a class of heath-jarrow-morton

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A CLASS OF HEATH-JARROW-MORTON TERM STRUCTURE MODELS

WITH STOCHASTIC VOLATILITY

CARL CHIARELLA AND OH KANG KWON

School of Finance and EconomicsUniversity of Technology Sydney

PO Box 123

Broadway NSW 2007

Australia

[email protected]

[email protected]

ABSTRACT. This paper considers a class of Heath-Jarrow-Morton term structure mod-

els with stochastic volatility. These models admit transformations to Markovian sys-

tems, and consequently lend themselves to well-established solution techniques for the

bond and bond option prices. Solutions for certain special cases are obtained, and com-

pared against their non-stochastic counterparts.KEY WORDS: stochastic volatility, interest rate modeling, Heath-Jarrow-Morton,

bond option

1. INTRODUCTION

The majority of the term structure models prior to Heath, Jarrow and Morton (1992)

were finite dimensional Markovian systems in which the interest rate economy was de-

termined by the spot rate and perhaps one or two additional state variables. This enabled

the use of standard arbitrage arguments, along the lines of Black and Scholes (1973)and Merton (1973), to derive the PDE for the bond and bond option prices which, in

turn, enabled the application of well-developed techniques from the theory of PDEs to

obtain analytic solutions, and numerical solutions in cases where this was not possible.

The progenitors of this approach could be regarded as Vasicek (1977) and Brennan and

Schwartz (1979).

Although these early models were useful from the viewpoint that analytic solutions were

often available, the calibration of model parameters to observed market data was a highly

non-trivial task. In particular, many models could not be calibrated consistently to the

initial yield curve, and the relationship between the model parameters and the market

observed variables were not always clear. Furthermore, it was not always possible to

incorporate observed market features, such as the humped volatility curve, into thesemodels.

By contrast, the Heath, Jarrow and Morton (1992) approach provides a very general

interest rate framework, capable of incorporating most, if not all, of the market observed

features. The HJM models are automatically calibrated to the initial yield curve, and the

Date: First version April 10, 1998. Current version November 1, 1999.

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connection between the model parameters and the market variables often emerge from

the theory.

The main drawback of the HJM framework is that it results in models that are non-

Markovian in general, and consequently the techniques from the theory of PDEs no

longer apply. For the general HJM model, Monte Carlo simulation, which can often be

time consuming, is the only method of solution. To overcome these problems, many

authors, including Carverhill (1994), Ritchken and Sankarasubramanian (1995), Bhar

and Chiarella (1997), Chiarella and Kwon (1998a), Chiarella and Kwon (1998b), andBhar, Chiarella, El-Hassan and Zheng (1999), have considered ways of transforming

the HJM models to Markovian systems. In these transformed systems, the desirable

properties of the earlier Markovian models and the HJM framework coexist, and provide

useful settings under which to study interest rate derivatives.

In the standard HJM framework, the uncertainty in the interest rate market is represented

by Wiener processes that drive the forward rate process. All other processes in the in-

terest rate market, including the forward rate volatilities, are thus also driven by the

same Wiener processes. Consequently, the standard HJM model does not incorporate

additional independent sources of stochastic volatility, as considered in Hull and White

(1987), Heston (1993), and Scott (1997).

The main contribution of this paper is the specification of volatility processes that are

driven by additional Wiener processes that are independent of those that drive the for-

ward rate process in the standard HJM framework. Such models will be referred to as

stochastic volatility HJM models. The stochastic HJM models considered in this paper

transform to Markovian systems, and hence enjoy the benefits enjoyed by such mod-

els. For certain special cases, explicit bond price formulae, in the spirit of Ritchken and

Sankarasubramanian (1995), Inui and Kijima (1998), and Chiarella and Kwon (1998a)

are given, along with numerical examples highlighting the effect of the additional Wiener

processes. The class of models constructed in this paper can, in some sense, be consid-

ered as analogues of the Hull and White (1987) and Heston (1993) stochastic volatility

models within the HJM framework, and provides one way of incorporating stochasticvolatility into the HJM framework, as alluded to in Jarrow (1997).

