one-factor Vasicek model, we obtain two separate ODEs for the

A and A :

1 0,Q Q QJA k B B

22 1

2Qr k B B B

0 0A B 0.

Integrating the two ODEs, we have the solutio for the CIR zero-coupon bond price with jumps in the short rate:

where

n

, ,A B rD r e

A 0

d ( ) 1 ,Q Q QJs k B s B s

2 1

,1 2Q

eB

e k

7We assume here that the process is already under EMM.

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A. PASEKA ET AL. 148

and

2 22 .Qk

3.3. A Two-Factor Vasicek Model with Jumps in the Long-Run Mean

In

of the short e,

this two-factor model, the two state variables are the short rate, r , and the central tendency (long-run mean)

rat . We assume as before that both risks are priced as follows:

dd d d dW J

Mr t W N t

M

where parameters have the same definitions as in (3.19). Under the EMM, the short rate follows a standard Va-sicek model, and the long run-mean of the short rate is given by a pure jump process:

d d QWr k r t Wk

d

and

d d .QJ N

The corresponding PDE for a zero-coupon bond price:

0,QD A D rD (3.16)

2

2

Q Wr rr

Q Q

A D k r D Dk

E

, , ,t D r J t D r t

1 .QJ

Since we are dealing with an exponential affine model once again, we can write the solution as

, , .A B r CD r e

Substituting this solution into (3.16), the PDE breaks down into three separate ODEs for functions A , B , and C :

2 12

QW JA B B C

2

0 1r k C kB

The functions

B B

0 0 0A B C 0

B and C are given by:

1 keB

k

and C B . Moreover, the

function A can be obtained num cally by using eri

2

2 Q

A

B s B s C

Note: When the jump size is normally distributed with mean J , variance 2

J , and MGF

221exp ,

2JC

J JEe C C

by using the following first-order Taylor series appr xi-mation as in [18]8,

o

2211

2J JJCEe C C

the function (3.1 s out to be: 7) turn

2 2

34J J WA

k2 2

2

2 2 3

2 22

3

2

1

2 6

.4

Q

Q QJ W J

QJ J Q

J

Qk J

k

k

k

k

ek

2 2

3 Q Q

k J W Jk k ke

k

4 4 3Qk k

ate solution for bond prices be-comes very useful for calibration, estimation and testing.

4. The Quadratic Model

nterest rate models is that interest rates can become negative (except in CIR- type models). Even though negative interest rates are not impossible, they are rare. This fact has led to the need to develop models of interest rates that guarantee positive rates. The models in this category are numerous.9 How-ever, most of these early models suffer from various problems. e.g., a rational log-normal model of [4 ] suf-fers from calibration problems in that bond prices and short rates are bounded both from below and above. As a result this model is arbitrage free only for a finite period of time. This restriction proves too binding in empirical work. Other ways of guaranteeing positive rates is to trans

ion tools, the two most s are [51,52]. However,

Under the normality assumption of the jump sizes, the closed-form approxim

The common problem with most i

8

form the state variable. Popular class of models that achieve that is log-r models. The short rate is the expo-nent of a state variable tY , tY

tr e , which follows an extended Vasicek process. Historically, due to availabil-ity of easy lattice implementatwidely used models in this clasthis transformation presents theoretical problems. The

0

1 d .2W J s s

(3.17)

8[18] argue that approximation is reasonable because jump sizes are typically small so that JC is expected to be small. 9[48] describes interest rate models based on a price kernel that guaran-tees positive rates. [49,50] provide a more detailed and general descrip-tion of the model. For more detailed discussion of these and other mod-els see [13].

Copyright © 2012 SciRes. JMF

A. PASEKA ET AL. 149

problem with these models is that the expected future value of the risk-free bank account

0

exp dt

t sV E r s

even for arbitrarily small 0t

(see [53]).

