Term Structure Models II:
Fixed-income Derivatives Pricing
Pierre Collin-Dufresne
UC Berkeley
Lectures given at Copenhagen Business School
June 2004
Contents
1
2
Bond Option pricing in the Gaussian case
4
1.1
Zero-coupon Bond option pricing in the Gaussian model
1.2
Three interpretations of the Forward measure .
1.3
Closed-form solution
1.4
Coupon-bond option pricing (Jamshidian) .
. . . . .
4
. . . . . . . . . . . . .
5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
. . . . . . . . . . . . . . .
Bond-Option Pricing in the Affine Framework
9
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
2.2.2
The Three-Factor Gaussian Model .
2.2.3
The Two-Factor CIR Model .
2.2.4
Appendix For Cumulant Expansion Technique
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1
. . . . . . . . . . . . . . . . . 18
. . . . . . . . . . . . . . . . . . . . . . 22
. . . . . . . . 24
2.3
3
. . . . . . . . . . . . . . . . . . . 29
Bond and Forward Rate models: The HJM approach
32
3.1
Absence of arbitrage and Equivalent Martingale Measure
3.2
Summary of the HJM approach: Fitting the term structure
without ‘tricks’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
4
General affine Jump-diffusion models
One-factor HJM Models .
. . . . 33
38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.1
The Gaussian Case
3.3.2
A one-Factor HJM Model with affine volatility structure
. 42
3.3.3
Hedging in one-factor HJM Model with affine volatility
structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 44
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4
Two-Factor HJM Model with ‘Unspanned’ Stochastic Volatility
3.5
Pricing Bond Options .
3.6
Markov Representation and Existence
3.7
The Multi-factor affine case
45
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
. . . . . . . . . . . . . . . . . . . 50
. . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Extending the affine framework to HJM and Random field models
60
4.1
Motivation
4.2
Traditional Affine Framework
. . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3
Time-Inhomogeneous Models
. . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4
Random Field or ‘String’ Models .
4.5
Generalized Affine Models .
4.6
Relative Pricing of Caps and Swaptions
4.7
Forward bond model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
. . . . . . . . . . . . . . . . . . . . . . 65
. . . . . . . . . . . . . . . . . . . . . . . . . . . 65
. . . . . . . . . . . . . . . . . . 68
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2
4.8
Optimal Portfolio Choice and Preferred Habitat
4.9
Preferred Habitat and Predictability in Bond Returns
4.10
Conclusion .
. . . . . . . . . . . . 71
. . . . . . . . 74
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 of
zero-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 standard approaches such as finite difference. In the present case, a closedfrom solution can be found by solving the expectation directly, using the
so-called Forward-Neutral change of measure (Jamshidian (1991), El
Karoui and Rochet (1989)). Define a new measure by the
4
likelihood ratio . Thus
. Note that 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
risk-neutral measure we have for :
. By definition of the
In other words the value of the security expressed using the money market
account as a numeraire is a martingale under the measure.
Performing the forward neutral change of measure we see that:
forward neutral measure the security ex-
We see that under the pressed using the zero-coupon bond price with maturity
is a martingale.
Correction for correlation
5
as a numeraire
Consider pricing a security that pays $X at date
risk-neutral measure we have:
. By definition of the
where we have used the definition of covariance. If is independent of
the risk-free rate we obtain:
However, in general, is not independent of the short rate so this is
not valid. The forward measure change allows to obtain almost the same
expression:
The measure change accounts for the correlation.
Making forward rates martingales
The forward neutral measure deserves its name because forward rates with
maturity are martingales under (i.e., the convexity bias disappears). Furthermore forward bond prices are martingales for
any maturity .
To prove the first,
To prove the second, recall 6
. Thus
Note that while the local expectation hypothesis holds under the riskneutral measure, forward rates are not equal to expected future spot rates
under . Indeed,
Thus, in general when interest rates are stochastic forward rates are biased
predictors 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 the
money under two different equivalent measures, e.g., for
. Girsanov gives:
Furthermore,
7
(2)
Thus,
Where we have defined:
The final solution is thus:
Remarks:
The structure is very similar to the Black and Scholes option formula except that the volatility is that of the log forward bond price
( ).
Following BS the natural candidate replicating portfolio involves two
zero coupon bonds with maturity and . However, this is not
necessary.
Since we only used the fact that the diffusion of bond prices is deterministic to get the closed-from solution above, a similar result will
hold for any model with deterministic bond price diffusion (e.g., any
Gaussian 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 by
Jamshidian 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 decreasing in we can define as the unique solution such that .
Clearly, the option will be exercised if and only if . Thus, we can
rewrite
(3)
(4)
Effectively the price of the coupon bond option can be rewritten as a sum
of zero-coupon bond options with different underlying and strikes given
by . Of course, this does not contradict the general result 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
(5)
2 Bond-Option Pricing in the Affine Framework
One of the nice characteristic of the Gaussian one-factor framework is that
we obtain closed-from solution for all bond option prices. We show in this
section that the tractability extends to the multi-dimensional affine case,
which allows for very efficient ‘almost’ closed-form solution for fixedincome derivative prices such as zero-coupon bond options and coupon
bond options.
We consider for this section a general -factor affine model of the term
structure by a vector of Markov processes whose dynamics
are such that the instantaneous drifts and covariances are linear in the
state variables. Further, the instantaneous short rate is defined as a linear
combination of the state variables: 1
Æ
Æ
Within an affine framework, we have seen previously that bond prices
possess 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
1
Duffie, Pan and Singleton (2000) provide the precise technical regularity conditions on the parameters for the SDE to be
well-defined. Dai and Singleton (2000) classify all factor affine term structure models into families depending on
how many state variables enter into the conditional variance of the state vector. Our approach is valid for each of these families
of models including the USV models covered previously.
