term structure models ii: fixed-income derivatives...

Term Structure Models II:Fixed-income Derivatives Pricing

Pierre Collin-DufresneUC Berkeley

Lectures given at Copenhagen Business SchoolJune 2004

Contents

1 Bond Option pricing in the Gaussian case 4

1.1 Zero-coupon Bond option pricing in the Gaussian model . . . . . 4

1.2 Three interpretations of the Forward measure . . . . . . . . . . . . . . 5

1.3 Closed-form solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Coupon-bond option pricing (Jamshidian) . . . . . . . . . . . . . . . . 9

2 Bond-Option Pricing in the Affine Framework 10

2.1 Zero-coupon bond option Pricing: Fourier Transform Approach. 10

2.2 Coupon bond option Pricing: Cumulant Expansion Technique 12

2.2.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2 The Three-Factor Gaussian Model . . . . . . . . . . . . . . . . . . 18

2.2.3 The Two-Factor CIR Model . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.4 Appendix For Cumulant Expansion Technique . . . . . . . . 24

1

2.3 General affine Jump-diffusion models . . . . . . . . . . . . . . . . . . . 29

3 Bond and Forward Rate models: The HJM approach 32

3.1 Absence of arbitrage and Equivalent Martingale Measure . . . . 33

3.2 Summary of the HJM approach: Fitting the term structurewithout ‘tricks’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 One-factor HJM Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.1 The Gaussian Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.2 A one-Factor HJM Model with affine volatility structure . 42

3.3.3 Hedging in one-factor HJM Model with affine volatilitystructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Two-Factor HJM Model with ‘Unspanned’ Stochastic Volatility 45

3.5 Pricing Bond Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6 Markov Representation and Existence . . . . . . . . . . . . . . . . . . . 50

3.7 The Multi-factor affine case . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Extending the affine framework to HJM and Random field mod-els 60

4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Traditional Affine Framework . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Time-Inhomogeneous Models . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4 Random Field or ‘String’ Models . . . . . . . . . . . . . . . . . . . . . . . 65

4.5 Generalized Affine Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.6 Relative Pricing of Caps and Swaptions . . . . . . . . . . . . . . . . . . 68

4.7 Forward bond model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2

4.8 Optimal Portfolio Choice and Preferred Habitat . . . . . . . . . . . . 71

4.9 Preferred Habitat and Predictability in Bond Returns . . . . . . . . 74

4.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3

1 Bond Option pricing in the Gaussian case

1.1 Zero-coupon Bond option pricing in the Gaussian model

A big advantage of affine models is their tractability for derivative pricing.We illustrate this within the Gaussian (Vasicek) model with the pricing ofzero-coupon bond options and coupon bond options.

The call option pays at � �� . Its price is

��

� ��

�This expectation can be solved four different ways:

1. Monte-Carlo Simulations.

2. Numerical Integration (e.g., Gaussian Quadrature).

3. PDE technique (e.g., finite difference scheme).

4. Closed-Form.

We will emphasize the last two approaches. Noticing that ��

is a �-martingale, we find the PDE that option prices must satisfy:

�

��

�� (1)

s.t. appropriate boundary conditions. This can then be solve using stan-dard approaches such as finite difference. In the present case, a closed-from solution can be found by solving the expectation directly, using theso-called Forward-Neutral change of measure (Jamshidian (1991), ElKaroui and Rochet (1989)). Define a new measure �� by the

4

likelihood ratio � � ��

� ��

�� . Note that ��

�� . Thus

� ��

��

and��

��

Also Bayes rule for conditional expectation gives

��

�

��

1.2 Three interpretations of the Forward measure

� Change of Numeraire.

Consider pricing a security that pays $X at date � . By definition of therisk-neutral measure we have for � � � � � :

��

��

� ��

��

��

��

��

��

�In other words the value of the security expressed using the money marketaccount �

� �� as a numeraire is a martingale under the � measure.

Performing the forward neutral change of measure we see that:

��

� ��

��

� � ��

��

��

�

�We see that under the �� forward neutral measure the security ex-pressed using the zero-coupon bond price with maturity � as a numeraireis a martingale.

� Correction for correlation

5

Consider pricing a security that pays $X at date � . By definition of therisk-neutral measure we have:

��

��

� ��

��

��

� ��

��

��

� ��

�where we have used the definition of covariance. If � is independent ofthe risk-free rate we obtain:

� � ��

��

� ��

��

However, in general, � is not independent of the short rate so this isnot valid. The forward measure change allows to obtain almost the sameexpression:

��

The measure change accounts for the correlation.

� Making forward rates martingales

The forward neutral measure deserves its name because forward rates withmaturity � are martingales under �� (i.e., the convexity bias disap-pears). Furthermore forward bond prices � �� are martingales forany maturity � �.

To prove the first,

��

� ��

To prove the second, recall � ��

. Thus

��

�

��

� � ��

� � ��

6

��

��

� � ��

Note that while the local expectation hypothesis holds under the risk-neutral measure, forward rates are not equal to expected future spot ratesunder �. Indeed,

��

��

� � ��

��

� � ��

Thus, in general when interest rates are stochastic forward rates are biasedpredictors of future spot rates under both � and �.

1.3 Closed-form solution

Back to option pricing:

��

��

��

� � ��

��

We are left with computing the probability of the option finishing in themoney under two different equivalent measures, e.g., �� for� � �� . Girsanov gives:

��

� � ��

��

��

��

�� (2)

Furthermore,

��

��

� � ��

��

�

�

��

��

��

��

� �

�

��

��

�

�

��

� � ��

��

�

�

��

��

��

��

��

��

�

� ��

��

�

��

��

�

�

7

Thus,

��

��

� �

� ��

��

�

��

� ��

Where we have defined:

��

� �

�

��

��

��

��

��

��

��

��

��

��

��

The final solution is thus:

��

��

��

��

��

��

��

��

��

Remarks:

� The structure is very similar to the Black and Scholes option for-mula except that the volatility is that of the log forward bond price(�� ).

� Following BS the natural candidate replicating portfolio involves twozero coupon bonds with maturity � � and � . However, this is notnecessary.

� Since we only used the fact that the diffusion of bond prices is de-terministic to get the closed-from solution above, a similar result willhold for any model with deterministic bond price diffusion (e.g., anyGaussian multi-factor model).

8

� For the special case of the Vasicek Model, � � �� ,thus �

�� and the option price is a function of the sole

state variable: �� . This insight was used further byJamshidian to derive a closed-form solution for coupon bond options.

1.4 Coupon-bond option pricing (Jamshidian)

A coupon bond option has a payoff at � of ��

�� . Noting that ��

� �� is decreas-

ing in �� we can define �� as the unique solution such that �� .Clearly, the option will be exercised if and only if �� . Thus, we canrewrite

��

��

�� (3)

� ��

��

�� (4)

�

��

�� (5)

Effectively the price of the coupon bond option can be rewritten as a sumof zero-coupon bond options with different underlying and strikes givenby �� . Of course, this does not contradict the general re-sult that a portfolio of options is worth more than an option on a portfolio(here the characteristics of the options in the portfolio are different).

9

2 Bond-Option Pricing in the Affine Framework

One of the nice characteristic of the Gaussian one-factor framework is thatwe obtain closed-from solution for all bond option prices. We show in thissection that the tractability extends to the multi-dimensional affine case,which allows for very efficient ‘almost’ closed-form solution for fixed-income derivative prices such as zero-coupon bond options and couponbond options.

We consider for this section a general �-factor affine model of the termstructure by a vector of Markov processes ��

�� whose dynamics

are such that the instantaneous drifts and covariances are linear in thestate variables. Further, the instantaneous short rate is defined as a linearcombination of the state variables:1

�� Æ�

��

Æ�

��

Within an affine framework, we have seen previously that bond pricespossess an exponentially-affine form:

� � ��

� ��

��

��

�� (6)

where the deterministic functions �� and �� satisfy a system of

ordinary differential equations known as Ricatti equations.

2.1 Zero-coupon bond option Pricing: Fourier Transform Approach.

For illustration, consider a call option on a discount bond with maturity� . The payoff of a European bond-option (or caplet) with exercise date �

1Duffie, Pan and Singleton (2000) provide the precise technical regularity conditions on the parameters for the SDE to bewell-defined. Dai and Singleton (2000) classify all � factor affine term structure models into � � � families depending onhow many state variables enter into the conditional variance of the state vector. Our approach is valid for each of these familiesof models including the USV models covered previously.

10

is��

��

�� (7)

The price of the bond-option at an earlier date-� before expiration can bewritten:

�� E �

��

� ��

��

��

�� E

�

��

� ��

��

�� E

�

��

� ��

� � ��

�� E�

�

��

�� E�

�

��

�(8)

� ��

��

�

�� (9)

where in going from the second line to the third line we have transformedfrom the risk-neutral measure to the so-called forward measures (Jamshid-ian (1991), El Karoui and Rochet (1989)) by using the relation:

E��

��

��

��

�� E�

�

��

��

�

�� (10)

Here, E� �� denotes expectation under the � -forward measure, whichtakes as numeraire the bond price whose maturity is � .2

It is convenient to introduce the Fourier-Stieltjes transform of �� :3�

�

��

�

��

� ��

�

��

� ��

�� !�

��"#� �

(11)where!�� is the characteristic function of the random variable ��

under the � -forward neutral measure. Following the insight of Hes-ton (1993), Duffie, Pan and Singleton (2000) we can use Levy inversion

2The Radon-Nikodym derivative is given by � � ��

� ��

�� , and the conditional likelihood ratio is given by

� ��

� ��

�� for � � � .

3To provide some intuition note that ��

�

��

��

��

�.

11

(Williams (1991)) to find

��

�

$

� �

�

�#Re

��!�

��"#�

" #

��

We may thus express bond option prices as:4

��

�

�

$

� �

�

�#Re

�� !�

��"#�

" #

��

��

�

�

$

� �

�

�#Re

�� !�

��"#�

" #

��(12)

The implication of equation (143) is that if the characteristic functionsof equation (11) can be written in closed-form, then so can the bond-option price. Note that the characteristic function is an expectation of anexponentially-affine function. Thus in the affine framework, it is itself anexponential affine function of the state variables �� (exercise).Thus computing zero-coupon bond options in a finite-dimensional affinemodel basically amounts to evaluating a single one-dimensional numer-ical integral, the Fourrier inversion, which analogously to the classicalBlack and Scholes formula can be compute in a fraction of a second.

2.2 Coupon bond option Pricing: Cumulant Expansion Technique

This section draws heavily on Collin-Dufresne and Goldstein (2001) AEuropean swaption gives its holder the right to enter a swap at some fu-ture date �� % �. As such, a swaption is readily interpreted as an optionon a coupon bond, where the strike is equal to the nominal of the con-tract, and the coupon rate is equal to the swap rate strike of the swaption.5

4The inverse Fourier transform technique is widely used. Duffie, Pan and Singleton (2000) provide a comprehensive expo-sition, examples and further references.

5Alternatively, a swaption can also be interpreted as a sum of options on the swap rate that must be exercised at the samedate (e.g., Musiela and Rutkowski (1997)).