The remainder of this paper is organised as follows. In

2 the HJM framework is briefly

outlined. The stochastic volatility model is then introduced in a simplified -dimensional

setting in 3, and the general stochastic model is described in 4. Numerical examples

illustrating the effect of stochastic volatility are given in

5, and the paper concludes with

6.

2. HEATH-JARROW-M ORTON FRAMEWORK

In this section, an overview of the general Heath-Jarrow-Morton framework is given.

For further details, the reader is referred to Heath, Jarrow and Morton (1992), Brace and

Musiela (1994), or Musiela and Rutkowski (1997).

2.1. The Risk-Neutral Framework. Let , and assume given a filtered prob-

ability space

satisfying the usual conditions, with filtration

generated by an -dimensional standard -Wiener process

.

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In the standard HJM interest rate framework, the time

instantaneous rate of return on

an investment contracted at time , denoted , is assumed to satisfy the stochastic

integral equation

(2.1)

where

,

, and

represents thedependence of the forward rate process on the Wiener path . For finite dimen-

sional Markovian specialisations of the HJM model, the path (

) dependence simplifies

to dependence on a finite number of state variables such as the spot rate

. Rel-

atively mild regularity assumptions are imposed on

so that the integrals are

well defined, and required manipulations are valid.

The spot rate process, , is obtained by setting in (2.1), so that

(2.2)

The money market account

, representing the time

value of unit investmentmade at time , is given by the equation

(2.3)

Finally, the time price of a maturity zero coupon bond, denoted , is defined

as

(2.4)

The differential forms of (2.1) and (2.2) are easily obtained and have the form

(2.5)

(2.6)

By using the stochastic Fubinis theorem and Itos lemma, HJM showed that

satisfies

(2.7)

where

.

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2.2. Derivative Prices as Conditional Expectations. It follows from Itos lemma and

(2.7) that the discounted bond price process satisfies

(2.8)

and is consequently an

-martingale under the equivalent martingale measure

.

Thus

(2.9)

Substituting for

and

, and imposing the condition

,

yields the important identity

(2.10)

which gives the bond price as the expected value of the discounted payoff. More gener-

ally, if

is a price process for a

-expiry option on ,

, with

payoff

, then

(2.11)

2.3. Feynman-Kac Theorem and Partial Differential Equations. If the interest rate

economy is Markovian, then an application of the Feynman-Kac Thoerem to (2.10) re-

sults in the partial differential equation

(2.12)

for the bond price, where is the infinitesimal generator associated with the Markovian

system resulting from (2.5) and (2.6). There are well-developed theoretical and numer-

ical techniques for solving such equations, and consequently many authors, including

Ritchken and Sankarasubramanian (1995), Inui and Kijima (1998), Chiarella and Kwon(1998a), Chiarella and Kwon (1998b), and Bhar, Chiarella, El-Hassan and Zheng (1999),

have studied conditions under which the HJM model transforms to Markovian systems.

3. ONE DIMENSIONAL HJM MODELS WITH STOCHASTIC VOLATILITY

Being path dependent, the volatility processes in the standard HJM framework are tech-

nically stochastic. However, in the literature (see Hull and White (1987), Heston (1993),

and Scott (1997)), the term stochastic volatility appears to be reserved only for those

volatility processes which are driven by Wiener processes linearly independent of the

Wiener processes that drive the underlying asset price process, or, as in this case, the for-ward rate process. It is in this sense that the standard HJM models fail to be stochastic

volatility models. A good discussion of stochastic volatility models in a non-stochastic

interest rate environment is Heston (1993) and Scott (1997) in a generalised setting.

In order to give a transparent exposition of the main ideas of this paper, the special case

of

-dimensional stochastic volatility HJM models are considered in this section. The

-dimensional generalisation is considered in 4.

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3.1. Embedding Stochastic Volatility in the HJM Framework. In the case of the

-dimensional HJM model, recall from (2.5) that the forward rate process evolves ac-

cording to the stochastic differential equation

(3.1)

Separable volatility processes of the form

(3.2)where is a deterministic function, have been considered by numerous authors (Ritchken

and Sankarasubramanian (1995), Inui and Kijima (1998), Bhar, Chiarella, El-Hassan and

Zheng (1999)), and have been shown to generate a useful class of Markovian interest

models in which a closed form formula for the bond price is available. It must be noted

that an additional assumption such as must be made in order to

obtain a Markovian model, although the bond price formula remains valid in the absence

of such assumptions. For example, in the Vasicek type model the additional assumption

, where

is a constant, is made, while for the CIR type model, the

assumption made is

.