4.1. Quadratic-Gaussian Interest Rate Models

Recently, more attention has been paid to a different specification, quadratic-Gaussian, that guarantees posi-tive rates, does not suffer from these problems, and still tractable enough to generate near closed form bond and in some cases even bond option prices. The simplest form of this model is t e so d square Gaussian model in which the short rate is simply the sum of squares of the state variable.10 A more general model is in the Quadratic-Gaussian (QG) class with the short rate a general quadratic form of the state variables, which

h -calle

del the short rate is follow a process with affine drift and constant volatility. In an n-factor Quadratic-Gaussian mo

1,b Y Y cY ( )

2t t t tr a 4.1

where , and . The state vari- n-

a , 1nb n nc ables Y follow an n-dimensional Gaussian process uder a risk neutral measure:

d d Σd ,t t tY Y t W (4.2)

where n n , 1n , Σ n n constant matrix, and 1n

tW is a vector of standard Wiener processes. We can always rewrite the short rate in (4.1) in the fol-lowing form after shifting the state variable tY by a constant vector:

1,

2t t tr a Y cY

The state SDE (4.2) does not change from this trans-formation except for obvious redefinition of parameters. Quadratic-Gaussian models have bee31-35,38,54,55]11 Similarly to affine m

coupon bond pr tion to rm:

n studied by [28, odels, we look for

the zero ice solu (2.10) in the fol-lowing fo

0 1

1, expt tD t Y Ω

2t t tY Y Y

(4.3)

where is matrix. The bond price satisfies wit

Ω.4)

n nh PDE (2 1 and 0u . Us

btain the foing (4.3) in (2.4),

we o llowing PDE:

0 1

1ΩY Y Y

1

21

( ) Ω Ω2

0.

t t t

t tY Y

Y

1 1 1

1

1 1 1Σ Ω Ω Ω Ω

2 2 2

1 1Ω Ω Ω Ω Ω Ω Σ

2 4

1

2

tr Y

Y YY

a Y c

Since this equation has to hold for any valuestate variables, it separates into three independent Riccati

or

s of the

ODEs f 0 , , a1 nd Ω :

0 1 1 1Σ Ω Ω Σ ΣΣ 0,4

tr a 1 1

2

1 1 1

1 1Ω Ω Ω Ω ΣΣ 0,

2 2

1Ω Ω Ω Ω Ω ΣΣ Ω Ω 0.

4c

Moreover, if the matrix is symmetric, the system reduces to

Ω

0 1

1 1

1Σ ΩΣ

21

ΣΣ 0,2

tr

a

1 1 1Ω ΩΣΣ 0,b

Ω 2 Ω ΩΣΣ Ω 0.c

Exampl . r a one 1 Fo e-factor square-Gaussian model of interest rates considered in [6] and [5]:

2 ,t tr Y

d dt tY t aY t W d .t (4.

For this model, the bond prices, as well as bond option prices, have closed form expressions given by the fol-lo

ro coupon bond price is

4)

wing: 1) The ze

2

0 1

, ,

exp , , Ω , ,

t t

t t

D t T Y

t T t T X t T X

where ,t t tX Y

00

d ;t

at ast e r e s s

10The models in this class have been studied by [28-30], among others.11See [56] for a general specification and a discussion of canonical form of a QG model.

2

2 2 20 1, , Ω ,

2

T

t

t T s T s T

d ,s s

Copyright © 2012 SciRes. JMF

A. PASEKA ET AL. 150

22

12

2

, d ,s s

sT

T

tTt

e a at T

e a a

2 1,

T tet T

2 T t a

and

e a

2 2 22 .a

1T 2) The European call option price with maturity e and strik K

s giwritten on a zero-coupon bond with

turity i ven by ma-

2T

1 2 2 1 2

1 3

, , ,

, ,

t t

t t

c T T D t T Y N d N d

4KD t T Y N d N d

with parameters given by the following expressions:

1

1

,,

h vd

C t T

2

1

,,

l vd

C t T

13

1

,,

,

h M t Td

C t T

14

1,C t T

1,v M t T

,,

l M t Td

, 21 1 2 1, ,T T C t T

21 1 21 2 , , ,C t T C T T

1 1 2,,

d T Th

C T T

1 22 ,

1 1 2

1 2

,,

2 ,

d T Tl

C T T

21 1 2

1 2 0 1 2

,

4 , , ln ,

d T T

C T T T T K

21

2

, , , , d ,

2 .