10
is
(7)
The price of the bond-option at an earlier date- before expiration can be
written:
E
E
E
E
E
(8)
(9)
where in going from the second line to the third line we have transformed
from the risk-neutral measure to the so-called forward measures (Jamshidian (1991), El Karoui and Rochet (1989)) by using the relation:
E
E (10)
Here, E denotes expectation under the -forward measure, which
takes as numeraire the bond price whose maturity is . 2
It is convenient to introduce the Fourier-Stieltjes transform of : 3
Ê
! "# (11)
where ! is the characteristic function of the random variable under the -forward neutral measure. Following the insight of Heston (1993), Duffie, Pan and Singleton (2000) we can use Levy inversion
, and the conditional likelihood ratio is given by
The Radon-Nikodym derivative is given by Ê
for .
½
.
3
To provide some intuition note that Ê
2
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 functions
of equation (11) can be written in closed-form, then so can the bondoption price. Note that the characteristic function is an expectation of an
exponentially-affine function. Thus in the affine framework, it is itself an
exponential affine function of the state variables (exercise).
Thus computing zero-coupon bond options in a finite-dimensional affine
model basically amounts to evaluating a single one-dimensional numerical integral, the Fourrier inversion, which analogously to the classical
Black 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) A
European swaption gives its holder the right to enter a swap at some future date
% . As such, a swaption is readily interpreted as an option
on a coupon bond, where the strike is equal to the nominal of the contract, and the coupon rate is equal to the swap rate strike of the swaption. 5
4
The inverse Fourier transform technique is widely used. Duffie, Pan and Singleton (2000) provide a comprehensive exposition, examples and further references.
5
Alternatively, a swaption can also be interpreted as a sum of options on the swap rate that must be exercised at the same
date (e.g., Musiela and Rutkowski (1997)).
12
In this section we propose a very accurate and computationally efficient
algorithm 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 by 6
Swn E
E
(13)
E
(14)
where we have defined to be the date-
(15)
price of the coupon bond:
(16)
In going from equation (14) to equation (15), we have transformed from
the 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 takes
as numeraire the bond price whose maturity is . 7
For each of the & relevant forward measures, 8 we estimate the probability distribution of the date- price of the coupon bond. We do this
6
Here we price a call option on a coupon bond which is identical to a receiver swaption (e.g., an option to enter a receive
fixed pay floating swap) when the strike is set to par and the coupon to the strike (rate) of the swaption. Similarly a payer
swaption could be priced as a put option on a coupon bond (or by put-call parity).
Ê
, and the conditional likelihood ratio is given by
7
The Radon-Nikodym derivative is given by Ê
for .
8
From 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, for
each 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 functions defined above. Note that depends only on the state variables 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 measure, we estimate % using a cumulant expansion of the distribution of . Cumulants are defined as the coefficients of a Taylor series
expansion of the logarithm
of the characteristic function. In other words,
defining ! ) " ) as the characteristic function of the
random variable , the cumulants are defined via:
" ! * 14
(21)
The +# order cumulant is uniquely defined by the first + moments of the
distribution (See, for example, Gardiner (1983)). As a reference, the first
seven cumulants are provided in the Appendix.
Armed with an explicit expression for the cumulants we can obtain the
probability density of by inverse Fourier Transform:
) $
"
! (22)
We can then make use of our cumulant expansion for the characteristic
function to obtain:
) $
$
$
"
"
(23)
" (24)
(25)
. Up to this point, the solution is exact. The
where approximation 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, to
any order , ) ) .
This expansion results in a sum of simple integrals which can easily be
solved by noting that:
15
(27)
where the last line defines the coefficients - .
The probability density can then be written
) $
where
. .)
(28)
, -
(29)
The coefficients . are provided in the Appendix for the case .
To price a swaption with strike , we need to compute the date-0 probability that will fall above the strike price. That is, we need to
compute the integral:
where
#
) ) .#
$
)
(30)
)
Note that all # can be solved in closed-form and involve, at worst, the
one-dimensional cumulative normal distribution function, for which there
exist standard numerical routines that do not require any numerical integration. We have thus obtained a very simple expression for the probability of the coupon bond price being in the money. It involves only simple
16
summations. In the Appendix we present the expressions for the coefficients . # 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 accuracy
of our proposed approach. Since the approach is model-independent, a
single program can be written for all models, needing only a call to a
subroutine for each specific model.
Below, we consider two models: a three factor Gaussian model, and a twofactor CIR model. We choose for the order of expansion, since
it appears to offer an excellent compromise between speed and accuracy. 9
For both cases we compute prices of swaptions for various strikes and
compare them to Monte-Carlo simulated prices for accuracy. Note that
the normalized highest order cumulant provides a good estimate of the
attained accuracy. We also list the CPU time required for our approximation.
In 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 -th
9
order approximation. Indeed, there are other terms which show up in the expansion which are products of lower-order
cumulants (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 following state variable dynamics:
/
/
0
where 0
0 1
, and form (Langetieg (1980)):
(32)
Æ /
.10 The bond prices take the
2 / $
(33)
where
%
1
Æ where we define 1
.
(34)
(35)
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 be
computed using the expression for the Laplace transform of the state variable under the forward neutral measure:
E
$
&
$
(37)
where 3 &
are given by:
&
3 10
!
%
1
!
% In fact, it can be shown that this model is ‘maximal,’ in the sense of Dai and Singleton (2000).
18
1
!