12

In this section we propose a very accurate and computationally efficientalgorithm for pricing swaptions in a general affine framework.

The date-� price of a swaption with exercise date-�� and with payments�

on dates �

" � �� & and strike price � is given by6

Swn�� E

�

��

� ��

��

��

��

��

�(13)

��

��

�� E �

��

� ��

��

� ��

��

�� E

�

��

� ��

��

� ��

�(14)

��

��

��

��

��

�� (15)

where we have defined �� to be the date-�� price of the coupon bond:

��

��

�� (16)

In going from equation (14) to equation (15), we have transformed fromthe risk-neutral measure to the so-called T-forward measure (Jamshidian(1991), El Karoui and Rochet (1989)) by using the relation:

��

�� % ��

��

� ��

� � ��

��

� E��

�� (17)

where E� denotes expectation under the � -forward measure, which takesas numeraire the bond price whose maturity is � .7

For each of the �& �� relevant forward measures,8 we estimate the prob-ability distribution of the date-�

�price of the coupon bond. We do this

6Here we price a call option on a coupon bond which is identical to a receiver swaption (e.g., an option to enter a receivefixed pay floating swap) when the strike is set to par and the coupon to the strike (rate) of the swaption. Similarly a payerswaption could be priced as a put option on a coupon bond (or by put-call parity).

7The Radon-Nikodym derivative is given by � � ��

� ��

�� , and the conditional likelihood ratio is given by

� ��

� ��

�� for � � � .8From equation (14), it follows that �� forward measures are of interest: � � corresponding to the exercise date � � ,

and � �� corresponding to the payment dates of the coupon-bond.

13

by determining the first � moments of the distribution. That is, foreach of the " � �� & forward measures, we determine the first' � �� moments of ��: E

�� . Note that for any ',

�� can be written as a sum of terms, each involving a product of 'bond prices:

��

��

��

� �

�

��

� �� (18)

Since all bond prices possess an exponential-affine structure, equation (18)takes the form

��

��

��

� �

�

��

��

��

��

(19)where the functions � �� and !

�� are sums of the �� and �

�� func-

tions defined above. Note that �� depends only on the state vari-ables�

�� in an exponentially-affine manner. This implies that the date-

� expectation of �� also possesses an exponentially-affine solution:

E��

��

��! �� (20)

where the deterministic functions �� and (�� satisfy a set of Ricatti

equations.

After obtaining the population moments of �� under each forward mea-sure, we estimate ��

�� % �� using a cumulant expansion of the distri-

bution of ��. Cumulants are defined as the coefficients of a Taylor seriesexpansion of the logarithm of the characteristic function. In other words,defining !� � �

��) �

��" ��

�)� as the characteristic function of therandom variable ��, the cumulants ��

are defined via:

��!� ��

��

�

�" ��

*�� (21)

14

The +�# order cumulant is uniquely defined by the first + moments of thedistribution (See, for example, Gardiner (1983)). As a reference, the firstseven cumulants are provided in the Appendix.

Armed with an explicit expression for the cumulants we can obtain theprobability density �

�� of �� by inverse Fourier Transform:

��

�)� �

��

$

��

��

� ��"!� � � (22)

We can then make use of our cumulant expansion for the characteristicfunction to obtain:

��

�)� �

��

$

��

��

� ��" �� (23)

�

��

$

��

��

� ��" ��

��

� � (24)

�

��

$

��

��

� ��" ��

� �� (25)

where � � ��

��

�� . Up to this point, the solution is exact. Theapproximation comes in when one truncates the Taylor series expansion� ��

��

� . Keeping all terms up to order �, we find:

��

�)� �

��

$

��

��

� ��

�

��"��

��

� � , � (26)

In the Appendix we provide the coefficients �, �� for the case� � �. We note that this expansion is ‘density preserving’ in that, toany order�,

��

��

�)��) � �.

This expansion results in a sum of simple integrals which can easily besolved by noting that:�

�

��

��

��

��

� ��

��

��

��

��

��

��

��

��

� ��

��

�

15

�

��

�

��

��

��

��

��

�

� ��

�

� ��

�

��

� ��

��

�� (27)

where the last line defines the coefficients - .

The probability density can then be written

��

�)� ��$��

��

�

��

��

.�� ) � ��

�� (28)

where

.��

��

, - �

�� (29)

The coefficients .�� are provided in the Appendix for the case � � �.

To price a swaption with strike �, we need to compute the date-0 prob-ability that �� will fall above the strike price. That is, we need tocompute the integral: � �

�

�)��

�)� �

��

.�#

(30)

where

#� ��$�

�

� �

�

�) � ��

�

�� )

Note that all #� can be solved in closed-form and involve, at worst, theone-dimensional cumulative normal distribution function, for which thereexist standard numerical routines that do not require any numerical inte-gration. We have thus obtained a very simple expression for the probabil-ity of the coupon bond price being in the money. It involves only simple

16

summations. In the Appendix we present the expressions for the coeffi-cients .�� #� for * � �� and � � �.

The swaption can then be written as:

Swn��

��

��

� ��

��

��

��

��

� ��

��

��

�

(31)

where . ��

� � #��

� are the various coefficients computed and each �-

forward-neutral measure with an approximation of order�.

2.2.1 Numerical Results

In this section we present numerical results on the speed and accuracyof our proposed approach. Since the approach is model-independent, asingle program can be written for all models, needing only a call to asubroutine for each specific model.

Below, we consider two models: a three factor Gaussian model, and a two-factor CIR model. We choose � � � for the order of expansion, sinceit appears to offer an excellent compromise between speed and accuracy. 9

For both cases we compute prices of swaptions for various strikes andcompare them to Monte-Carlo simulated prices for accuracy. Note thatthe normalized highest order cumulant provides a good estimate of theattained accuracy. We also list the CPU time required for our approxima-tion.

9In fact, for the examples considered we need only calculate up to the fifth cumulant. That is, we find that we can set ��

and��

to zero without significantly affecting the numerical results. Note, however, that this is not the same as using a � �-thorder approximation. Indeed, there are other terms which show up in the � expansion which are products of lower-ordercumulants (e.g., terms proportional to ��

and � �). Hence, our expansion is not an � � expansion.

17

2.2.2 The Three-Factor Gaussian Model

Here we consider the three-dimensional Gaussian model, with the follow-ing state variable dynamics:

�/� �

/��

�0�

� (32)

where �0��0�

� 1

��, and � � Æ

�� /

.10 The bond prices take theform (Langetieg (1980)):

� � �2� /��

�� $

�� (33)

where

�� %�

(34)

�� Æ� ��

1

��

��

��

(35)

where we define 1�� .

Under the � -forward measure, the state variables have the dynamics

�/��

��/�

��

1��

��

�� 0�

� (36)

The expectation of products of bond prices at some future date can becomputed using the expression for the Laplace transform of the state vari-able under the forward neutral measure:

�� E��

��

�� $��

�� &��

�� $�� (37)

where 3�& are given by:

&�� !

��%�

3��

1

!

��

�� % ��

��

1!

!

�

10In fact, it can be shown that this �� model is ‘maximal,’ in the sense of Dai and Singleton (2000).

18

Parameters$�� $�� $�� Æ %� %� %� '� '� '� (�� (�� (��.01 .005 -.02 .06 1.0 0.2 0.5 .01 .005 .002 -.2 -.1 .3

Table 1: Parameters chosen for the Gaussian three factor model numerical results.

These formulas allow us to readily compute all the moments of the couponbond price at the maturity date ��. We can thus compute the relevant

cumulants (see the Appendix) and the parameters to .�

� #

�

to be used in

the formula 31 above. We list the selected parameter values in Table 1.Figures 1 and 2 show respectively the absolute and relative deviation ofour approximation relative to a Monte-Carlo solution. The Monte Carloprices are obtained using the exact (Gaussian) distribution of the statevariable at maturity to avoid any time discretization bias. The numberof simulations is set to obtain standard errors of order �� (2,000,000random draws with standard variance reduction techniques). As the figureshows our approximation is excellent: The absolute error relative to thetrue solution is less than a few parts in ��. The relative error is verysmall: less than a few parts in �� with the biggest errors for highly outof the money options, which have negligible values, thus making this typeof metric somewhat misleading. Again our approximation takes less than�� seconds to compute all �� swaption prices (corresponding to differentstrikes).

Another advantage of the Edgeworth expansion approach is that the orderof magnitude of the error term can be predicted by looking at the ‘scaled

cumulants’

��

��

�11 In Table 2, we present the mean, variance, and

the 3��-5�# scaled cumulants for each of the (N+1)=21 measures. Twonotable features are apparent from the table. First, the scaled cumulantsdecay quickly, which provides an indication of the appropriateness of theEdgeworth expansion approach. Further, it also provides an estimate of

11That the scaled cumulants are the appropriate measures for estimating the error can be seen from the Appendix. Seeequations (54)-(58).

19

Figure 1: Difference between cumulant approximation and Monte-Carlo swaption prices for various strike prices. Theparameters are as in table 1 above. Monte-Carlo are run using the exact (Gaussian) distribution of the state variable at maturityto avoid a time discretization bias, and the standard error of the Monte-Carlo prices are less than � ��.

Figure 2: Relative difference between cumulant approximation and Monte-Carlo swaption prices for various strike prices.The parameters are as in table 1 above. Monte-Carlo are run using the exact (Gaussian) distribution of the state variable atmaturity to avoid a time discretization bias, and the standard error of the Monte-Carlo prices are less than � �� .

20

measure mean variance ��

��

��

��

��

��

0 1.291480 0.00073789 0.01166506 0.00036705 0.00001361 1.291569 0.00073801 0.01166509 0.00036705 0.00001362 1.291647 0.00073810 0.01166513 0.00036706 0.00001363 1.291714 0.00073819 0.01166515 0.00036706 0.00001364 1.291773 0.00073826 0.01166518 0.00036706 0.00001365 1.291826 0.00073833 0.01166521 0.00036706 0.00001366 1.291872 0.00073839 0.01166523 0.00036706 0.00001357 1.291914 0.00073844 0.01166525 0.00036706 0.00001368 1.291951 0.00073849 0.01166527 0.00036706 0.00001379 1.291985 0.00073853 0.01166528 0.00036707 0.0000136

10 1.292015 0.00073857 0.01166530 0.00036707 0.000013611 1.292042 0.00073861 0.01166531 0.00036707 0.000013612 1.292066 0.00073864 0.01166532 0.00036707 0.000013513 1.292088 0.00073867 0.01166533 0.00036707 0.000013614 1.292108 0.00073869 0.01166534 0.00036707 0.000013515 1.292126 0.00073871 0.01166535 0.00036707 0.000013616 1.292142 0.00073873 0.01166536 0.00036707 0.000013617 1.292157 0.00073875 0.01166537 0.00036707 0.000013718 1.292170 0.00073877 0.01166537 0.00036707 0.000013619 1.292182 0.00073879 0.01166538 0.00036707 0.000013620 1.292193 0.00073880 0.01166538 0.00036707 0.0000136

Table 2: Mean, variance, and scaled cumulants for the 20 forward measures and the risk-neutral measure for the 3-factorGaussian model.

the truncation error. Indeed, at the rate at which the scaled cumulants aredecaying, one can guess that the ��# scaled cumulant, and hence the error,is indeed of the order of ��. Second, the 5�# scaled cumulants are nearlyidentical across measures. Hence, for time efficiency, one only needs tocalculate the 5th scaled cumulant for a single measure.