T

t

T t

T t

,M t T N t T X N s T s T s N t T

e

e a a

4.2. Linear-Quadratic Interest Rate Model

The difficulty of fitting term structures of interest rates with diffusion models has led to the study he impact

of jumps in the interest rates on spot rate properties. [25, 26,37,57,58] point out that including mps in a model helps better explain properties of various interest rates. When extending a model in any way, we are also con-cerned about whether the model will still have a closed form solution for bond prices, bond option (cap, floor, swaption) prices, and whether the extension helps im-prove the fit of volatility term structures to tho e implied by swaptions and caps when the simpler model has failed to do so. In the case of including jumps into diffusion models, [59] use three years of interest rate cap price data and show that within a three factor SVJ DTSM signifi-cant negative jumps in interest rates are required to fit implied volatility smile.

[36] argue that most of the previous research is too re-al as-er re-

MM (IS-GMM) procedure of

of t

ju

s

strictive in the way it treats jump intensity. Ususumptions range from constant intensity to a rathstrictive functional form of underlying state variables that may include economic variables as in [37]. [36] intro-duce a new class of interest rate models, linear-quadratic term structure model (LQTSM). The new feature is that jump intensity is now a separate state variable that fol-lows it own SDE. In their empirical implementation, they use essentially a 3-factor model with the state vector that includes the short rate, the random volatility, and the stochastic intensity. The rate on 3-month T-bills serves as a proxy for the short rate. To estimate parameter values [36] relies on generalized method of moments (GMM) approach of [60]. At each iteration of the GMM optimi-zation routine, however, they have to compute the re-maining two latent state variables, volatility and intensity,for every date of their dataset. They do it by resorting to the so-called implied-state G

[61]. For given parameter values at each iteration they use high frequency futures market data on the changes in T-Bill yields to compute conditional variance and kurto-sis. Stochastic volatility and intensity are then identified from these two higher moments. They find that intro-ducing random intensity substantially improves both the model fit (through better fit of short term kurtosis) as well as the dynamics of the interest rate volatility term structure. In a VAR model run on daily data they also establish that jumps are not only associated with macro-economic announcements or FOMC meeting dates but also with unanticipated news.

Below, we show how LQTSM class is designed to have a closed-form solution, and we give a simple exam-ple of a special case closely related to the SVJT model that [36] studies. We start with the extended transform (2.6). LQTSM class is designed (or defined) to have the following exponential LQ form for extended transform12: 12[36] assume that the jump size distribution is Bernoulli taking a posi-tive value μ+ > 0 with probability p and a negative value μ− < 0 with probability 1 − p. We do not make any specific assumption on the jump size distribution and keep the discussion general.

Copyright © 2012 SciRes. JMF

A. PASEKA ET AL. 151

, exp ,

t t t

T

t s Tt

A B Y Y C Y

f t Y E r s Y ds u Y

e

(4.5)

d d Σd dt JY t W J N (4.6)

, r r rr t Y a b Y Y c Y (4.7)

where 1nY is a vector of state variables, 1n , is jump intensity, jump size vector 1nJ with time-homogeneous MGF u J

J u Ee , the mean of the jump size vector 1n

J E J , Σ n n , 1nW is a vector of standard Wiener processes,

Poisson process N is independent of jump size and of the diffusion vector W, ,ra A , 1, n

rb B , and , .n n [36]

ters basedrc C

parame deri on

ves ro

isk-neutwer-utili

ral correctity pricing kern

ons to model p

lowing the approach of [4,38]. We skip this stderivation and assume that all processes are already ex-pressed under an EMM. Extended transform (4.5) must satisfy the following PDE:

el by fol-age in the

,f Af r t Y f 0 (4.8)

, u Yf T Y e )

(4.9

Σ Σ2 YY

1

, ,

J Y

t

Af f

Y E f t Y f t Y

)

where Y is the value ate mediate. Similar to affine JD models, the expectation

in the integro-differential Equation (4.8) should break up into the product of the MGFs for continuous and discon-tinuous parts of the state vector process as these parts are assumed independent in this specification. The only complication is the quadrati

tr f (4.10

of the st vector im ly after a ju

c form in the definition of extend nsform for LQTSM class in (around t complication by assuming tvector ly a subvector of, say, the is disco uous, i.e., follows a JD processing components are diffusi

mp

ed trahis

onntin

n N

4.5). [36] works hat in the state

first 1n components . The remain-

2 1n ons:

1

2

1

10

n

n

JJ

1

2

1

1

1

2

n

n

YY

Y

The extended transform (4.5) must be exponential quadratic only in 2Y :