! $ $ $ .01
.005
-.02
Æ
%
Parameters
%
.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 coupon
bond 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 of
our approximation relative to a Monte-Carlo solution. The Monte Carlo
prices are obtained using the exact (Gaussian) distribution of the state
variable at maturity to avoid any time discretization bias. The number
of simulations is set to obtain standard errors of order (2,000,000
random draws with standard variance reduction techniques). As the figure
shows our approximation is excellent: The absolute error relative to the
true solution is less than a few 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 have negligible values, thus making this type
of metric somewhat misleading. Again our approximation takes less than
seconds to compute all swaption prices (corresponding to different
strikes).
Another advantage of the Edgeworth expansion approach is that the order
of magnitudeof theerror 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. Two
notable features are apparent from the table. First, the scaled cumulants
decay quickly, which provides an indication of the appropriateness of the
Edgeworth expansion approach. Further, it also provides an estimate of
11
That the scaled cumulants are the appropriate measures for estimating the error can be seen from the Appendix. See
equations (54)-(58).
19
Figure 1: 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 at maturity
to 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 at
maturity to avoid a time discretization bias, and the standard error of the Monte-Carlo prices are less than .
20
measure
mean
variance
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1.291480
1.291569
1.291647
1.291714
1.291773
1.291826
1.291872
1.291914
1.291951
1.291985
1.292015
1.292042
1.292066
1.292088
1.292108
1.292126
1.292142
1.292157
1.292170
1.292182
1.292193
0.00073789
0.00073801
0.00073810
0.00073819
0.00073826
0.00073833
0.00073839
0.00073844
0.00073849
0.00073853
0.00073857
0.00073861
0.00073864
0.00073867
0.00073869
0.00073871
0.00073873
0.00073875
0.00073877
0.00073879
0.00073880
0.01166506
0.01166509
0.01166513
0.01166515
0.01166518
0.01166521
0.01166523
0.01166525
0.01166527
0.01166528
0.01166530
0.01166531
0.01166532
0.01166533
0.01166534
0.01166535
0.01166536
0.01166537
0.01166537
0.01166538
0.01166538
0.00036705
0.00036705
0.00036706
0.00036706
0.00036706
0.00036706
0.00036706
0.00036706
0.00036706
0.00036707
0.00036707
0.00036707
0.00036707
0.00036707
0.00036707
0.00036707
0.00036707
0.00036707
0.00036707
0.00036707
0.00036707
0.0000136
0.0000136
0.0000136
0.0000136
0.0000136
0.0000136
0.0000135
0.0000136
0.0000137
0.0000136
0.0000136
0.0000136
0.0000135
0.0000136
0.0000135
0.0000136
0.0000136
0.0000137
0.0000136
0.0000136
0.0000136
Table 2: Mean, variance, and scaled cumulants for the 20 forward measures and the risk-neutral measure for the 3-factor
Gaussian model.
the truncation error. Indeed, at the rate at which the scaled cumulants are
decaying, one can guess that the # scaled cumulant, and hence the error,
is indeed of the order of . Second, the 5# scaled cumulants are nearly
identical across measures. Hence, for time efficiency, one only needs to
calculate the 5th scaled cumulant for a single measure.
To investigate whether these results are specific to the Gaussian case we
now apply the same approach to a second example where the state variables 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. The
spot 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 simple
extension to the original CIR bond pricing formula:
2 / 2 / 2 $ $ (41)
where
Æ .
)
.
)
.
)
.
(42)
.
.
and where we have defined . (43)
.
¿From equation (41), we note that products of bond prices (with differing
maturities) will take the form:
$ $ (44)
As in the Gaussian case, we can compute (for all relevant measures) the
moments of the distribution of a coupon-bond by noting
E
$ E E
$ $ (45)
$ (46)
$ $ (47)
22
where , !
!
that the solution to this expectation takes the form:
& $ . It is well known
$ where the functions 3 & & satisfy the Riccati Equations
(48)
&
&
Æ
&
&
3
(49)
(50)
with ‘initial conditions’ & !
, 3 . We find
Æ ! " ! ! !
where we have defined . #
! !
(51)
"
(52)
%
)
'
.
We can thus determine the relevant cumulants (see the Appendix) and parameter inputs . # that are needed to price the swaption via equation (31). The selected parameter values are provided in Table 3. Figures 3 and 4 show respectively the absolute and relative deviation of our
approximation relative to a Monte-Carlo solution. The Monte Carlo prices
are 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 errors of order less than (e.g. 5,000,000 paths with standard variance
reduction techniques). 12 As the figure shows, our approximation is excellent. The absolute error relative to the true solution is less than a few
12
We also used a third pricing approach, a standard numerical integration technique, with similar results, thus not reported.
23
$ $ 0.04
0.02
Æ
Parameters
%
%
*
*
'
'
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 have
negligible values thus making this type of metric somewhat misleading.
Our approximation takes less than seconds to compute all swaption
prices (corresponding to different strikes).
In Table 4, we present the mean, variance, and the 3 -5# scaled cumulants for each of the (N+1)=21 measures. Note that the third cumulant
is now negative. This can be understood as follows: under the square
root process, higher interest rates lead to higher volatility, in turn leading
to an upward skew in interest rates, which produces a downward skew
for (coupon) bond prices. Also note that the cumulants do not decay as
quickly as in the Gaussian case, leading to a slightly larger error for this
case.13 Finally, note that the 5th scaled cumulants are not as similar as
they were in the Gaussian case. As such, for numerical efficiency one
can choose to compute only two of them, corresponding to the shortest
and longest forward-measure maturities, and then estimate the others via
interpolation as a function of forward-measure maturity.
2.2.4
Appendix For Cumulant Expansion Technique
Relation between Cumulants and moments
For reference, here we provide the first seven cumulants , in terms of
the moments 4
. The formula that relates cumulants and moments can
Note that it would be appropriate to go to the level, even if we still set to zero. Indeed, one can expect
a 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 is
computationally costly and grows exponentially in the order.