To investigate whether these results are specific to the Gaussian case wenow apply the same approach to a second example where the state vari-ables do not follow a Gaussian process.

21

2.2.3 The Two-Factor CIR Model

We choose a standard two-factor CIR model of the term structure. Thespot rate is defined as � � Æ/

�/

�where the two state variables follow

independent square root processes:

�/�

��

� /

� ��

�/�0�

� (40)

where the Brownian motions are independent. Bond prices are a simpleextension to the original CIR bond pricing formula:

� � �2� /��2�� /��2�� $��$�� (41)

where

�� Æ �

��

�

�

.�

� � �

�

��

��

.

��)� � �� .

.

��(42)

��

��)� � ��

� .

��)� � �� .

� (43)

and where we have defined .��

.

¿From equation (41), we note that products of bond prices (with differingmaturities) will take the form:

� ��

�� $��$�� (44)

As in the Gaussian case, we can compute (for all relevant measures) themoments of the distribution of a coupon-bond by noting

�� E��

�� $��$��

�(45)

��

�� E�

�

��

�� $��$��

�(46)

��

�� E�

�

��

� ��

��$��$��

��

��(47)

22

where � � � � �� , !�� !

�

�� . It is well known

that the solution to this expectation takes the form:

��

�

�� &��$��$�� (48)

where the functions 3�&�� &� satisfy the Riccati Equations

�& �

� �� &

�

& �

(49)

3 � � �Æ ��

�&

� (50)

with ‘initial conditions’ &�� !

�, 3�� . We find

�� Æ � �

��

��

��

�

� � ��

��

� �

��

�� !��

��

��

� !�

"��

��(51)

��

��

��!��

��

"��

�!��

��!��

��!��

�� (52)

where we have defined .��

� #

�� %�)'�

.

We can thus determine the relevant cumulants (see the Appendix) and pa-rameter inputs �.�

� #

�

that are needed to price the swaption via equa-

tion (31). The selected parameter values are provided in Table 3. Fig-ures 3 and 4 show respectively the absolute and relative deviation of ourapproximation relative to a Monte-Carlo solution. The Monte Carlo pricesare obtained using a standard Euler Discretization scheme of the SDE.To reduce the time discretization bias we choose a very small time step:�� . The number of simulation is set to obtain standard er-rors of order less than �� (e.g. 5,000,000 paths with standard variancereduction techniques).12 As the figure shows, our approximation is ex-cellent. The absolute error relative to the true solution is less than a few

12We also used a third pricing approach, a standard numerical integration technique, with similar results, thus not reported.

23

Parameters$�� $�� Æ %� %� *� *� '� '�0.04 0.02 0.02 0.2 0.2 0.03 0.01 0.04 0.02

Table 3: Parameters chosen for the two factor CIR model numerical results.

parts in ��. The relative error is very small less than a few parts in ��

with the biggest errors for highly out of the money options which havenegligible values thus making this type of metric somewhat misleading.Our approximation takes less than �� seconds to compute all �� swaptionprices (corresponding to different strikes).

In Table 4, we present the mean, variance, and the 3��-5�# scaled cumu-lants for each of the (N+1)=21 measures. Note that the third cumulantis now negative. This can be understood as follows: under the squareroot process, higher interest rates lead to higher volatility, in turn leadingto an upward skew in interest rates, which produces a downward skewfor (coupon) bond prices. Also note that the cumulants do not decay asquickly as in the Gaussian case, leading to a slightly larger error for thiscase.13 Finally, note that the 5th scaled cumulants are not as similar asthey were in the Gaussian case. As such, for numerical efficiency onecan choose to compute only two of them, corresponding to the shortestand longest forward-measure maturities, and then estimate the others viainterpolation as a function of forward-measure maturity.

2.2.4 Appendix For Cumulant Expansion Technique

� Relation between Cumulants and momentsFor reference, here we provide the first seven cumulants ��

, in terms of

the moments �4. The formula that relates cumulants and moments can

13Note that it would be appropriate to go to the � � level, even if we still set � � � �� to zero. Indeed, one can expecta contribution of the order of the third scaled-cumulant to the third power, divided by 3!, which is of the order of �� .Note, going to higher orders of is computationally very inexpensive – it is determining the higher order moments which iscomputationally costly and grows exponentially in the order.

24

Figure 3: Difference between cumulant approximation and Monte-Carlo swaption prices for various strike prices. Theparameters are as in table 3 above. Monte-Carlo are run using �� paths and setting �� . The standard errorof the Monte-Carlo prices are less than � ��.

Figure 4: Relative difference between cumulant approximation and Monte-Carlo swaption prices for various strike prices.Theparameters are as in table 3 above. Monte-Carlo are run using �� paths and setting �� . The standard errorof the Monte-Carlo prices are less than � ��.

25

measure mean variance ��

��

��

��

��

��

0 1.23511950 0.00160541 -0.04087912 0.00205713 0.000126431 1.23530353 0.00160333 -0.04089828 0.00206201 0.000125822 1.23546977 0.00160145 -0.04091559 0.00206642 0.000125263 1.23561989 0.00159976 -0.04093121 0.00207041 0.000124764 1.23575544 0.00159823 -0.04094532 0.00207401 0.000124295 1.23587782 0.00159684 -0.04095805 0.00207725 0.000123886 1.23598829 0.00159560 -0.04096954 0.00208019 0.000123517 1.23608800 0.00159447 -0.04097991 0.00208284 0.000123178 1.23617799 0.00159346 -0.04098927 0.00208523 0.000122859 1.23625919 0.00159254 -0.04099772 0.00208739 0.00012259

10 1.23633246 0.00159172 -0.04100534 0.00208933 0.0001223411 1.23639857 0.00159097 -0.04101222 0.00209109 0.0001221012 1.23645821 0.00159030 -0.04101842 0.00209268 0.0001219013 1.23651202 0.00158969 -0.04102402 0.00209411 0.0001217214 1.23656055 0.00158915 -0.04102907 0.00209540 0.0001215515 1.23660433 0.00158865 -0.04103363 0.00209656 0.0001214016 1.23664382 0.00158821 -0.04103774 0.00209761 0.0001212817 1.23667944 0.00158781 -0.04104144 0.00209856 0.0001211618 1.23671156 0.00158745 -0.04104479 0.00209941 0.0001210319 1.23674053 0.00158712 -0.04104780 0.00210018 0.0001209620 1.23676666 0.00158683 -0.04105052 0.00210088 0.00012085

Table 4: Mean, variance, and scaled cumulants for the 20 forward measures and the risk-neutral measure for the two-factorCIR model.

26

be found in, for example, Gardiner (1983).

��

��

��

� �

� �

� � �

��

��

� � �

� ��

��

��

��

��

� ��

��

��

� ��

��

��

��

� � ��

��

��

� � ��

��

��

��

��

� �

��

� ��

� ��

� � ��

��

�

��

�

��

��

� ��

� ��

� ��

� ��

� The coefficients in the approximation of order � � �

� ��

��

��

��

��

��

�

��

� �

��!� ��

��

��

!��

��

�

�� !�

��

�

��

�� !�

��

��

Define " ��

��

�� "� �# � �� . Then, the probability density can then

be written

� �"� ��

��

�

��

��

��!�

� (53)

where

��

��

��

� "� � �"

�(54)

��

��

��

� "� � �"� �

�(55)

��

��

��

� "� � ��"� ��"

�(56)

��

��

��

�

�

��

��

��

��

�� "� � ��"� ��"� � ��

�(57)

��

��

��

��

��

��

��

�� " � ��"� ��"� � ��"

� (58)

27

For pricing options, we eventually want to integrate this density above some strike price� . Defining � � ��

�, we have:� �

��

��

��

� �

��

��

�

� ��

� �� (59)

�

��

��

(60)

All of these terms can be written, at worst, in terms of the cumulative normal function, forwhich there are excellent approximations to, without the need of numerical integration.The first seven are:

�� N

��

�(61)

��

��

�

� ��

�

��

��

�(62)

��

�N

��

�

��

�

� ��

�

��

��

��

(63)

� �

��

�

� ��

�

��

��

��

�

�(64)

��

�N

��

�

��

�

� ��

�

��

��

��

��

��

(65)

��

��

�

� ��

�

��

��

��

��

��

�

�(66)

��

�N

��

��

�

��

�

� ��

�

��

��

��

��

��

��

��(67)

��

��

�

� ��

�

��

��

��

��

��

��

��

�

� (68)

The relevant coefficients ��

for the ��

are obtained by collecting terms of the samepowers in equations (54)-(58). They are

��

�

��

��

��

��

��

��

��

�

��

��

�(69)

��

��

��

��

��

��

��

��

��

��

��

� ��

��

�(70)

28

��

��

��

��

��

��

��

��

�

��

��

�(71)

� �

�

��

��

��

��

��

��

��

��

��

��

� ��

��

�(72)

��

�

��

��

��

��

��

��

��

�

��

��

�(73)

��

�

��

��

��

��

��

��

� ��

��

�(74)

��

�

��

��

��

�

��

��

�(75)

��

�

� �

��

��

� ��

��

�(76)

Remark: Alternative approaches are:

� Piece-wise linear approximation of the exercise boundary combinedwith using Fourier inversion of Umantsev and Singleton (2003).

� Wei (1997) for single-factor case and Munk (1999) for multi-factorcase show that one can approximate the coupon bond option price bythe price of a zero-coupon bond option with maturity equal to stochas-tic duration of CIR (1981).

� Monte-carlo simulation.

2.3 General affine Jump-diffusion models

Below we present the more general framework of the Affine Jump-Diffusionmodel of Duffie, Pan and Singleton (2000).

Fix a filtered probability space �� and assume � is aMarkov process in some state space ( � �

that solves the followingSDE:

�� 4�� 5� (77)

29

where � is a vector of + independent Brownian motions, and � is a purejump process with jump size distribution 6 on �

and jump intensity�#�� , and

� 4�/� � �� / for � � ��

� ��/��/�� 7�� 7�� /, for 7 � �7�� 7��

�

� #�/� � 8� 8� /, for 8 � �8�� 8��

� ��/� � Æ� Æ� /, for Æ � �Æ�� Æ��

Under some technical condition given in DPS, the transform

��9��

� �� +��

�has a closed form solution given by:

��9� /� �� ,��-��$

where :� , satisfy the complex-valued ODEs:14

:�� Æ� �� :��

�

:��7�:�� 8��:�� (78)

,�� Æ� �� :��

:��7�:�� 8��:�� (79)

with boundary conditions :�� 9 and ,�� and where we havedefined the ‘jump transform’ ��

��

��.�6�0�.