2 2, tA B Y Y C Yf t Y e (4.11)

In order to obtain separate ODEs for three functions ,A B , and C , the next obvious requirement is

that once we substitute (4.11) into (4.8) th re must be no terms other than affine in Y (i.e.,

eA Y ) and quadratic

in Y2 (i.e., 2Y ). Then, the functional fo Q, in

rms of the short rate, r tensity, , an

Eq

dieo

d dri diffusion coef-uation re dictated by xam cond term in nt of sform and the

ft and (4.6) a

ple, the se the tran

ficients of ththe above PDE (4.8) cojump part is

e tranrequirem

ntaiprop

sitione

ns the ortional

nt. For e

gra t :

2 2 1 2

YY n

1 1 1Y nf Constf

f A Y

These functional forms of the gradienterm jointly imply that the first comto

t and the jump ponents of vec-1

r n

in (4.6) and intensity must be 1A Y and 2Q Y and the last 2 components of vector n must

be 2A Y :

1 1

2 2

1 1 1

1 2

,n J Y

J Yn Y

ff

f

2 2 ,a b Y Y c Y

1 ,k

a b Y c Y 2 kY

2 2 ,k k

d e Y

1 2

Just like

1, , 1, .k n k n

, the short rate (last erm in the PDE) must be

t 1A Y and 2 Q Y :

2 2, ,r r rr t Y a b Y Y c Y

with us d

obvio efinitions for . Further, the trace part of the PDE has the fo :

, ,r r ra b cllowing form

1 1 1 2 2

2 2

11 12

21 22

11 12 22

* * *

Σ Σ ΣΣ Ω

Ω Ω

Ω Ω

Ω 2Ω Ω

YYYY YY

YY

Y Y Y Y Y Y

const f A Y f Q Y f

tr f tr f tr f

tr f

tr f f f

2

where Ω ΣΣ is a symmetric ma ix. The above ex-pression implies that

tr

11 1Ω A Y

and

,Ω ,Ω .Q Y A Y const 2 12 2 22

Substituting (4.11) into PDE (4.8) we have the fol-lowing identity that must hold for all Y :

Copyright © 2012 SciRes. JMF

A. PASEKA ET AL.

Copyright © 2012 SciRes. JMF

152

2 2 2 2C C Y B C C Y C C

11 1 2 1 22 2

2 22

11 1 1 12 1 2 2 2

2 2 1

1Ω 2Ω Ω

2

1

J J J

J r r

Ba a b Y Y c c YB C Cd e Y

tr B B B B C C Y B

a b Y Y c Y B a b Y

2 2 0rY c Y where

Terms independent of , affine in , affine in and quadratic in must each be e o zero inidentity. This restriction lts in a system of four for functions

bA B Y Y C Y

Y

2

2

2

2

11 12

21 22

d1

Ω Σ* var d Σd

d

Ω Ω .

Ω Ω

r

v

rv v r

rv v v v v

r v v

W

Wt

W

v v v

v

v

2

1

2 1n

B

1 1n

BB

Y

resu

1Yqual t

2Y , this

ODEs 2Y

A , 1B , 2B , and C . Foritions

details, the read are

1

er can consult [36]. The initial cond

11 10 0, 0 ,nA B u

We look for a zero coupon bond price in the following form:

2 22 1 20 , 0 0n nB u C 2.n

The zero coupon bond price has the same form as in (4.5) with

1 2 2 2

, ,

exp .t

D t Y

2A B r B Y Y C Y

1 2, 0Nu u u

e (LQTSM of [3 (4.13)

As in [36] we assume that

. Exampl 6]). Assume that the state

ve n mctor consists of short rate, volatility, a d random ju p intensity. These three state variables are assumed to solve the following transition equations:

1 1 1d d dt

t

rY

Y v

C lowing

2 2 1

2 2 1 2

2 2

Ψ d

Ψ

d

t

r rr rJ

r

Y

r Y Y Yt

Y

W

is symmetric. In the Appendix we derive the fol ODEs for functions A , 1B , 2B , and C

1 2 1 12

22 2 2

Ω

1Ω 2 0,

2

r 2A B B B

dN Σ d 0 ,

d 0vW J

W

(4.12)

,

with the following definitions and notation:

0 , ,v

rJJ

1 2

00Ψ ,Ψ ,

00 0vvvv

0 0

Σ 0 0

0 0v

v

d 1

d 1

r rv r

r v

W

W

var d 1 d ,v rv vW t

B

tr B B C

(4.14)

1 1 1 0,rrB B (4.15)

2 1 2 2

1 12 22 21

Ψ 2

02 Ω 2 Ω 0,

1

rJ

J

B B B C

B C C BB

(4.16)

21

1 1 2 22

0Ψ 2Ψ 2 Ω 0,2

0 0

BC B C C C

(4.17)

subject to the following initial conditions:

10 0, 0 0A B , (4.18)

2 2 1 20 0 , 0 0B C 2. (4.19)

Here, 11

B JJ B Ee is the moment generating func-

tion of the jump size. In order to conclude the discussion of LQTSMs, we

derive the price of a European call option from aalized transform using a technique similar to the one used by [39,62,63]. We are looking for the price of a European ca ond. Th

gener-

ll option on a zero coupon b e call has

A. PASEKA ET AL. 153

T t years to maturity and a strike price, K . The ond maturi e, zero-coupon b has ty dat DT

provior LQTSM

, the face valuee de

from

d the current price . We de th riva-

e option price for th [36] given in the above example. We assumdynamics in (4.12) are given under an EMM. The value of the European call option is given by the following risk neutral expectation with respect to (

where

of $1 , antion of th

tDe 3-fact

e that the state

4.12):

exp d ln lnT

t t s T Tt

c E r s D K D K

, (4.20)

is the Heaviside function defined as

In order to handle the truncation under the expectation tegral repre-

1, if

ln ln .0, if

TT

T

D KD K

D K

in (4.20), we use a well known complex insentation of the Heaviside function

ln lndln ln

2πTi D K

T

iD K e

i

Using th we write the call price as (4.21) where

is representation

ln ln, exp d ln ,T

i Kk t s k T

t

e E r s g D

(4.22)

and 1kg k i , 0,1.k To complete the deriva-tion, we use a well-known result from the theory gen-eralized functions:

of

1,

P

iπi

(4.23)

where P

is the principal value of the integral over

1

:

0

lim d .P

Each term in (4.23) is understood as a linear fu

acting on a finite infinitely differentiable function

nctional

:

0lim d

i

0π 0 lim di

0π 0 d ,i

where we use the fact that the principal va e of the inte-gral over

lu vanishes as this function is odd. Sub- 1

tracting 0

from the integrand makes it regular, does

not require principal value specification, and is suitable for numerical integration.

The expression for the call price can now be written as follows:

0 1Π Π ,tc K

where

0, , 0,dΠ ,

2 2π

0

k i

k ,1.

k k k

on

Functi ,k in (4.22) can be viewed as a gen-eralized tr the zero coupon bond yield. Since an

the expectation in (4.22) in th

sform forwe already have the result for the bond price for the LQTSM class, we look for

e exponential quadratic form: (4.24) Here, DT T

at the timis the time remaining to the bond’s

maturity e the option matures. Recall that T t

option as is the tim of today’s

e remaining to the maturity of the date . For a given , the four

functions in (4.24),

t k

,Λm k ( 0,3m ), satisfy the same

Riccati ODEs (11)-(16) as A , 1B , 2B , and C , respectively, and the tions: following initial condi

0,Λ ,k k0 g A 1, 1Λ 0 ,k kg B

2, 2Λ 0 ,k kg B 3,Λ 0 ,k kg C

exp d ln ln exp ln ln

d

T T

t t s T T t

s T Tt t

c E r s D D KE D K

i

d

ln exp (ln ln )

s Tt

T

t s T

r s

K KE r ds i D K

dexp ln ln

2π

t

T

t

iE r ds D i D

i

2π

K

0 1, , ,K i

(4.21)

ln1 2 2 2 2

0, 1, 2, 2 2 3, 2

, exp d

exp ln Λ Λ Λ Λ .