13
24
Figure 3: Difference between cumulant approximation and Monte-Carlo swaption prices for various strike prices. The
paths and setting . The standard error
parameters are as in table 3 above. Monte-Carlo are run using of the Monte-Carlo prices are less than .
Figure 4: Relative difference between cumulant approximation and Monte-Carlo swaption prices for various strike prices.The
parameters are as in table 3 above. Monte-Carlo are run using paths and setting . The standard error
of the Monte-Carlo prices are less than .
25
measure
mean
variance
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1.23511950
1.23530353
1.23546977
1.23561989
1.23575544
1.23587782
1.23598829
1.23608800
1.23617799
1.23625919
1.23633246
1.23639857
1.23645821
1.23651202
1.23656055
1.23660433
1.23664382
1.23667944
1.23671156
1.23674053
1.23676666
0.00160541
0.00160333
0.00160145
0.00159976
0.00159823
0.00159684
0.00159560
0.00159447
0.00159346
0.00159254
0.00159172
0.00159097
0.00159030
0.00158969
0.00158915
0.00158865
0.00158821
0.00158781
0.00158745
0.00158712
0.00158683
-0.04087912
-0.04089828
-0.04091559
-0.04093121
-0.04094532
-0.04095805
-0.04096954
-0.04097991
-0.04098927
-0.04099772
-0.04100534
-0.04101222
-0.04101842
-0.04102402
-0.04102907
-0.04103363
-0.04103774
-0.04104144
-0.04104479
-0.04104780
-0.04105052
0.00205713
0.00206201
0.00206642
0.00207041
0.00207401
0.00207725
0.00208019
0.00208284
0.00208523
0.00208739
0.00208933
0.00209109
0.00209268
0.00209411
0.00209540
0.00209656
0.00209761
0.00209856
0.00209941
0.00210018
0.00210088
0.00012643
0.00012582
0.00012526
0.00012476
0.00012429
0.00012388
0.00012351
0.00012317
0.00012285
0.00012259
0.00012234
0.00012210
0.00012190
0.00012172
0.00012155
0.00012140
0.00012128
0.00012116
0.00012103
0.00012096
0.00012085
Table 4: Mean, variance, and scaled cumulants for the 20 forward measures and the risk-neutral measure for the two-factor
CIR model.
26
be found in, for example, Gardiner (1983).
The coefficients in the approximation of order Define " be written
where
!
!
" # .
" !
! Then, the probability density can then
! (53)
" "
(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, for
which there are excellent approximations to, without the need of numerical integration.
The first seven are:
N
N
(61)
N
N
(62)
(64)
(65)
(63)
(66)
(67)
(68)
The relevant coefficients for the are obtained by collecting terms of the same
powers in equations (54)-(58). They are
28
(69)
(70)
(71)
(72)
(73)
(74)
(75)
(76)
Remark: Alternative approaches are:
Piece-wise linear approximation of the exercise boundary combined
with using Fourier inversion of Umantsev and Singleton (2003).
Wei (1997) for single-factor case and Munk (1999) for multi-factor
case show that one can approximate the coupon bond option price by
the price of a zero-coupon bond option with maturity equal to stochastic 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-Diffusion
model of Duffie, Pan and Singleton (2000).
Fix a filtered probability space and assume is a
Markov process in some state space ( Ê that solves the following
SDE:
4 5
(77)
29
where is a vector of + independent Brownian motions, and is a pure
jump process with jump size distribution 6 on Ê and jump intensity
# , and
4/ / for Ê Ê //
7 7 /, for 7 7 7 Ê Ê #/ 8 8 /, for 8 8 8 Ê Ê / Æ Æ /, for Æ Æ Æ Ê Ê 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
: 7 : 8 : (78)
: 7 : 8 : (79)
Æ : with boundary conditions : 9and , and where we have
defined the ‘jump transform’ Ê . 6 0 .
: , Æ
: Most option pricing requires solving the following expectation:
!/
0 ) 14
/
Note that since % is a tensor, & % & is a vector with ' element
30
#$
& & %
This can be done by noting that the Fourier-Stieltjes transform of ! is
given by:
Ê
!) "
- ";< / 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 derivatives.
31
3 Bond and Forward Rate models: The HJM approach
Assume uncertainty is generated by -dimensional vector of independent
Brownian motions on a standard filtered probability space where the filtration is the natural filtration generated by the Brownian motion.
We assume a family of zero-coupon bond prices processes following well-defined SDEs:
have
4 (82)
Then by definition we obtain:
( 4( 4( ( 4 (83)
(84)
(85)
and (if it exists) the short rate follows:
4 Proof: Since 4 ( 4( (86)
(87)
(88)
Itô gives:
4 32
¾
Alternatively, had we started from a family of forward rates satifying (83)
then bond prices would satisfy (82) with:
4 Proof: Since where =
= ( 2
(89)
4( 2 ( 2
(90)
Itô gives
= = = 2.
The result follows. ¾
4( 2 ( 2 Let us emphasize that the relations between the drift and diffusions of
forward rate and bond prices derived in equations ?? are simple consequences of Itô’s lemma and the defining relation between forward rates
and 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 different
maturities: + . 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 the
dollar amount invested in each security as $ + . Itô 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, then
by 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 also
orthogonal to the vector of excess returns. Standard algebra then shows
that 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 of
risk satisfies additional technical restrictions, then it defines an equivalent
so-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 prices
is 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 maturities. In that case, the martingale measure is unique and markets are
complete (i.e., all fixed income derivatives can be replicated by an
appropriate trading strategy involving only bonds). We will see later
this is not necessarily the case.