Most option pricing requires solving the following expectation:

!/0�)��

� �� /��

#�$��

�14Note that since % is a tensor, &�%& is a vector with ��' element

�� &�&�%��

30

This can be done by noting that the Fourier-Stieltjes transform of ! isgiven by:�

�

��"�!�)��

� �� /��0��

�(80)

� ��- ";<�� (81)

Since we have a closed-form for the latter, we can find! by Levy-inversion.Under some technical conditions:

!/0�)�� -��

��

$

� �

�

Im��- ";<�� "

�;

�;

The above outlines the approach in a quite general framework. DPS(2000) provide several applications to equity and fixed-income deriva-tives.

31

3 Bond and Forward Rate models: The HJM approach

Assume uncertainty is generated by �-dimensional vector of independentBrownian motions� on a standard filtered probability space ��

where the filtration is the natural filtration generated by the Brownian mo-tion.

We assume a family of zero-coupon bond prices �� haveprocesses following well-defined SDEs:

��

� � �� 4�

��

�� (82)

Then by definition we obtain:

�� 4�(��

(�� (83)

�(��

�

�� (84)

4�(��

��

�

�� 4

�

�� (85)

and (if it exists) the short rate follows:

�� 4�� (86)

�� (�� (87)

4�� 4(�� (88)

Proof: Since � � �� Ito gives:

��

��

� � ��

��

��

��4�

��

��

��

��

32

�

Alternatively, had we started from a family of forward rates satifying (83)then bond prices would satisfy (82) with:

��

� �

�

�(��2 (89)

4��

� �

�

4�(��2

�

��

�

�(��2�� (90)

Proof: Since � � �� Ito gives

��

� � �� =� �

��=� = ��

where =� �� 2.

�=� �

��

� �

�

4�(��2

��

��

�

�(��2

��

The result follows. �

Let us emphasize that the relations between the drift and diffusions offorward rate and bond prices derived in equations ?? are simple conse-quences of Ito’s lemma and the defining relation between forward ratesand bond prices. There is no economic content in these relation.

3.1 Absence of arbitrage and Equivalent Martingale Measure

Consider a self-financing portfolio of � zero coupon bonds with differentmaturities: � ��

�� +��

��. Where we define � �� as

the value of investing $1 and rolling it over at the instantaneous risk-free

33

rate (the money market account), and � �� " � � Define thedollar amount invested in each security as $� � +��

�. Ito gives:

��

�$��

��

$��4��

��

��

$��

��

�� $��4� ��

�� $�� (91)

where we define $� 4�� respectively as the vector/matrix of stacked $ �� 4��

�� . Suppose we can pick the portfolio to be locally risk-free, thenby absence of arbitrage it should earn the risk-free rate of return:

$�� $��4� ��

In words, any vector $ orthogonal to all the column vectors of � is alsoorthogonal to the vector of excess returns. Standard algebra then showsthat the excess return vector must be in the span of �, i.e. � .� s.t. 4 �� .. Or alternatively

4��

��

.�� " � ��

Plugging this into the SDE for bond price gives:

��

� ��

��

��

where we have defined �� .��. If the market price ofrisk satisfies additional technical restrictions, then it defines an equivalentso-called risk-neutral probability measure for bond prices. Indeed, under� defined by Girsanov’s theorem, � � �� is a martingale.

A few remarks are in order:

� In general it is assumed that the diffusion matrix, �, of bond pricesis well-behaved (e.g. invertible). This implies that all sources of risk(brownian motions) that drive bond prices can be perfectly hedged by

34

selecting a sufficiently large number of bonds with different maturi-ties. In that case, the martingale measure is unique and markets arecomplete (i.e., all fixed income derivatives can be replicated by anappropriate trading strategy involving only bonds). We will see laterthis is not necessarily the case.

� The equivalent martingale measure is defined by the Radon-Nykodim

derivative: �� )��1��

� ��

�� )��

��.This defines an equiv-alent probability measure if and only if ��

�� . Liptser and

Shiryaeve provide necessary and sufficient conditions on . for whatthey call the diffusion case (theorem 7.19 p. 294)). For the generalcase, a sufficient condition is the so-called Novikov condition.

� A priori, in our argument above the equivalent probability measuremay depend on the choice of the � maturities selected. In fact, thechoice of maturities may change over time (Bjork et al (1995)andHeath, Jarrow and Morton (1992) discuss this point further). How-ever, once such a martingale measure is selected it should, in princi-ple, price all other bond prices. In other words, the price processes of� reference bonds define a risk-neutral measure and all other bondscan be perfectly replicated by an appropriate trading strategy involv-ing only these reference bonds (in the complete markets case at least).

The importance of finding an equivalent martingale measure resides in the“fundamental theorem of asset pricing” which states that the existenceof an EMM precludes the existence of arbitrage opportunities. Indeedconsider any self-financing portfolio � ��

�� +

�� then 2 ��

�

� ��

��

� �� +

��2��

��.

Using the EMM we have15 � �� 2 ��

�. And we see that if we

15Technically, we also need to further restrict the trading strategy to those that are martingale generating, i.e. we want thestochastic integral to keep the martingale proprety of the integrator. See Dybvig and Huang (1989)

35

have an arbitrage strategy such that � �� and �� % �� % � then� �� % � (because � � �).

Now using the EMM we obtain the HJM-restrictions for bond prices andforward rate processes under the equivalent martingale measure.

Bond prices are given by:

��

� � ��

��.��

�� (92)

� �� (93)

Using 4��

��.�� in (83)-(85) we obtain:

��

��

�� .��

�

�� (94)

� ��

��

��

�

�� (95)

Alternatively, had we started from a family of forward rates. Using (89)we would have:

�� (��

��

�

�(��2 .��

��

(�� (96)

� �(��

� �

�

�2�(��

(�� (97)

and bond prices would satisfy (82) with:

��

� �

�

�(��2 (98)

4��

� �

�

�(��2.�� (99)

The above relations were first derived in full generality by Heath Jarrowand Morton (1992). They show that absence of arbitrage imposes strong

36

restrictions on the drift of forward and bond prices. The most strikingresult is the fact that under the EMM Q-measure, the drift of forward ratesis entirely specified by its diffusion term. Of course, this is equivalent tothe more traditional result that the drift of bond prices be equal to theshort term rate under Q (the local expectation hypothesis holds under theQ measure).

This HJM restriction implies that given a specification of forward rate (orbond) volatilities, once we have fixed the drift of � arbitrary maturities for-ward rates (or bond prices) under the historical measure, then the marketprice of risk vector, ., is fully determined. In turn, the martingale measureand hence all forward rates (or bond prices) are uniquely determined bythe no arbitrage condition (and initial term structure).

Another often confusing point, is that these restrictions must be satisfiedby any arbitrage-free model. In that sense all (reasonable) models areHJM models. However, market practice has widely associated the termHJM-model to designate models that have (i) the ability to fit the observedterm structure at some particular date perfectly, and (ii) model the dynam-ics of forward rates as a starting point (instead of bond prices or futuresprices for example). We will perpetuate this (unfortunate) tradition.

The ‘HJM model’ as presented above is an ‘empty shell’. It is the weakestrestriction one must impose on a term structure model for it to be sensi-ble. To obtain more implications for derivative pricing we need to imposemore structure on the model.

37

3.2 Summary of the HJM approach: Fitting the term structure with-out ‘tricks’

We have shown above that all term structure models should satisfy theHJM restriction. In that sense all models are HJM models. However, fol-lowing a now standard terminology we define in the following as ‘HJM-models’ all models that (i) fit the term structure, (ii) specify the dynamicsof forward rates as primitives. Such models are designed to price deriva-tives relative to observed bond prices. Starting from:

�� 4�(��

(�� (100)

��

� � �� 4�

��

�� (101)

where �� . The definition � � ��

� �� implies

��

� �

�

�(��2 (102)

4��

� �

�

4�(��2

�

��

�

�(��2�� (103)

Assuming a family of bond prices �� is traded and provided�

(or equivalently (

is well-behaved) we have many more securitiesthen needed to hedge the � sources of risks (Brownian motions). Picking� bonds with different maturities and the money market will, in general,be enough to hedge all sources of risk. In turn, this implies the existenceof a market price of risk .�� such that

4��

��.�� (104)

Provided . verifies some regularity conditions (such as the Novikov con-ditions) then ��

� � �� .�� defines a brownian motion under � �� . Under � we have

��

� � ��

�� (105)

38

�� 4�(��

(��.��

(�� (106)

� �(��

� �

�

�(��2��

(�� (107)

Indeed combining 104, 102 and 103 we obtain

��

� �

�

�(��2�.��

� �

�

4�(��2

�

��

�

�(��2��

and differentiating with respect to � we obtain

4�(��

(��.��

(��

� �

�

�(��2�

As such the model is an ‘empty shell.’ We need to impose more structureon the model to go further. All the action to price derivatives comes from(i) the number of factors driving the term structure �, (ii) the specificationof the volatility structure of forward rates �

(�� which entirely determines

the dynamics of forward rates under the risk-neutral measure, and (ii) theinitial forward rate term structure:

� � ��

� ��

��

�

�

��

��

��

� �

�

��

��

�

�(108)

3.3 One-factor HJM Models

3.3.1 The Gaussian Case

We first investigate a simple deterministic volatility structure for forwardrates. Consider the one-factor model:

�� 2� � ��-�2� � ��2� � ��2 �-�2� � � �0�

��2� (109)

39

where �0��2� is a regular Brownian motions, -�2� �� is deterministic, and

we have defined ��2� � � �� -�2� 9��9.

The drift of the forward rate dynamics under the risk-neutral measure isdetermined by the HJM restriction.

Since bond prices are defined by

� � �2� � �!

��

�

�;� ��2�

�� (110)

Ito’s lemma implies the bond price dynamics are

�� 2�

� � �2�� 2� ��2� � ��0�

��2� � (111)

Since there is only a single Brownian motion �0��2� in equation (111),

all bond price innovations are perfectly correlated. For general functions-�2� � � it is not possible to obtain a Markov representation for the modelproposed above. However, as we show below in a more general case, it isstill possible to price derivative securities such as bond options.

While we are able to provide closed-form solutions for a large numberof derivatives, we cannot in general propose simple algorithms to pricepath-dependent instruments such as American options. In this sectionwe show that by specializing further the choice of -�2� � � we can ob-tain a Markov representation of the term structure and hence price all as-sets using partial differential equations techniques. Indeed, if we choose-�2� � � � �!�� 2�� in equations 109, then the short rate is one-factor Markov and bond prices are exponentially affine in ��. All fixed-income derivatives are solutions to a partial differential equation, subjectto appropriate boundary conditions.

Proof: Integrating the forward rate dynamics we obtain:

�� = �� (112)

40

where we have defined

��

� �

�

�2 ��%��

� �

�

�0��2� ��%�� (113)

= ��

� �

�

�2 ��%�� (114)

Applying Ito’s lemma we obtain the dynamics of ��, = ��:

��

��

� ��

�� 0

�� (115)

�= ��

��

� = ��

�� (116)

Using equation (112) we obtain the dynamics of the short-term rate:

��

�� = ��

�� 0

�� (117)

Since � is deterministic, it is clear that �� is one-factor Markov.