Ti K

k t s k Tt

k k t k k

e E r s g A B r B Y T Y T C Y T

i K r Y Y Y

(4.24)

Copyright © 2012 SciRes. JMF

A. PASEKA ET AL. 154

where ,1.1 , 0kg k i k

umerical analysis of Since n option prices and Greeks for hedging purposes is beyond the scope of this paper, we refer the reader to the following work on this topic. In [64,65], authors compute European option prices and the Greeks. For a model of an underlying asset process they use a theoretically attractive Variance-Gamma process. They estimate gradients of a European call option by Monte Carlo simulation methods. In computing the gra-dients, they compare the efficiency of indirect methods (finite difference techniques) and direct methods (infini-tesimal perturbation analysis and likelihood ratio).

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Appendix

Derivation of Bond Price for Cox-Ingersoll-Ross Model

Proof. The bond price process is

, E exp dT

t

B t T r u u t

.

Because

0 0

exp d , E exp dt T

r u u B t T r u u t

the tower property implies that this is a martingale. The Markov property implies that , ,B t T B r t t T ,

.

Because is a martingale, 0

exp d , ,t

r u u B r t t T

its differential has no drift term. We compute

0

0

2

0

d exp d , ,

1exp d , , d , , d , , d , , d

2

1exp d d d d d d

2

t

t

r rr t

t

r r rr t

r u u B r t t T

r u u r t B r t t T t B r t t T r t B r t t T r t B r t t T t

r u u rB t B r t B r W B r t B t

Setting the drift term to zero, we obtain the partial dif-

ferential equation

2

, , , ,

1, , , , 0

2

t

r rr

rB r t T B r t T

r B r t T rB r t T

The terminal condition is . We look for a solution of the form , where

, , 1, 0B r T T r , , rC t TB r t T e , ,A t T

, 0, ,C T T A T T t t tB rC A B rB C

0B 2

rrB C B. Then we have

, , , so that the partial differential equation becomes

2 210 1

2t trB C C C B A C

.

The ordinary differential equations become

2 211 , , ,

2tC t T C t T C t T 0,

0

d ,

with , and ,C T T

, ,T

t

A t T C u T u

where ,A T T 0 and ,t ,A t T C t T . This is equivalent to solving

2 2 2 2

2 2

2

d ,

2, ,

d .2

C s T

C s T C s T

s

2

Integrating both sides of the above equation with re-spect to s, we obtain

2 2 2 2

2 2

2 2

d , d ,

2 2, ,

2 .

T T

t t

C s T C s T

C s T C s T

T t

Letting 2 2 2 , we have

2

2

,.

,T tC t T

eC t T

Therefore, ,C t T

is given by

2

2 22

2

2 2 )

1,

1

2 1since

2

2 1by letting 2

2 1 1

sinh.

1cosh sinh

2

T t

T t

T t

T t

T t

T t T t

et T

e

e

e

e

e e

T t

T t T t

Derivation of ODEs (4.14)-(4.17).

To arrive at the result (4.14)-(4.17), we need some in-termediate expressions for partial derivatives of the zero coupon bond price given by (4.13) with respect to the state vector:

2 2 22Y D B CY D

Copyright © 2012 SciRes. JMF

A. PASEKA ET AL. 158

2 2 2 2 2 22 2 2Y Y D B CY B CY C D

Insert zero price expression (4.13) in to PDE (4.8) with

2 1 2 22Y r D B B CY D

21

22 2 1

11 1 2 2

0Ω

0 0

rr

rr

D B D

BD v B D Y Y D

and 3 10u : 1

2 2 1 2 11 2 2 2 2

2 2 2 2

21

2 2 12 1 2 2 22 2 2 2 2 21

Ψ

Ψ 2

0012Ω 2 Ω 2 2 2

12 0 0

r rr rJ

J

r Y Y Y BA B r B Y Y CY

Y B CY

Btr Y Y B B CY B CY B CY C Y r

B

0.

The above equation must hold for arbitrary values of

th

e state vector. This condition gives rise to a system of four Riccati ODEs for functions A , 1B , 2B , and C :

1 2 1 12

22 2 2

2

1Ω 2 0

2

r ΩA B B

tr B B C

B B

2

1 1 1 0rrB B

2 1 2 2 1 1

22 21

Ψ 2 2 Ω

02 Ω 0

1

rJ

J

B B B C B C

C BB

21

1 1 2 22

0Ψ 2Ψ 2 Ω 02

0 0

BC B C C C

The system is subject to initial conditions (4.18)-(4.19).

Copyright © 2012 SciRes. JMF