The equivalent martingale measure is defined by the Radon-Nykodim
) .This defines an equiv) 1
derivative: alent probability measure if and only if . Liptser and
Shiryaeve provide necessary and sufficient conditions on . for what
they call the diffusion case (theorem 7.19 p. 294)). For the general
case, a sufficient condition is the so-called Novikov condition.
A priori, in our argument above the equivalent probability measure
may depend on the choice of the maturities selected. In fact, the
choice of maturities may change over time (Bjork et al (1995)and
Heath, Jarrow and Morton (1992) discuss this point further). However, once such a martingale measure is selected it should, in principle, price all other bond prices. In other words, the price processes of
reference bonds define a risk-neutral measure and all other bonds
can be perfectly replicated by an appropriate trading strategy involving 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 existence
of an EMM precludes the existence of arbitrage opportunities. Indeed
consider any self-financing portfolio + then 2 +
2 .
Using the EMM we have 15 15
2 . And we see that if we
Technically, we also need to further restrict the trading strategy to those that are martingale generating, i.e. we want the
stochastic integral to keep the martingale proprety of the integrator. See Dybvig and Huang (1989)
35
have an arbitrage strategy such that % (because ).
and % % then
Now using the EMM we obtain the HJM-restrictions for bond prices and
forward 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 . ( 2( ( (96)
(97)
and bond prices would satisfy (82) with:
4 ( 2
( 2. (98)
(99)
The above relations were first derived in full generality by Heath Jarrow
and Morton (1992). They show that absence of arbitrage imposes strong
36
restrictions on the drift of forward and bond prices. The most striking
result is the fact that under the EMM Q-measure, the drift of forward rates
is entirely specified by its diffusion term. Of course, this is equivalent to
the more traditional result that the drift of bond prices be equal to the
short term rate under Q (the local expectation hypothesis holds under the
Q measure).
This HJM restriction implies that given a specification of forward rate (or
bond) volatilities, once we have fixed the drift of arbitrary maturities forward rates (or bond prices) under the historical measure, then the market
price of risk vector, . , is fully determined. In turn, the martingale measure
and hence all forward rates (or bond prices) are uniquely determined by
the no arbitrage condition (and initial term structure).
Another often confusing point, is that these restrictions must be satisfied
by any arbitrage-free model. In that sense all (reasonable) models are
HJM models. However, market practice has widely associated the term
HJM-model to designate models that have (i) the ability to fit the observed
term structure at some particular date perfectly, and (ii) model the dynamics of forward rates as a starting point (instead of bond prices or futures
prices for example). We will perpetuate this (unfortunate) tradition.
The ‘HJM model’ as presented above is an ‘empty shell’. It is the weakest
restriction one must impose on a term structure model for it to be sensible. To obtain more implications for derivative pricing we need to impose
more 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 the
HJM restriction. In that sense all models are HJM models. However, following a now standard terminology we define in the following as ‘HJMmodels’ all models that (i) fit the term structure, (ii) specify the dynamics
of forward rates as primitives. Such models are designed to price derivatives relative to observed bond prices. Starting from:
4( ( (100)
4 (101)
. The definition implies
( 2
(102)
4 4( 2 (103)
( 2
Assuming a family of bond prices is traded and provided
where (or equivalently ( is well-behaved) we have many more securities
then 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 existence
of a market price of risk . such that
4 . (104)
(105)
Provided . verifies some regularity conditions (such as the Novikov conditions) then . defines a brownian motion under . Under we have
38
4( ( ( . ( (106)
( 2 ( (107)
Indeed combining 104, 102 and 103 we obtain
( 2
. and differentiating with respect to
4( 4( 2 ( 2
we obtain
( . ( ( 2
As such the model is an ‘empty shell.’ We need to impose more structure
on the model to go further. All the action to price derivatives comes from
(i) the number of factors driving the term structure , (ii) the specification
of the volatility structure of forward rates ( which entirely determines
the dynamics of forward rates under the risk-neutral measure, and (ii) the
initial 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 forward
rates. Consider the one-factor model:
2 -2
2 2 -2 02
39
(109)
where 0 2 is a regular Brownian motions, -2 is deterministic, and
we have defined 2 -2 99.
The drift of the forward rate dynamics under the risk-neutral measure is
determined by the HJM restriction.
Since bond prices are defined by
2 !
; 2 (110)
Itô’s lemma implies the bond price dynamics are
2
2
2
2
02 (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 model
proposed above. However, as we show below in a more general case, it is
still possible to price derivative securities such as bond options.
While we are able to provide closed-form solutions for a large number
of derivatives, we cannot in general propose simple algorithms to price
path-dependent instruments such as American options. In this section
we show that by specializing further the choice of -2 we can obtain a Markov representation of the term structure and hence price all assets using partial differential equations techniques. Indeed, if we choose
-2 ! 2 in equations 109, then the short rate is onefactor Markov and bond prices are exponentially affine in . All fixedincome derivatives are solutions to a partial differential equation, subject
to appropriate boundary conditions.
Proof: Integrating the forward rate dynamics we obtain:
40
= (112)
where we have defined
2 = 2 % % 0 2 % (113)
(114)
Applying Itô’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:
% Thus bond prices satisfy:
% = (118)
$ % (119)
$ % & $% % ' & (120)
& (121)
Finally, let us consider a path-independent European contingent claim
that has a payoff at time that is a functionof the entire term structure
at time , i.e. > > . The price of that security
is > ! 22> where the second
equality follows from the Markov-property. Moreover a standard argument (which requires some regularity conditions on and its derivatives,
41
is a martinsee Duffie Appendix E p.296) shows that gale and that its drift must vanish, or equivalently E Using Itô’s lemma we obtain the partial differential equation for the price
of the European contingent-claim:
= (122)
¾
Note that
In this model all path-dependence is captured by the stochastic integral 5 % 0 2 which follows a one-factor Markov
Gaussian process: 5 5 .