More generally, the forward rates may be written as:

� �� %�� %�� = �� (118)

Thus bond prices satisfy:

� � ��

��

�

�$ % ��

�(119)

� ��

��

�

�$ % ��

�� & ��

�(120)

� ��

��

�

�$% ��

�% �� '�� & ��

��

�� & ��

��(121)

Finally, let us consider a path-independent European contingent claimthat has a payoff at time � that is a function of the entire term structureat time � , i.e. >�� >

��

. The price of that security

is >�� !��

��2��2�>��

�� where the second

equality follows from the Markov-property. Moreover a standard argu-ment (which requires some regularity conditions on � and its derivatives,

41

see Duffie Appendix E p.296) shows that �� is a martin-

gale and that its drift must vanish, or equivalently��

E�

��

��

Using Ito’s lemma we obtain the partial differential equation for the priceof the European contingent-claim:

� � �� = ��

�

�� (122)

�

Note that

� In this model all path-dependence is captured by the stochastic in-tegral 5� �

� ��

�%��0�� 2� which follows a one-factor Markov

Gaussian process: �5� � �5�� .

� Note that it is straightforward to generalize the approach to functionsof the form -�2� � � � -�� -�2� (as we show below in a multi-factorversion).

� The model is identical to the extended Vasicek (Hull and White)model studied previously (compare short rates).

� European zero-coupon and coupon bond option prices can be derivedin closed-form as shown in section 1.1 and 1.4 above.

3.3.2 A one-Factor HJM Model with affine volatility structure

We extend the previous model to allow for level dependence in volatilityas follows:

42

�� 2� � -�2� � ��2� � � ��2� �2 -�2� � ��

��2� �0��2� (123)

where:��2� � ?� ?� �� (124)

In other words the volatility is driven by the short rate. In that case, fol-lowing the previous derivation one can show that if -�2� �� -�2��-��

the short rate becomes two-factor Markov in �� =�. Indeed, =� is notdeterministic anymore and becomes an additional state variable. Sincethe analysis is similar and this model is nested in the one presented nextwe do not derive the results.

The results can be extended to more general diffusion functions

� Ritchken and Sankarasubramaniam (1995) show that two-factor Markovrepresentation obtains for

(�2� ��

�%��3. This is a partic-

ular case of Cheyette (1995).

� �(�� %�

/�%� �� is also of the form analyzed above and leads

to one-factor Markov representation (Rebonato (1998)). It allowshumps in the volatility structure, but leads to non-stationary volatilitystructure (exercise).

� �(�� is stationary, allows for hump-shaped

volatility structure, but requires two state variables for Markov repre-sentation of the term structure (exercise).

� A slight extension of the above shows that if �(�� is a polynomial of

order + then an + � Markov representation obtains (exercise).

� Keeping the affine (square root form), we could model �� as linear in(1) constant maturity forward rates (�� , (2) fixed maturity for-

ward rates (� �� , (3) integrals of forward rates� ��

�� 2 and still

43

obtain closed-from solution for derivatives even in the non marko-vian case. With separable -�2� ��, a Markov representation would alsoobtain.

� The affine (square-root) diffusion allows as we show below to derivethe Fourier transform of the log of bond prices in closed-form andthus by inversion get derivative prices.

3.3.3 Hedging in one-factor HJM Model with affine volatility struc-ture

All one-factor (Brownian motion) models, whether or not they are Markov,require a single bond and the money market to hedge any fixed-incomederivative.

For the particular examples analyzed above where a Markov represen-tation exists in terms of �� =� and where =� is locally deterministic,hedging is simple to implement.

Consider again a path-independent European contingent claim that hasa payoff at time � that is a function of the entire term structure at time� , i.e. >�� >

��

. The price of that security is

>�� !��

��2��2�>��

�� = �� where the sec-

ond equality follows from the Markov-property.

Since markets are complete any contingent claim can be replicated by aself-financing admissible trading strategy �,� �� with corresponding port-folio �� ,��

��

� ��. Thus �� >� -�2�. Using the Risk neutralmeasure and the fact that the strategy is self-financing, we have:

,��

� ��

��

� ��

� ��

>��

�� = ��

� ��

44

. Applying Ito and matching diffusion terms we find:

�� =��

�� =��

And the self-financing condition gives:

,� �� =�� =��

��

The hedging portfolio corresponds to the relative sensitivity to the solesource of (instantaneous) risk of the asset and the hedging instrument.

3.4 Two-Factor HJM Model with ‘Unspanned’ Stochastic Volatility

We now present a two-factor model with affine volatility structure whichnests the one factor Vasicek and affine model. It is interesting because

� it remains tractable as it allows to price derivatives in a non-Markovianframework,

� it accommodates level-effects in volatility, item it captures the notionof a volatility-specific factor that cannot be hedged with bond prices(USV).

We also show that with some further restrictions on the functions -�2� � �

we obtain a simple Markov representation of the term structure. For thatcase we consider existence issues in greater detail.

Consider the two-factor model:

�� 2� � -�2� � ��2� � � ��2� �2 -�2� � ��

��2� �0��2� (125)

��2� � "��2� �2 <�

��2��1 �0�

��2�

�� 1� �0�

��2�

��(126)

45

where �0��2� and �0�

��2� are independent Brownian motions, and we have

defined

"��2� � @��

@��2� @�

��2� (127)

��2� � ?� ?��2� ?� ��2� (128)

��2� � � ��

�

�; -�2� ;� � (129)

The drift of the forward rate dynamics under the risk-neutral measure isdetermined by the HJM restriction.16

Since bond prices are defined by

� � �2� � �!

��

�

�;� ��2�

�� (130)

Ito’s lemma implies the bond price dynamics are

�� 2�

� � �2�� 2��2� � �

��2� �0�

��2� � (131)

Since there is only a single Brownian motion �0��2� in equation (131),

all bond price innovations are perfectly correlated. However, the Brow-nian motion �0�

��2� affecting volatility dynamics in equation (126) will

generate innovations in fixed-income derivatives that cannot be hedgedby portfolios consisting solely of bonds.

As specified, the model leads to very general dynamics for the term struc-ture of forward rates, and hence also for the risk-free rate. In particular,when the functions -�2� � � are chosen arbitrarily, the dynamics of the thesystem

�� 2��

will in general be non-Markov.17 Regardless, the

16As noted in Andreasen, Collin-Dufresne and Shi (1997), the HJM restriction alone does not identify the process of � underthe risk-neutral measure, since the Girsanov factor associated with )

�cannot be identified from changes in bond prices alone.

To determine the market price of risk associated with volatility-specific risk )�, either the prices of other interest-rate sensitive

securities in addition to bond prices must be taken as input to the model, or some equilibrium argument must be made.17Cheyette (1995) demonstrates that a sufficient condition for the forward rates to be Markov in two state variables (the

risk-free rate and the cumulative quadratic variation) is *�� *�� *�� and +�� . Jeffrey (1995) demonstrates that for

the short-rate to be one-factor Markov the functions *�� must satisfy a very specific functional form (his equation (18) p.631).

46

chosen specification for both the volatility-structure of forward rates andthe dynamics of the volatility-specific state variable generates a dynamicalsystem that can be characterized as an ‘extended’ affine system.18 Thatis, even though this system falls outside the class of models investigatedby DPS 2000, we are nevertheless able to derive closed-form solution forderivatives using an approach introduced by Heston (1993) and general-ized by DPS 2000. Indeed, to determine the date-2 price of a bond-optionwith exercise date-� on a bond that matures at date-� , we have shownabove that only the characteristic function of the date-� bond price underthe � and � forward measures is required as input.

The forward rate dynamics under the forward-� measure (denoted by�� ) are given by:

�� 2� � ��2�-�2� � ��2� � �� 2�� 2 -�2� � ��

��2� �0��

�2�(132)

��2� � "� �2� �2 <�

��2��1 �0�

��2�

�� 1� �0�

��2�

�� (133)


"� �2� � "��2�� 1 <��2�� 2� � (134)

We can solve for characteristic function under the � -forward measure:

��#�� E��

��

� �� E�

�

��

� ��

��

We find that it is exponentially affine in the current forward rates � ��2�

and current volatility-specific state variables ��2�:

��#�� &�� &��

��&�� (135)18Our use of the word ‘affine’ is consistent with the terminology of DK 1996 and DPS 2000. We coin the phrase ‘extended’

affine because, in contrast to their model, we do not have a finite-dimensional Markov system, and, in particular, the short rateis not Markov in a finite number of affine state variables.

47

where 3�� * � �� are solutions to

�3 ��2� � � � ?

��2� � � 3��2� � �@�

�* � �� (136)

3��2� � � � ?� ��2� � � 3��2� � �@��

�2 � � (137)

3��2� � � � �# �2 % �� (138)

where

��

�

��

�

�$��$� � � �� $�

��

()��

� �

�

�$��$� � � �� $�

�

��)�

��

� �

�

�$��$� � � �� $� �*�� $��*��+ �� (139)

and the following boundary conditions hold:19

3�� 3�� (140)

� Intuition:Duffie, Pan and Singleton (2000) find a closed-from for:

>�9� � E��

��

��

Where: –� multivariate-Markov State Variables, � scalars

Here, ��#� � E�

��

� ��

�� E�

��

� ��

‘sum’ of Multi-variate Markov state variables with dynamics modeledwithin an affine structure.

Proof:

It is sufficient to show that �� given in equation (135) above is a martin-gale. Indeed, if �� is a martingale, then

�� E��

�� E�

�

��

� ��

��

19Notice that eq. 138 gives � �� +�� +

�� ,� �� ,��

�.

48

Applying Ito’s lemma to ��, and using equations (136)-(138), we obtain:

��

� �

�

-�2� �0� �2� �

With sufficient regularity conditions the stochastic integral is a martingale.Hence, the result follows. �

Using the closed-form solution for the characteristic function, we shownext that the price of a European bond option with strike price � caneasily be obtained following the approach introduced by Heston (1993)and extended by, among others DPS (2000). In turn, the prices of capsand floors are derived as portfolios of zero-coupon bond options (see, forexample, Hull (2000) p.539).

3.5 Pricing Bond Options

� Closed-form characteristic function�Closed-form bond option prices

The payoff of a European option with maturity � , whose underlying assetis a discount bond with maturity � , is

��2 � ��

�� (141)

Under the � - and � -forward measures, the bond-option price at an earlierdate-2 can be written:

��2� �� 2� E��

��

�� 2� E�

�

��

�� 2� E�

�

��

� ��

�� 2� E�

�

��

� ��

This can be solved by following the approach of Heston (1993) or DPSabove.