Stochastic volatility may be introduced into the standard HJM model in several ways,

but the method adopted in this paper is to assume a volatility process of the form

-

(3.3)

where

is a vector of a finite set of fixed

tenor forward rates with

, and are deterministic

functions, and

is a stochastic process satisfying

(3.4)

where is a constant, , , and are deterministic functions, and is a

standard Wiener process such that

(3.5)

Appropriate choice of the parameter - allows the volatility or variance to be modeled as

a stochastic process. An example of such a volatility specification is

with

satisfying

(3.6)

where

,

,

, and

are constants.

To apply the techniques of Heath, Jarrow and Morton (1992), it is convenient to replacethe correlated Wiener processes and with uncorrelated processes, and it is

easily seen that

and

, defined by

(3.7)

(3.8)

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have this property. Inversion of the pair of equations above yields

(3.9)

(3.10)

and substituting (3.10) into (3.1) gives

Now define

(3.11)

(3.12)

(3.13)

Then (3.1) can be rewritten as

(3.14)

which is the starting point in Heath, Jarrow and Morton (1992), for a -dimensional

model with volatility processes

. Assuming the existence of market prices of

risk

, , associated with the two sources of uncertainty, and following

the arbitrage arguments of Heath, Jarrow and Morton (1992), the forward rate process is

seen to satisfy

(3.15)

where

, and

are standard Wiener processes under the equivalent martingale measure

. The gen-eral framework developed in Heath, Jarrow and Morton (1992) applies verbatim to the

stochastic volatility model introduced in this section, and in particular, expressions such

as (2.2), (2.6), and (2.10) remain valid for the stochastic volatility model.

3.2. Markovian System. The volatility process of the form (3.3) satisfies the Markov-

ian condition given in Inui and Kijima (1998) and Chiarella and Kwon (1998 a), and,

consequently, the corresponding HJM model transforms to a Markovian system. Since

the transformation in the case of stochastic volatility has not been considered in previous

literature, the argument is briefly outlined below.

Define state variables

and

by

(3.16)

(3.17)

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and for

define

(3.18)

(3.19)

Note that and .

Lemma 3.1. The variables and satisfy the equations

(3.20)

(3.21)

where

(3.22)

(3.23)

Proof. See Appendix A.

Since the volatility process (3.3) is a function of forward rates

,

,

the equations governing the evolution of the forward rates must also be computed. For

this, note that, from (3.15), the stochastic integral equation for

is

Lemma 3.2. The stochastic differential equation for the forward rate is

(3.24)

where

.

Proof. See Appendix B.

Proposition 3.3. Let the market price of risk

be a function of

,

,

,

and

, where

. Then the set

forms an

-dimensional Markovian system.

Proof. From Appendix C, the state variables and satisfy the equations

and from (3.4),

satisfies the equation

(3.25)

Recall that

.

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Now, if

is a function of

,

,

, and

, then together with (3.2),

these equations determine a Markovian system having

, , , and

as the state variables.

Note the dependence of the drift term in (3.25) on the market price of risk

. This is

analogous to the corresponding situation in the stochastic volatility models of Hull and

White (1987), Heston (1993), and Scott (1997), and arises from market incompleteness,

which is in turn a result of the absence of a traded asset related to volatility.

3.3. State Variables and as Functions of Forward Rates. The economic

significance of the state variables and is not at all clear from (3.16) and (3.17).

In this subsection, it is shown that they can, in fact, be expressed as linear combinations

of forward rates, and hence a useful interpretation of the state variables in terms of

economically meaningful variables is established.

Setting in (2.1), and using Lemma 3.1, the equation for the forward rate

can be written

(3.26)

For notational convenience, let

and

(3.27)

Then (3.26) can be written

(3.28)

An important feature of (3.28) is that

and

are deterministic functions, and

so, for any , (3.28) gives the value of

, and hence the value of , as

linear combination of the state variables

and

.