Note that it is straightforward to generalize the approach to functions
of the form -2 - -2 (as we show below in a multi-factor
version).
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 derived
in 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 volatility
as follows:
42
2 -2
2 2 2 -2 2 02
where:
2 ? ?
(123)
(124)
In other words the volatility is driven by the short rate. In that case, following the previous derivation one can show that if -2 -2-
the short rate becomes two-factor Markov in =. Indeed, = is not
deterministic anymore and becomes an additional state variable. Since
the analysis is similar and this model is nested in the one presented next
we do not derive the results.
The results can be extended to more general diffusion functions
Ritchken and Sankarasubramaniam (1995) show that two-factor Markov
representation obtains for ( 2 % 3 . This is a particular case of Cheyette (1995).
( / %% is also of the form analyzed above and leads
to one-factor Markov representation (Rebonato (1998)). It allows
humps in the volatility structure, but leads to non-stationary volatility
structure (exercise).
( is stationary, allows for hump-shaped
volatility structure, but requires two state variables for Markov representation 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 markovian case. With separable -2 , a Markov representation would also
obtain.
The affine (square-root) diffusion allows as we show below to derive
the Fourier transform of the log of bond prices in closed-form and
thus by inversion get derivative prices.
3.3.3
Hedging in one-factor HJM Model with affine volatility structure
All one-factor (Brownian motion) models, whether or not they are Markov,
require a single bond and the money market to hedge any fixed-income
derivative.
For the particular examples analyzed above where a Markov representation exists in terms of = and where = is locally deterministic,
hedging is simple to implement.
Consider again a path-independent European contingent claim that has
a payoff at time that
is a function ofthe entire term structure at time
, i.e. > > . The price of that security is
>
!
22>
= where the sec-
ond equality follows from the Markov-property.
Since markets are complete any contingent claim can be replicated by a
self-financing admissible trading strategy , with corresponding portfolio , . Thus > -2. Using the Risk neutral
measure and the fact that the strategy is self-financing, we have:
, = >
44
. Applying Itô and matching diffusion terms we find:
=
=
And the self-financing condition gives:
, =
=
The hedging portfolio corresponds to the relative sensitivity to the sole
source 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 which
nests the one factor Vasicek and affine model. It is interesting because
it remains tractable as it allows to price derivatives in a non-Markovian
framework,
it accommodates level-effects in volatility, item it captures the notion
of 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 that
case we consider existence issues in greater detail.
Consider the two-factor model:
-2 2 2 2 -2 2 02 (125)
2 " 2 2 < 2 1 0 2 1 0 2 ( 126)
2
45
where 0 2 and 0 2 are independent Brownian motions, and we have
defined
"2 2 2 @ @ 2 @ 2
? ? 2 ? 2
; -2 ; (127)
(128)
(129)
The drift of the forward rate dynamics under the risk-neutral measure is
determined by the HJM restriction. 16
Since bond prices are defined by
2 !
; 2 (130)
Itô’s lemma implies the bond price dynamics are
2
2
2
2
2 02 (131)
Since there is only a single Brownian motion 0 2 in equation (131),
all bond price innovations are perfectly correlated. However, the Brownian motion 0 2 affecting volatility dynamics in equation (126) will
generate innovations in fixed-income derivatives that cannot be hedged
by portfolios consisting solely of bonds.
As specified, the model leads to very general dynamics for the term structure of forward rates, and hence also for the risk-free rate. In particular,
when the
of the the
functions -2 are chosen arbitrarily, the dynamics
system 2 will in general be non-Markov. 17 Regardless, the
As noted in Andreasen, Collin-Dufresne and Shi (1997), the HJM restriction alone does not identify the process of under
the 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.
17
Cheyette (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).
16
46
chosen specification for both the volatility-structure of forward rates and
the dynamics of the volatility-specific state variable generates a dynamical
system that can be characterized as an ‘extended’ affine system. 18 That
is, even though this system falls outside the class of models investigated
by DPS 2000, we are nevertheless able to derive closed-form solution for
derivatives using an approach introduced by Heston (1993) and generalized by DPS 2000. Indeed, to determine the date-2 price of a bond-option
with exercise date- on a bond that matures at date- , we have shown
above that only the characteristic function of the date- bond price under
the 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(132)
2
2 " 2 2 < 2 1 0 2 1 0 2 (133)
where we have defined
" 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:
# & & 18
& (135)
Our 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 rate
is not Markov in a finite number of affine state variables.
47
where 3 *
are solutions to
3 2 ? 2 3 2 @
* 3 2 ? 2 3 2 @ 2 3 2 # 2 % where
$ $ $
) () (136)
(137)
(138)
$ $ $
$ $ $ * $ * + (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 modeled
within an affine structure.
Proof:
It is sufficient to show that given in equation (135) above is a martingale. Indeed, if is a martingale, then
E E
19
Notice that eq. 138 gives + + ,
48
, .
Applying Itô’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.
¾
Using the closed-form solution for the characteristic function, we show
next that the price of a European bond option with strike price can
easily be obtained following the approach introduced by Heston (1993)
and extended by, among others DPS (2000). In turn, the prices of caps
and floors are derived as portfolios of zero-coupon bond options (see, for
example, 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 asset
is a discount bond with maturity , is
2 (141)
Under the - and -forward measures, the bond-option price at an earlier
date-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 DPS
above.