49

Define !�� E��

�where for simplicity we write ��

�� and � ��. Then the Fourier-Stieltjes transform of ! is(note that �!� ��

� �� ):��

��!��

�� "#��

We can thus find ! by Levy inversion (see Williams (1991)):

!��

� �

$

� �

�

�##�

��"#��

�#

�

Whence, the bond option price can be written as

��2� �� 2�

��

�

$

� �

�

�#Re

�� "#� ��

" #

�� 2�

��

�

$

� �

�

�#Re

�� "#� � �

" #

��(143)

where the characteristic function of �� under different forwardmeasures has been defined above. The implication of Equation 143 isthat, if the characteristic function can be written in closed-form, then socan the bond-option price. This result shows how tractable the generalizedaffine representation is, even in the absence of a Markov representation.

3.6 Markov Representation and Existence

As mentioned previously, for general functions -�2� � � it is not possible toobtain a Markov representation for the model proposed above. While weare able to provide closed-form solutions for a large number of derivatives,we cannot in general propose simple algorithms to price path-dependentinstruments such as American options. In this section we show that byspecializing further the choice of -�2� � � to the widely used exponential

50

case20 we can obtain a Markov representation of the term structure andhence price all assets using partial differential equations techniques. Weclaim:

Proposition 1 Define = �� 2 �

��%�� 2�. If -�2� � � � �!��2�� in equations 125-126, then the model possesses a Markov representa-tion in the three state variables �� = ��. The state vector isaffine, and bond prices are exponentially affine functions of the subset�� of the state vector. All fixed-income derivatives are solutionsto a partial differential equation, subject to appropriate boundary condi-tions.

Proof: Integrating the forward rate dynamics we obtain:

�� = �� (144)


��

� �

�

�2 ��%�� 2�

� �

�

�0��2� ��%��

��2� � (145)

= ��

� �

�

�2 ��%�� 2� � (146)

Applying Ito’s lemma we obtain the dynamics of ��, = ��:

��

��

� ��

��

�� 0�

�� (147)

�= ��

��

� = ��

�� (148)

Using equation (144) we obtain the dynamics of the short-term rate:

��

��

��

��

�� = ��

��

�� 0�

�� (149)

20See for example: Carverhill (1994), Jeffrey (1995) , Cheyette (1995), Ritchken and Sankarasubramaniam (1995) and deJong and Santa-Clara (1999). It was pointed out to us by a referee that De Jong and Santa-Clara (1999) actually obtain the sameclosed form we derive below, without however identifying the link with True Stochastic Volatility models.

51

Recalling the definition of ��, it is clear that �� = �� form aMarkov system.

More generally, the forward rates may be written as:

� �� %�� %�� = �� (150)

Thus bond prices satisfy:

� � ��

��

�

�$ % ��

�(151)

� ��

��

�

�$ % �� & ��

�(152)

� ��

��

�

�$% ��

�% �� '�� & ��

�

�� & ��

��(153)

where we have defined �� ,

.

Finally, let us consider a path-independent European contingent claim thathas a payoff at time � that is a function of the entire term structure at time� , i.e. >�� >

��

. The price of that security is >��

�� !��

��2��2�>��

�� = �� where the second

equality follows from the Markov-property. Moreover a standard argu-ment (which requires some regularity conditions on � and its derivatives,

see Duffie Appendix E p.296) shows that �!��

��2 ��2�� = ��

is a martingale and that its drift must vanish, or equivalently��

��

�E�

�� =�� =��

Using Ito’s lemma we obtain the partial differential equation for the priceof the European contingent-claim:

� � ��

��

�

��

�� & ��

� �

�

�% ��

�% �� & �� '��

��

��

��

��

��)� �

��()

�� '�� (154)

52

�

This result illustrates the impact of true stochastic volatility state vari-ables. From equation (153), we see that bond prices are exponential-affinefunctions of � and = alone. Since = is locally deterministic, the innova-tions of any bond can be hedged by a position in any other bond and themoney-market fund. However, �=� � are not jointly-Markov. As a con-sequence, the dynamics of bond prices over a finite time period dependon the dynamics of the additional state variable � as well. An implicationis that bond prices alone do not permit a complete hedge against changesin �, since ��2� �0��2� � �� Also, we note that it is straightforward togeneralize the model functions of the form -�2� � � � -�� -�2�.21

In general, it is not possible to guarantee that the above stochastic differ-ential equations for � and � are well-defined. Indeed, for general initialterm structures and parameter choices, �� may take on negative val-ues.22 The following lemma demonstrates that there exists a feasible setof parameters such that � remains strictly positive (almost surely) and theSDE’s are well-defined.

Proposition 2 If the parameters and the initial forward rate curve satisfy:

1. � �, ?� � �

2. ?�@�� ?� � @�

�?�

3. �� % �

4. ?��?��?�@

��?�@

��

�

�?�

� ?�

�<� 1<?�?�

��

21In the appendix ?? we provide an example of such an extension in a multi-factor version of the model presented in thissection.

22Moreover, the square root diffusion coefficients does not verify the standard Lipschitz conditions at zero, but see Duffie(1996) appendix E p.292 and DK 1996.

53

then �� % � �� a.s. and the SDE’s for the forward rates � �� ; andthe stochastic volatility �� are well-defined.

Proof: Note that under condition 1 of the proposition

�� ,

%�� ,

�% �� ,

"�� ,

�-

�� ,

�-

� -

��

�� .

�� + ��

�"�� "��

where : ��?�

� ?�

�<� 1<?�?�

and

�� -

��?� 1 ?�<� �0��

�� 1� ?� < �0��

�is a standard Brow-

nian motion and 5�� . A minor adaptation of the proof of the SDETheorem in DK 1996 (which extends Feller (1951) to a vector of affineprocesses) to account for deterministic coefficients in the drift of �, allowsus to conclude that the SDE for forward rates and stochastic volatility statevariables are well-defined. �

Note that the above proposition puts joint restrictions on both the feasibleset of parameters and the initial curve of forward rates. Also note that aMarkov representation exists for this special choice of volatility structure,implying this model has an affine structure in the sense of DK 1996 orDPS 2000, but with two distinct features: (i) it is consistent with the initialterm structure and (ii) it results in only a subset of the state variablesentering the bond prices exponentially (i.e. the loading of the log-bondprice is zero for the state variable �).

This approach provides a straightforward and efficient method to con-struct HJM affine models with stochastic volatility. It can be extendedto the infinite dimensional ‘string’ models analyzed by Kennedy (1994),Goldstein (2000), Santa-Clara and Sornette (2000), while retaining theanalytical tractability of the finite dimensional models proposed above(Collin-Dufresne and Goldstein (2000)).

54

We provide a generalization to multiple factors of the previous HJM-model. We consider the affine multi-factor case directly, since it neststhe Gaussian case.

3.7 The Multi-factor affine case

We specialize the model to obtain a simple Markov representation of theterm structure in terms of a finite number of state variables by setting-��2� � � � -�� -��2�.23 This result extends the analysis of Cheyette(1995) to a more general framework and provides a simple illustration of‘true’ stochastic volatility variables.24 The specialized setup can be writ-ten as follows. Assume the forward rate dynamics and stochastic volatilityare specified for all � � 2 � � � � as:

�� 2� �

��

-��

-��2�

�� 2�

-��2�

��2� �2

��

-��

-��2�

��2� �0�

�2� �(155)

��2� � "��2� �2

��

�$��2� �0�

�2�

��

��2��0�

�2� � (156)

"��2� � @��

@�� 2� @�

��2� @�

�� 2� � (157)

��2� � ?�� ?�

�� 2� ?�

��2� ?�

�� 2� � (158)

$��2� � .�� .�

�� 2� .�

��2� .�

�� 2� � (159)

��

�

�; -��;� � (160)

and all parameters �@� ?

� .

* � �� are assumed to be at most

deterministic functions of time. Also, it is straightforward to generalize23Assuming *�� *�� , Frachot and Lesne (1993) and Frachot, Janci and Lacoste (1993) identify the restric-

tions these functions must satisfy to obtain a linear-factor representation of the term structure. In their model, the factorsare constant-maturity forward rates. Note that, while quite general, our separability assumption precludes for example theone-factor extended CIR model.

24In the absence of ‘true stochastic’ volatility state variables, our volatility structure is a special case of Cheyette’s generalclass of volatility structures that allow a Markov representation (see also Ritchken and Sankarasubramaniam (1995) and DeJong and Santa-clara (1999)). It turns out that the affine structure, even in the presence of ‘true’ stochastic volatility statevariables, considerably simplifies when a Markov representation is possible.

55

to a vector of �, � , and %� . The ' Brownian motions ��0��2��

��are

independent of the +-Brownian motions �0�

.

The functions -��2� are general deterministic functions. We claim:

Proposition 3 In the framework given by equations (155-160) the sys-tem comprising ��2� and the vector � � �= �

��2�� = �

��2��

��is Markov,

where we define:

= ��

� �

�

�9-��

�-��9��9� (161)

= ��

� �

�

�9-��9�

�-��9��9�

� �

�

�0��9�

-��

-��9�

��9� �(162)

The bond prices are exponential affine in the state variables �. Further-more the price of any fixed-income derivative securities solves a secondorder partial differential equation.

Proof: Note that

��

�= �� = �

�� (163)

and more generally:

� ��

��

�-��;��;�

-��= ��

-��;�

-��= ��

�� (164)

Bond prices are given by the formula:

� � �� !��

�

�; � �� (165)

� �!

��

�

�; � ��

�3 �

�� = �

�� 3 �

�� = �

��

(166)

56

where � " � �� + we have defined:

3 ��

� �

�

�;-��;��;�

-�� (167)

3 ��

� �

�

�;-��;�

-�� (168)

Clearly, bond prices are exponentially affine in the state variables �.It remains to be shown that the system formed by ��2� and the vector� � �= �

��2�� = �

��2�� is Markov. Applying Ito’s lemma we obtain the

dynamics for the new state variables �" � �� +:

�= ��

��

-��

�-��

�� -��

��

�= ��

�� (169)

�= ��

��

-��

�� -��

��= ��

��

�� 0�

�� (170)

Given that forward rates are linear combinations of � and by the defini-tion of �, it is clear that the above system is Markov.25

Finally, let us consider a path-independent European contingent claim thathas a payoff at time � that is a function of the entire term structure attime � , i.e. >�� >

��

. The price of that security

is >�� !��

��2��2�>��

�� where the

second equality follows from the Markov-property. Moreover a standardargument (which requires some regularity conditions on � and its deriva-tives, see Duffie Appendix E p.296) shows that �!��

��2 ��2��

is a martingale and that its drift must vanish, or equivalently��

��

�E�

��

Using Ito’s lemma and expressing �� in terms of the state variableswe obtain the partial differential equation for the price of the European

25In addition the drift and diffusion should satisfy some regularity conditions, essentially Lipschitz and Growth conditions,for the Stochastic differential equations to have well-defined (e.g. unique strong) solutions and for the Markov property to hold.We refer the reader to Duffie’s SDE theorem, appendix E p. 292.