Another useful feature of (3.28) is that it can be used to express the state variables

and as linear combinations of a finite set of forward rates. For this, fix two tenors

. Then setting

in (3.28), for

, gives rise to the system

(3.29)

Inverting this system of equations, the state variables and can be written as

linear combinations of forward rates

and

in the form

(3.30)

(3.31)

3.4. Pricing Partial Differential Equation. Since the stochastic volatility model intro-duced in this section is Markovian, the Feynman-Kac Theorem can be applied to obtain

the pricing PDE for interest rate contingent claims. Recall that if the state variables

,

, for a Markovian system satisfies

(3.32)

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where

, then the associated infinitesimal generator is given by

(3.33)

where

(3.34)

For the case at hand, the state variables are given by Proposition 3.3, and the equations

determining the drift and diffusion coefficients are given in Lemma 3.2 and Proposi-

tion 3.3. To clarify the above discussion, the infinitesimal generator in the case where

and

is given below. For this case, the only forward rate contained in the

set of state variables is the spot rate, and the relevant equations are

(3.35)

where

The infinitesimal generator is then given by

(3.36)

and the pricing PDE for a contingent claim is given by

(3.37)

subject to appropriate boundary conditions.

3.5. Bond Price as a Function of the State Variables. The bond price formula for

HJM models driven by separable volatility processes was obtained by Ritchken and

Sankarasubramanian (1995) for the one dimensional case, and extended to the general

case by Inui and Kijima (1998). The formula was then further generalised by Chiarellaand Kwon (1998a) to forward rate dependent volatility processes. In this subsection, a

brief outline of the derivation of the bond price formula is given. Note that from (2.4)

and (3.15), the price of a

-maturity bond is given by the equation

(3.38)

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From Appendix D,

(3.39)

where

, and consequently

(3.40)

Note that the state variables and can be replaced by forward rates, as shown in

3.3.

3.6. Pricing of European Bond Options. To compute the price of a

-maturity

European option on a -maturity bond, with payoff

, the system of stochastic

differential equations (3.35) must be solved numerically for the values of state variables

,

,

, and

, and the discount factor

, at option maturity

. The bond price

can then be computed using (3.40), and this, together with

the payoff function

, determines the option price

. Repeating

the simulation a suitable number of times, and taking the average, then gives the value of the European option. Results from implementing this procedure are given in 5.

4. GENERAL HJM MODELS WITH STOCHASTIC VOLATILITY

In this section, the introduction of stochastic volatility into the

-dimensional HJM

model, presented in 3, is extended to the general -dimensional HJM framework.

4.1. Embedding Stochastic Volatility in the HJM Framework. Recall from (2.5)

that in the

-factor risk-neutral HJM framework, the instantaneous forward rate process

evolves according to the equation

(4.1)

where

are independent standard Wiener processes. Let be a positive integer,

and fix a sequence

. Define a vector of forward rates

by

(4.2)

As in the -dimensional case, to introduce stochastic volatility into the HJM model, the

are assumed to be of the form

-

(4.3)

where

and

are deterministic functions, -

, and

satisfy

(4.4)

Note that since , the variable is redundant.

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where

,

, and

are deterministic functions, and

, for

. It is

further assumed that

if

if (4.5)

where

,

, and

for all

. Then the Wiener processes,

,

defined by

(4.6)

(4.7)

are independent, and inversion of the the above system yields

(4.8)

(4.9)

Substituting (4.9) into (4.1) gives

Now define

(4.10)

(4.11)

(4.12)


(4.13)

which is the starting point in Heath, Jarrow and Morton (1992) with volatility processes

. Assuming the existence of market prices of risk

,

, and

following the arguments of Heath, Jarrow and Morton (1992), one obtains

(4.14)

where

and

are

standard Wiener processes under the equivalent martingale measure

. The results ob-

tained in Heath, Jarrow and Morton (1992) remain valid for the stochastic volatility

model introduced in this section modulo the obvious modifications, viz. replacing

by

and

by

.

This restriction is imposed purely to simplify exposition. The analysis in this section remains valid

for any correlation structure.

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4.2. Markovian System. The volatility processes given by (4.11) and (4.12) satisfy the

Markovian condition given in Inui and Kijima (1998) and Chiarella and Kwon (1998a),

and the corresponding HJM model transforms a finite dimensional Markovian system.