49
Define ! E where for simplicity we write and . Then the Fourier-Stieltjes transform of ! is
(note that ! È ):
Ê
! "# We can thus find ! by Levy inversion (see Williams (1991)):
! $
# "#
#
#
Whence, the bond option price can be written as
2
"#
2
# Re
$
"#
"#
(143)
2
# Re
$
"#
where the characteristic function of under different forward
measures has been defined above. The implication of Equation 143 is
that, if the characteristic function can be written in closed-form, then so
can the bond-option price. This result shows how tractable the generalized
affine 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 to
obtain a Markov representation for the model proposed above. While we
are able to provide closed-form solutions for a large number of derivatives,
we cannot in general propose simple algorithms to price path-dependent
instruments such as American options. In this section we show that by
specializing further the choice of -2 to the widely used exponential
50
case20 we can obtain a Markov representation of the term structure and
hence price all assets using partial differential equations techniques. We
claim:
Proposition 1 Define = 2 % 2. If -2 ! 2 in equations 125-126, then the model possesses a Markov representation in the three state variables = . The state vector is
affine, and bond prices are exponentially affine functions of the subset
of the state vector. All fixed-income derivatives are solutions
to a partial differential equation, subject to appropriate boundary conditions.
Proof: Integrating the forward rate dynamics we obtain:
where we have defined
2 = 2 % 2 % 2 = 0 2 (144)
% 2 (145)
(146)
Applying Itô’s lemma we obtain the dynamics of , = :
0 = = (147)
(148)
Using equation (144) we obtain the dynamics of the short-term rate:
= 20
0
(149)
See for example: Carverhill (1994), Jeffrey (1995) , Cheyette (1995), Ritchken and Sankarasubramaniam (1995) and de
Jong and Santa-Clara (1999). It was pointed out to us by a referee that De Jong and Santa-Clara (1999) actually obtain the same
closed form we derive below, without however identifying the link with True Stochastic Volatility models.
51
Recalling the definition of
Markov system.
, it is clear that = form a
More generally, the forward rates may be written as:
% % = (150)
Thus bond prices satisfy:
$ % (151)
$ % & $% (152)
% ' & where we have defined & (153)
.
,
Finally, let us consider a path-independent European contingent claim that
that is a function
has a payoff at time
of the entire term structure at time
, i.e. > > . The price of that security is > !
= where the second
22>
equality follows from the Markov-property. Moreover a standard argument (which requires some regularity conditions
on andits derivatives,
see Duffie Appendix E p.296) shows that !
2 2 = is a martingale and that its drift must vanish, or equivalently
E = = Using Itô’s lemma we obtain the partial differential equation for the price
of the European contingent-claim:
)
& ()
52
'
% % & '
(154)
¾
This result illustrates the impact of true stochastic volatility state variables. From equation (153), we see that bond prices are exponential-affine
functions of and = alone. Since = is locally deterministic, the innovations of any bond can be hedged by a position in any other bond and the
money-market fund. However, = are not jointly-Markov. As a consequence, the dynamics of bond prices over a finite time period depend
on the dynamics of the additional state variable as well. An implication
is that bond prices alone do not permit a complete hedge against changes
in , since 2 0 2 Also, we note that it is straightforward to
generalize the model functions of the form -2 - -2. 21
In general, it is not possible to guarantee that the above stochastic differential equations for and are well-defined. Indeed, for general initial
term structures and parameter choices, may take on negative values.22 The following lemma demonstrates that there exists a feasible set
of parameters such that remains strictly positive (almost surely) and the
SDE’s are well-defined.
Proposition 2 If the parameters and the initial forward rate curve satisfy:
1. , ?
2. ? @ ? @ ?
3. % 4. ? ? ? @ ?
@
21
? ? < 1<? ? In the appendix ?? we provide an example of such an extension in a multi-factor version of the model presented in this
section.
22
Moreover, 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 ; and
the stochastic volatility are well-defined.
Proof: Note that under condition 1 of the proposition
, % , % , " , - , - - . + " " ? ? < 1<? ? and
- ? 1 ? < 0 1 ? < 0 is a standard Brownian motion and 5 . A minor adaptation of the proof of the SDE
where :
Theorem in DK 1996 (which extends Feller (1951) to a vector of affine
processes) to account for deterministic coefficients in the drift of , allows
us to conclude that the SDE for forward rates and stochastic volatility state
variables are well-defined. ¾
Note that the above proposition puts joint restrictions on both the feasible
set of parameters and the initial curve of forward rates. Also note that a
Markov representation exists for this special choice of volatility structure,
implying this model has an affine structure in the sense of DK 1996 or
DPS 2000, but with two distinct features: (i) it is consistent with the initial
term structure and (ii) it results in only a subset of the state variables
entering the bond prices exponentially (i.e. the loading of the log-bond
price is zero for the state variable ).
This approach provides a straightforward and efficient method to construct HJM affine models with stochastic volatility. It can be extended
to the infinite dimensional ‘string’ models analyzed by Kennedy (1994),
Goldstein (2000), Santa-Clara and Sornette (2000), while retaining the
analytical tractability of the finite dimensional models proposed above
(Collin-Dufresne and Goldstein (2000)).
54
We provide a generalization to multiple factors of the previous HJMmodel. We consider the affine multi-factor case directly, since it nests
the Gaussian case.
3.7 The Multi-factor affine case
We specialize the model to obtain a simple Markov representation of the
term 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 written as follows. Assume the forward rate dynamics and stochastic volatility
are specified for all 2 as:
2
2
"2 2 2 $2 "2
-2
-
2 2 2 - 2
0
(155)
2 2
-2
$2 0
2 20
2 2
@
2
@
2 ? ? 2 ? 2 ? 2 @ @ .
.
2 . 2 .
; -; 2 (156)
(157)
(158)
(159)
(160)
and all parameters @ ? . * are assumed to be at most
deterministic functions of time. Also, it is straightforward to generalize
Assuming * * , Frachot and Lesne (1993) and Frachot, Janci and Lacoste (1993) identify the restrictions these functions must satisfy to obtain a linear-factor representation of the term structure. In their model, the factors
are constant-maturity forward rates. Note that, while quite general, our separability assumption precludes for example the
one-factor extended CIR model.