57

contingent-claim:

� � �� "��

��

�. �

��

-��

�-��

�� -��

��

�= ��

�

��

��

. �

��

-��

�� -��

��= ��

�

��

�

. �. �

�. ��

�

�

�

��

� ��

��

$��

��

��

�= �� = �

��

�� (171)

�

� This again illustrates the impact of unspanned stochastic volatilitystate variables. From equation (166), we see that bond prices are ex-ponential affine functions of �. Thus the instantaneous dynamicsof bond prices are determined by the innovations of these state vari-ables. Thus, any combination of + distinct bonds allows to hedgeagainst innovations in these state variables. However, we emphasizethat� alone is not Markov. As a consequence, the dynamics of bondprices over a finite time period depend on the dynamics of the addi-tional state variables ��2� as well. An implication is that bond pricesalone do not permit a complete hedge against changes in ��2�, since��2� �0 � � � ".

� Note that the ‘separability assumption’ -��2� �� /��

/��considerably

simplifies the framework and reduces it to a finite-dimensional (2n+1)affine model. Indeed the �� variables are all affine in the tradi-tional sense. Note also, that the state variables = �

�" � �� + are

actually locally deterministic, which simplifies the numerical compu-tations even further.

� The multi-factor Gaussian case is obtained by setting ? �� ?��

?�� ". In that case the number of state variables necessary to

58

obtain a Markov representation is equal to the number of factors + .Indeed all the = �

� state variables are deterministic (since � is), so thesystem is Markov in the + Variables = �

� " � �� +. In fact, from thedefinition of = �

� we see that in the Gaussian case all pass-dependence

is summarized by the + stochastic integrals� ��0

��9� /

��

/�+�

��9� �",

which form a Markov system and could be taken as alternative to the= �� state variables chosen above.

� In the Gaussian case European zero-coupon bond option prices canbe solved in closed-from as in section 1.1. However the approach ofsection 1.4 for pricing coupon coupon bond options does not workanymore.

� For very fast and accurate approximations for prices of zero-couponas well as coupon bond options in the generalized-affine setup (i.e.,‘square-root’ type volatility function including the Gaussian case),the Cumulant-expansion technique developed previously applies inthe HJM framework as well.

� For hedging at least + instruments are necessary. Note that the num-ber of hedging instruments corresponds to the number of factors whichis in general smaller than the number of state variables required to de-scribe the term structure dynamics.

59

4 Extending the affine framework to HJM and Random field models

4.1 Motivation

� Affine framework is widely used because of its analytic tractability.

– Time homogeneous models with finite number of factors and statevariables.

Vasicek; Cox, Ingersoll and Ross.

– Analytic solutions for bond prices and various derivatives.Duffie and Kan.

– Explicit solution for optimal bond portfolio choice.Liu.

� Recent empirical research challenges its ability to capture:

– Joint dynamics of bonds and derivatives.Jagannathan, Kaplin and Sun.

– Predictability of bond returns.Duffee; Cochrane and Piazzesi.

– Low correlation of non-overlapping forward rates.Dai and Singleton; Lekkos.

� HJM approach takes bond prices as inputs; prices fixed income deriva-tives.

– In general, spot rate dynamics are non-Markov.Derivative pricing computationally intractable.

– Identify volatility structures with finite state space representation.Mostly restricted to ‘separable’ volatility structures.Cheyette; Ritchken and Sankarasubramaniam.

60

� Random field or String models can capture arbitrary term structureshapes (present and future), and arbitrary variance covariance struc-tures.

– Infinite factor, infinite state variable models.no analytical or numerical tractability.Kennedy; Goldstein; Santa-Clara and Sornette.

– In general, resort to finite factor approximations.Longstaff, Santa-Clara and Schwartz; Han.

� Below we introduce a ‘generalized affine’ framework.

– Extends standard affine framework to HJM and Random Fieldmodels.

– Accommodates arbitrary term structures and correlation structures.

– Provides analytic solutions for derivatives (option on bonds, yields,futures).

– Offers explicit and unique solution to optimal portfolio choiceproblem.

Outline

1. Background:

(a) Traditional (time-homogeneous) affine models

(b) HJM-type models

(c) Random field models

2. Generalized affine models

3. Application: Relative pricing of caps and swaptions

61

4. Application: Optimal bond portfolio choice and ‘preferred habitat’

5. Conclusion

4.2 Traditional Affine Framework

The affine Framework:

� N jointly Markov State variables �� compose the state vector ��

� Dynamics �� 4 �� 5 such that:

– 4� -

<

�

–��

�

� ��

��

� � � Æ� Æ� �

– Duffie and Kan (1996)

– Duffie, Pan and Singleton (2000)

– Chacko and Das (2002)

– Dai and Singleton (2000)

– Duffie, Filipovic and Schachermayer (2002)

– Piazzesi (2002)

Advantages of traditional affine framework:

� Characteristic Function has exponential-affine solution (in fact, thisdefines affine models - DFS)

@� ��

��

� ��

�� /�

�� !

��

��

��

��

�

62

� Characteristic function related to probability distribution (e.g., mo-ments).

Derivative prices depend on probability that asset ends up ‘in themoney.’

� Many derivatives priced by Fourier inversion of characteristic func-tion.

Heston (1993)

� Analytic solutions for bonds and many derivativesCaps, Floors, Options on zero-coupon bonds,

Options on baskets of yields, Options on futures...

� Analytic solutions for moments of bond portfoliosFast, accurate algorithms for pricing options on coupon bonds (i.e.,

swaptions).

Recent research presents several challenges for the traditional affine model:

� Joint dynamics of bonds and derivatives (Jagannathan, Kaplin andSun (2001); Collin-Dufresne and Goldstein (2001); Heiddari and Wu(2001)).

– Some factors driving implied volatilities of Caps and Floorsare distinct from factors driving yields (USV).

– In standard affine model any factor driving volatilities also drivesyields.

� Predictability of bond returns (Duffee; Cochrane and Piazzesi).

– CP show that a specific linear combination of all available forwardrates has predictive power for future bond returns.

63

– In N-factor affine framework, a linear combination of any N for-ward rates would work.

� Low correlation of non-overlapping forward rates (Dai and Singleton;Lekkos).

4.3 Time-Inhomogeneous Models

� In contrast to standard affine models, so-called ‘arbitrage-free’ ap-proach takesbond prices as ‘initial condition,’ and directly models their dynamics

�� 2� � ��2�

��

�

/�2� �9

��2

��2� �0��2�

� Dynamics specified by ‘volatility structure’Ho and Lee (1986); Heath Jarrow and Morton (1992)

� For most specifications (even finite factor) bonds cannot be expressedin terms of finite state vector.

– Spot rate dynamics are not Markov.

– Pricing derivative securities typically becomes intractable.

� Motivated search for HJM models with finite state variable represen-tations

– Typically requires separable volatility structure.

Carverhill (1994); Jeffrey (1995); Ritchken and Sankarasubramaniam(1995);Cheyette (1995); Bhar and Chiarrella (1997); Inui and Kijima (1998);de Jong and Santa-Clara (1999); Bjork and Svensson (2001)

64

4.4 Random Field or ‘String’ Models

� Infinite-dimensional HJM models are ‘finite factor’ in that the rank ofthe covariance matrix of bond price innovations is finite.� Can hedge risk of one bond using N other bonds.

� Random field models allow each bond (with a continuum of maturi-ties) to have its own ‘idiosyncratic risk’: ‘Infinite-factor’ model(Kennedy (1994, 1997); Goldstein (2000); Santa-Clara and Sornette(2001))

� Random fields cannot possess a finite state variable representation,# state variables � # factors

� Typically implemented by considering finite factor approximationsLongstaff, Santa-Clara and Schwartz (2001); Han (2002)

4.5 Generalized Affine Models

� Combines Tractability of Traditional Affine Models with Flexibilityof HJM and Random Field Models

� Takes log-bond (�� ) prices as state variables:

��

� � ��

�� 0�

��

� Volatility state variables �� follow square root processes:

�� 6�

�� ? ��

65

� Brownian field associated with deterministic correlation structure:

�0� �� 0�

��

� Volatility risk cannot be hedged by trading in bonds:

�0� �� ?

��

� In contrast to the traditional affine framework, log bond prices can-not be expressed as linear functions of the (finite-dimensional) statevector. Thus no bond price is redundant in this model.

� Each bond price is itself a state variable: The dimension of the statevector is infinite, as it consists of a continuum of maturities of bonds(and �)

� No restrictions placed on the deterministic volatility and correlationstructures � �� and ��.

� Both volatility and correlation of bond prices are stochastic, and volatility-risk (�?) cannot be hedged by trading in bonds.

� Possible extensions:

– multiple Brownian fields,

– multiple stochastic volatility state variables,

– correlation between volatility and bond return innovations,

– regime shifts in correlation structure.

� The model is ‘generalized affine’ because the characteristic function:

@��,� � ��

�

��

� ��

,� ��

�� , ��

�

� � �� !

��3 �� & ��

,��

��

� ��

��66

has an analytic solution which is exponentially-affine in ��

� Same tractability as traditional affine framework, even though:

1. no finite state-variable representation,

2. no need to restrict volatility structures to be ‘separable’,

3. no need to restrict to finite state variable, or even finite-factor,model.

� Typically to price options need to evaluate expression:

��

��

��

� ��

�� *� ��

�� *� ��

��#� ��

�� #� ��

�

� Fourier inversion theorem gives:

"��*�#

� � �@�

��-�

� �

$

� �

�

#��@�

��- " ;<� ��

�;

�;�

� Evaluating "��*�#

� � requires only a single numerical integral.

� Example: Zero-coupon option.

�� E�

�

��

� ��

��

��

�� E�

�

��

� ��

��

��

�� E�

�

��

� ��

��

�� "�

��*�*�� "�

��*��

with - � �� .

� Similarly obtain closed-form for caps, floors, options on yields, for-wards, futures, coupon bonds, convexity bias.

67

4.6 Relative Pricing of Caps and Swaptions

� Previous Studies document relative mispricings between caps andswaptions:� either arbitrage opportunities or model misspecification.

Longstaff, Santa-Clara and Schwartz; Jagannathan, Kaplin and Sun;Fan, Gupta and Ritchken

� Claim: Mispricing due to strong restrictions traditional models placeon joint evolution of:

1) interest rates2) volatility3) correlation

� As is well-known:Cap (Floor) � Portfolio-of-OptionsSwaption� Option-on-a-Portfolio

� Relative pricing of caps and swaptions driven by correlation struc-ture

� Generalized affine class provides framework to disentangle effects ofcorrelation, volatility and interest rates.

� Indeed, consider for example the simple model:

�� 2�

� � �2�� 2� ��2� �5�

�2�

�5�

�2� �5�

�2� � �� 2��2 �

� The caplet price (maturity ��, tenor Æ):

� ��

��

� ��

� �

��

��

��

��

�

��

��

� ��

��

��

��

��

��

�

��68

where �� Æ , �� Æ�.

� The implied volatility is negatively related to correlation:

�� Var��

��

� ��

�

�2��2�� 2��2� ��2� ��2��

��

� “Numeraire Effect”: what matters is volatility of forward bond: ��Æ��

��

.

If Æ is small, then ’numeraire effect’ is large.

� For caps, Æ = 6 months.� Thus Caps are strongly negatively related to changes in bond yieldcorrelations

� For Swaptions, Æ can be large. In addition, increase in correlation in-creases volatility of portfolio of bonds, generating a partially-offsettingeffect.� Swaptions are relatively insensitive to changes in correlation ofbond yields.