Since the method for transforming the model to a Markovian system closely parallels the

method employed in

3, the proofs are kept brief.

For

, define variables

and

by

(4.15)

(4.16)

and for define

and

by

(4.17)

(4.18)

(4.19)

Lemma 4.1. The variables

and

can be expressed in the form

(4.20)

(4.21)

where

(4.22)

(4.23)

Proof. See Appendix A.

Lemma 4.2. The stochastic differential equation for the forward rate is

(4.24)

where

.

Proof. See Appendix B.

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Lemma 4.3. The state variables

and

satisfy the stochastic differential equa-

tions

(4.25)

(4.26)

Proof. See Appendix C.

Proposition 4.4. Let

be a function of

,

,

, and

, where

and

, for each

. Then the set

forms an

-dimensional Markovian system.

Proof. From (4.4), the SDE governing the evolution of

with respect to Wiener

processes

is

(4.27)

Since

are assumed to be functions of the state variables, the result now follows

from Lemma 4.2 and Lemma 4.3.

4.3. State Variables

and

in Terms of Forward Rates. As in the -dimensional

case, the state variables

and

can be expressed in terms of a finite number of

fixed tenor forward rates.

Setting

in (2.1), and using Lemma 4.1, the equation for the forward rate can

be written

(4.28)

Let

and

(4.29)


(4.30)

Once again, since

and

are deterministic functions, (4.30) gives the value

of

, and hence the value of

, as linear combination of a finite number ofstate variables

and

.

Now, fix tenors

. Then setting

in (4.30), for

, gives rise to the linear system

(4.31)

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where

...

...

...

Inverting this system of equations, the state variables

and

can be expressed as

linear combinations of forward rates

,

.

4.4. Bond Price as a Function of the State Variables. From Appendix D, the price of

a -maturity bond, in the stochastic volatility HJM model introduced in this section, is

(4.32)

where

. Note from

4.3 that the state variables

and

can be expressed in terms of a finite number of fixed tenor forward rates, and so the

stochastic volatility model of this paper falls under the exponential affine class of models

in the sense of Duffie and Kan (1996).

5. NUMERICAL EXAMPLE

In this section, a special case of the general stochastic volatility framework introduced in

4 is considered to illustrate the effect of stochastic volatility on the spot rate, the bond

price, and the European call price.

5.1. Model Specification. The model used for numerical simulation in this section is

the

-dimensional model of

3, with volatility process given by

(5.1)

where

and

are constants. The dynamics of

is restricted to be of the form

(5.2)

where , , and are constants.

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5.2. System of Stochastic Differential Equations. The model specified by (5.1) and

(5.2) is a -dimensional Markovian system with state variables , , and

. The system of SDEs governing the dynamics are:

(5.3)

(5.4)

(5.5)

5.3. Parameters and Simulation. The set of parameters used for the simulation are as

follows:

(5.6)

Antithetic variable method was used to compute

(i) time

distribution of the spot rate and the

-year bond price,

(ii) time price of an at-the-money -month call option on the -year bond,

for various correlation coefficients

and the volatility of volatility

.

5.4. Distribution of Spot Rate and Bond Price. The effect of varying the correlation

between the Wiener process driving the forward rate process and that driving the sto-

chastic scaling factor is illustrated in Figure 5.1 for the distribution of the spot rate, and

Figure 5.2 for the distribution of the bond price.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.0492 0.0494 0.0496 0.0498 0.05 0.0502 0.0504 0.0506 0.0508

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

FIGURE 5.1 . Distribution of Spot Rate with

and Varying

The effect of varying the volatility of volatility,

, on these distributions is illustrated in

Figure 5.3 and Figure 5.4.

It is interesting to observe from Figure 5.1 and Figure 5.2 that increasing

from negative

to positive values tends to skew the spot distribution to the left and the bond distribution

to the right. Both distributions tend to peak around zero correlation between the two

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0

0.01

0.02

0.03

0.04

0.05

0.06

0.87 0.8705 0.871 0.8715 0.872 0.8725 0.873

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

FIGURE 5.2. Distribution of Bond Price with and Varying

0

0.01

0.02

0.03

0.04

0.05

0.06

0.0492 0.0494 0.0496 0.0498 0.05 0.0502 0.0504 0.0506 0.0508

0.0

0.2

0.4

0.6

0.8

FIGURE 5.3 . Distribution of Spot Rate with and Varying

Wiener processes. When there is positive correlation between the two Wiener processes,

Figure 5.3 and Figure 5.4 show that an increase in volatility of volatility skews the spot

rate distribution to the left and the bond price distribution to the right.