� Consistent with empirical observation of Fan, Gupta, Ritchken (FGR),but apparently inconsistent with statements in the literature

Driessen, Klaasen, and Melenberg (DKM); Rebonato.

� In fact, there is no discrepancy:

– FGR model under risk-neutral measure, and hence estimatebond-yield volatilities and correlation

– In contrast DKM and Rebanato use ‘market model,’dynamics specified under a ‘forward measure’.

69

4.7 Forward bond model

� Consider forward bond price

� �

(��

� ��Æ��

� � �� (172)

� Under � -forward measure � �(�� is a martingale with general affine

dynamics:��

(��

� �(��

� �(�� 5�

�� (173)

where �5��

is a Brownian field under the � -forward measure with

� �54

�� 5�

�� 4�

(�� and �?

��2� � �5�

�� (174)

� Bond dynamics are consistent with forward bond dynamics iff:

�(�� 5�

��

��5�

�� Æ�

��5��Æ�

�� (175)

� In terms of quadratic variations we find

�(��

��

��Æ�

��Æ�� Æ�

�� (176)

and thus �� Æ� �� (�2�� 2�

� In Forward bond model, cap prices depend only on forward bondvolatilities

(and not on their correlation structures ��

(��.

� However, the forward bond volatility (

is a function of bond yieldcorrelationstructure ��. Thus caps are affected by bond correlations, but notby forward bond correlations.

� Cannot compare ‘correlation effect’ directly across these two differ-ent frameworks!

70

� The generalized affine framework (whether expressed in terms of for-ward bonds or bonds) can capture by construction the relative pricesof caps and swaptions.(

can be used to calibrate cap prices perfectly and �(

can be used tocalibrate swaption prices perfectly and independently.

� With truly stochastic correlation and volatility, need both caps andswaptions to estimate (or calibrate) model.

– In forward bond model, swaption are more sensitive to changes incorrelations (�

() than are caps . In fact, caps are only functions of

volatility (.

– In bond model, caps are more sensitive to correlation (�) than areswaptions.

4.8 Optimal Portfolio Choice and Preferred Habitat

� Consider Dynamic portfolio choice when agents can invest in contin-uum of assets (Heaney and Cheng (1984), Bjork et al. (2000))

� Consider general affine model under the historical measure:

�� 2�

� � �2�� ,� �2�� 2� ��2�

�2� �

�

��2� �0� �2�� (177)

� the stochastic volatility state variable has � -measure dynamics givenby:

��2� � �� 2�� 2 6�

��2� �?�2�� (178)

where ? is a Brownian motion, and �?�2� �0� �2� � � for all maturi-ties � .

� Risk-premia are given by �0� �2� � �0� �2�� ,� �2�

��2� �2�

71

� The correlation structure is specified as: �0� �2� �0�

�2� � ��2��2�

� In this model, no bond is redundant.

� Consider an agent who can invest in a continuum of zero-couponbonds withmaturities � � �� to maximize his expected utility of terminalconsumption:

�)��

�

��!��) if . % �

�� if . � �(179)

� The current wealth of the agent can be written

� ��

� ��

�9 �+�� +�� (180)

where �+�� is the number (density) of shares of bond with maturity-9held.

� Define the fraction of wealth scaled by volatility: $+�� */��/�� +��.

Thus: �$� �� 9 $+��+��

� Define log-discounted-wealth as state variable: ��

� ��

with dynamics:

��2� �

��$� ,�� ,� �2�� 2��

�$��$��

�

�� 2��

��2� �2

�

��2�

�� 2� �0� �2��

� ��

�9 $+�2��0+�2�

��

where, we introduce the notation

�$� ,��

�;$+�2�,+�2� and ��$�+��

�;�+��2�$��2�(181)

72

� Impose that trading strategies be admissible (i.e., such that SDE for�� iswell-defined, and � �� 1�� % � -�2�.

� Further restrict $ to be in the set � defined by: �� $+��

�$��$�� We write $ � �� if $ is admissible and �� 9 � � $+��

��.

� Given these conditions, the optimal control problem of the agent as:

�� 5 ��

��)��

1�� (182)

has the following solution:

��

�� .��)�1�� (183)

where the functions �� solve the system of ODE (s.t. ��

�� ):

�� (184)

�� 6�

�� (185)

�� .�

�,� �� .��

� �� .�(�$��

(186)

where: (�$� �� )��$��$�� $� A�� and A+�� ,+��

.�� +��

73

� The optimal trading strategy solves:

��5��

(�$� �� s.t. �$��

��

� The actual portfolio holding &$+�� $+

��+

�are:

&$+��

�

.+�

��,

#

�

�+�

� �� .

.�/��

� (187)

where #� is the Lagrange multiplier associated with the portfolio con-straint given by:

#� �� ,� �

'��

�� '� �'��

(188)

� In contrast to finite dimensional model we obtain a unique optimalportfolio choice.

� Agents invest more in assets with higher mean return or lower corre-lation with overall portfolio (Merton (1973)).

� In contrast to Merton (1973), preferred habitat for maturity � trans-lates in over (under) investment in preferred habitat bond dependingon whether agent is more (less) risk-averse than log.

� Since Sharpe ratio (,) on bonds is not stochastic, no hedging demand.

4.9 Preferred Habitat and Predictability in Bond Returns

� Results of Cochrane and Piazzesi (2002) suggest that bond returns arepredictable and driven by a single factor which is a portfolio of for-ward rates.

� This is easily captured in generalized affine framework.

74

� Consider, for example, the dynamics:

�� 2�

� � �2�� 2� ,� �2�'�2� �2� � �2� �0� �2�� (189)

� Risk-premia are driven by innovations of all forward rates:

�'�� 4� 4�'��

� ��

6��0��2� (190)

� The instantaneous sharpe ratio (or risk-premium) on bonds are givenby ,�

�'�2�.

� Consistent with CP, each forward rate contributes information to pre-dictability regressions since:

Corr�� '�� 6��

��6�6��

differs across maturities.

� Consider similar optimization problem as before.The optimal control problem of the agent as:

�� 5 ��

��)��

1�� (191)

has the following solution:

��

�� .��)�1�� (192)

where the functions ��& solve the system of ODE (s.t. �� &�� ):

�*��

�� /� /� ��

��/��

�

��

��

�� 0��

�� (193)

�� /��/� �� /��

��

�� 0�� 0��

�� /��/� �� 0��

�� (194)

75

� The optimal trading strategy $�� $� $�'�� solves��5�� (�$� �� s.t. �$� �

'�� , where

(�$� �� .

�$��$�� $� A��

and

A+�� .�� +��6�+� �,+

�� &��6�+

��'��

� A+��

A+��'��

� The optimal trading strategy is thus given by

�/�

� ��

��!��

� ��

�//� ��

�� /��

��

�

��

��

��

�/

�

� ��

�//�

� �/�� /

��

where #� � #�� #��'�� is the Lagrange multiplier associated withthe portfolio constraint given by:

#� �. � ��A��

�'��

�� '� �'��

� '��

��A��

�'��

�� '� �'��

�

4.10 Conclusion

� Introduce generalized affine framework which is infinite factor,infinite state variable model, but

– Retains tractability of traditional affine class,

– Increases significantly the flexibility of the affine class

� Investigate the relative pricing of caps and swaptions

– Demonstrate that caps, not swaptions, are very sensitive to changesin correlation of bond-yields

76

– However, in forward-bond model (or market model), swaptions(and not caps) are sensitive to forward bond correlation(for appropriate choice of tenor).

– Generalized affine model which can accommodate truly stochas-tic correlation and volatility changes, can fit relative pricing ofcaps and swaptions by construction. Both instruments are re-quired to calibrate/estimate the model.

� Investigate optimal bond portfolio choice for agent with preferredhabitat who can invest in a continuum of bonds.

– Closed form solution for agents value function with CRRA utility.

– Since no bonds are redundant unique portfolio choice obtains.

– Investor over (under) invests in bond with preferred habitat matu-rity depending on whether more (less) risk-averse than log.

– Can accommodate time varying sharpe ratios consistent with pre-dictability regressions of Cochrane and Piazzesi (2002). In thatcase, optimal bond portfolio holdings are time varying due tohedging demand.

References

[1] A. Carverhill. When is the short rate markovian. Mathematical Fi-nance, 4 (Oct):305–312, 1994.

[2] O. Cheyette. Markov representation of the Heath-Jarrow-Mortonmodel. BARRA Inc. working paper, 1995.

[3] P. Collin-Dufresne and R. S. Goldstein. Generalizing the affineframework to HJM and random field models. Carnegie Mellon Uni-versity Working Paper, 2002.

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[4] Q. Dai and K. J. Singleton. Specification analysis of affine termstructure models. Journal of Finance, 55:1943–1978, 2000.

[5] F. de Jong and P. Santa-Clara. The dynamics of the forward interestrate curve: A formulation with state variables. JFQA, 34:131–157,1999.

[6] D. Duffie, J. Pan, and K. Singleton. Transform analysis and optionpricing for affine jump-diffusions. Econometrica, 68:1343–1376,2000.

[7] Dybvig and Huang. Non-negative wealth, absence of arbitrage, andfeasible consumption plans. The Review of Financial Studies, 1:377–401, 1989.

[8] W. Feller. Two singular diffusion problems. Annals of Mathematics,54:173–182, 1951.

[9] A. Frachot, D. Janci, and J.-P. Lesne. Factor analysis of the termstructure: A probabilistic approach. Banque de France Working Pa-per, 1993.

[10] A. Frachot and J.-P. Lesne. Econometrics of the linear gaussianmodel of interest rates. Banque de France Working Paper, 1993.

[11] C. W. Gardiner. Handbook of Stochastic Methods. Springer-Verlag,1983.

[12] D. Heath, R. Jarrow, and A. Morton. Bond pricing and the termstructure of interest rates: A new methodology for contingent claimsevaluation. Econometrica, 60:77–105, 1992.

[13] S. L. Heston. A closed form solution for options with stochasticvolatility. Review of financial studies, 6:327–343, 1993.

[14] J. Hull. Options, Futures and Other Derivative Securities. PrenticeHall, 2000.

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[15] F. Jamshidian. Contingent claim evaluation in the gaussian interestrate model. Research in Finance, 9:131–170, 1991.

[16] A. Jeffrey. Single factor Heath-Jarrow-Morton term structure modelsbased on markov spot interest rate dynamics. Journal of Financialand Quantitative Analysis, 30n04:619–642, 1995.

[17] N. E. Karoui and J. Rochet. A pricing formula for options on coupon-bonds. Cahier de Recherche du GREMAQ-CRES,no. 8925, 1989.

[18] T. C. Langetieg. A multivariate model of the term structure. Journalof Finance, 35:71–97, 1980.

[19] P. Ritchken and L. Sankarasubramaniam. Volatility structure of for-ward rates and the dynamics of the term structure. MF, 5:55–72,1995.

[20] D. Williams. Probability with Martingales. Cambridge UniversityPress, 1991.

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