5.5. Call Price as Function of

and

. The effect of varying the correlation

andthe volatility of volatility on the price of a -month call option on a -year bond is

illustrated in Figure 5.5 and Figure 5.6 respectively.

Figure 5.5 shows that the call option value increases with increasing correlation , and

this is to be expected given the skewing to the right of the bond price distribution in this

situation. Similarly, Figure 5.6 shows that the call option value increases with increasing

volatility of volatility.

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0

0.01

0.02

0.03

0.04

0.05

0.06

0.87 0.8705 0.871 0.8715 0.872 0.8725 0.873

0.0

0.2

0.4

0.6

0.8

FIGURE 5.4. Distribution of Bond Price with and Varying

6e-005

8e-005

0.0001

0.00012

0.00014

0.00016

0.00018

0.0002

-1 -0.5 0 0.5 1

FIGURE 5.5. Call Price against with

6. CONCLUSION

A fairly broad class of forward rate volatility processes within the HJM framework has

been considered in this paper. These forward rates depend on a set of fixed tenor forward

rates, time dependent quantities, and stochastic quantities driven by Wiener processesindependent from those driving the forward rate dynamics.

It is shown how the stochastic dynamics of the resulting system can be reduced to a Mar-

kovian form, and that many of the subsidiary state variables introduced by the Markovian

reduction procedure can be expressed in terms of a set of fixed tenor forward rates. The

only non-market traded quantities in the stochastic dynamics are the stochastic volatility

quantities.

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0.0001

0.00012

0.00014

0.00016

0.00018

0.0002

0 0.2 0.4 0.6 0.8 1

FIGURE 5.6. Call Price against with

It is possible to obtain an explicit formula for the bond prices in this framework so thatMonte Carlo simulation is required only for the calculation of option prices. This fact

makes option pricing feasible with the class of stochastic volatility models presented in

this paper.

Some numerical results have been given indicating how the level of the volatility of

volatility, and the correlation between the noises driving the forward rates and the sto-

chastic volatility, affect the spot rate, bond price, and European call option values. These

calculations indicate the computational feasibility of the approach developed in this pa-

per, at least as far as off-line calculations are concerned. No doubt further research

on numerical methods tailored to these stochastic volatility models would also make

feasible calculations in trading time.

It has also been shown how the models developed in this paper may be viewed as a

subclass of the exponential affine class of interest rate models considered by Duffie and

Kan (1996).

APPENDIX A. PROOF OF LEMMA 3.1 AN D LEMMA 4. 1

Consider firstly Lemma 4.1. It follows easily from definition (4.3) that

Using this identity,

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Next, for

, let

(A.1)

so that

,

. Then for

,

and likewise

Addition of the previous two equations yields the required identity

Lemma 3.1 is obtained by setting .

APPENDIX B. PROOF OF LEMMA 3.2 AN D LEMMA 4.2

Consider the more general case of Lemma 4.2, and, for notational convenience, write for

each

(B.1)

(B.2)

Then the integral equation for the forward rate process can be written

(B.3)

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Assuming volatility processes of the form (3.3), and differentiating w.r.t.

(B.4)

(B.5)

for

. Similar computations establish

(B.6)

(B.7)

These equations combine to yield the expression

and the result follows from Appendix A. Setting

yields Lemma 3.2 as a specialcase.

APPENDIX C. PROOF OF LEMMA 4.3

The key identity

(C.1)

follows from (4.3). Now, first consider (4.25).

by (C.1)

Next consider (4.26). From (B.1) and (B.2),

(C.2)

and from (B.4)-(B.7),

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APPENDIX D. PROOF OF (3.39) AN D (4.32)

Recall the definitions of

and

from Lemma 3.1, and define

(D.1)

Then

and for ,

Analogous computation yields

Now, using (A.1) and Fubinis Theorem

for

, and similar computations yield

Addition of the previous two equations implies

21


22/22

Summation over

results in the equation

and hence

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