24
In the absence of ‘true stochastic’ volatility state variables, our volatility structure is a special case of Cheyette’s general
class of volatility structures that allow a Markov representation (see also Ritchken and Sankarasubramaniam (1995) and De
Jong and Santa-clara (1999)). It turns out that the affine structure, even in the presence of ‘true’ stochastic volatility state
variables, considerably simplifies when a Markov representation is possible.
23
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 system comprising 2 and the vector = 2 =2
is Markov,
where we define:
- = 9 9
(161)
9
9
- = 9 9 0
9 9 (162)
- 9
- 9
The bond prices are exponential affine in the state variables . Further
more the price of any fixed-income derivative securities solves a second
order partial differential equation.
Proof: Note that
= = (163)
and more generally:
; ; = -; = - - (164)
Bond prices are given by the formula:
!
!
; ; (165)
56
3
= 3 = (166)
where " + we have defined:
3 3
-; ; ; -; ; - (167)
(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 Itô’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 definition of , it is clear that the above system is Markov. 25
Finally, let us consider a path-independent European contingent claim that
has a payoff at time that
is a function of the entire term structure at
time , i.e. > > . The price of that security
is > ! 22> where the
second equality follows from the Markov-property. Moreover a standard
argument (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 Itô’s lemma and expressing in terms of the state variables
we obtain the partial differential equation for the price of the European
25
In 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
¾
This again illustrates the impact of unspanned stochastic volatility
state variables. From equation (166), we see that bond prices are exponential affine functions of . Thus the instantaneous dynamics
of bond prices are determined by the innovations of these state variables. Thus, any combination of + distinct bonds allows to hedge
against innovations in these state variables. However, we emphasize
that alone is not Markov. As a consequence, the dynamics of bond
prices over a finite time period depend on the dynamics of the additional state variables 2 as well. An implication is that bond prices
alone 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 traditional sense. Note also, that the state variables = " + are
actually locally deterministic, which simplifies the numerical computations 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 the
system is Markov in the + Variables = " +. In fact, from the
definition 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 can
be solved in closed-from as in section 1.1. However the approach of
section 1.4 for pricing coupon coupon bond options does not work
anymore.
For very fast and accurate approximations for prices of zero-coupon
as 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 in
the HJM framework as well.
For hedging at least + instruments are necessary. Note that the number of hedging instruments corresponds to the number of factors which
is in general smaller than the number of state variables required to describe 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 state
variables.
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 derivatives.
– 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 structure
shapes (present and future), and arbitrary variance covariance structures.
– 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 Field
models.
– 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 choice
problem.
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, this
defines affine models - DFS)
@
/
62
! Characteristic function related to probability distribution (e.g., moments).
Derivative prices depend on probability that asset ends up ‘in the
money.’
Many derivatives priced by Fourier inversion of characteristic function.
Heston (1993)
Analytic solutions for bonds and many derivatives
Caps, Floors, Options on zero-coupon bonds,
Options on baskets of yields, Options on futures...
Analytic solutions for moments of bond portfolios
Fast, 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 and
Sun (2001); Collin-Dufresne and Goldstein (2001); Heiddari and Wu
(2001)).
– Some factors driving implied volatilities of Caps and Floors
are distinct from factors driving yields (USV).
– In standard affine model any factor driving volatilities also drives
yields.
Predictability of bond returns (Duffee; Cochrane and Piazzesi).
– CP show that a specific linear combination of all available forward
rates has predictive power for future bond returns.
63
– In N-factor affine framework, a linear combination of any N forward 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’ approach takes
bond 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 expressed
in 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 representations
– 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 of
the 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 maturities) 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 approximations
Longstaff, Santa-Clara and Schwartz (2001); Han (2002)
4.5 Generalized Affine Models
Combines Tractability of Traditional Affine Models with Flexibility
of 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 cannot be expressed as linear functions of the (finite-dimensional) state
vector. Thus no bond price is redundant in this model.
Each bond price is itself a state variable: The dimension of the state
vector is infinite, as it consists of a continuum of maturities of bonds
(and )
No restrictions placed on the deterministic volatility and correlation
structures and .
Both volatility and correlation of bond prices are stochastic, and volatilityrisk (? ) 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, forwards, futures, coupon bonds, convexity bias.
67
4.6 Relative Pricing of Caps and Swaptions
Previous Studies document relative mispricings between caps and
swaptions:
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 place
on joint evolution of:
1) interest rates
2) volatility
3) correlation
As is well-known:
Cap (Floor) Portfolio-of-Options
Swaption Option-on-a-Portfolio
Relative pricing of caps and swaptions driven by correlation structure
Generalized affine class provides framework to disentangle effects of
correlation, volatility and interest rates.
Indeed, consider for example the simple model:
2
2
2
2 5 2
5 2 5 2 The caplet price (maturity
22 , 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:
If Æ is small, then ’numeraire effect’ is large.
.
Æ
For caps, Æ = 6 months.
Thus Caps are strongly negatively related to changes in bond yield
correlations
For Swaptions, Æ can be large. In addition, increase in correlation increases volatility of portfolio of bonds, generating a partially-offsetting
effect.
Swaptions are relatively insensitive to changes in correlation of
bond 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 estimate
bond-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 bond
volatilities ( and not on their correlation structures ( .
However, the forward bond volatility ( is a function of bond yield
correlation
structure . Thus caps are affected by bond correlations, but not
by forward bond correlations.
Cannot compare ‘correlation effect’ directly across these two different frameworks!
70
The generalized affine framework (whether expressed in terms of forward bonds or bonds) can capture by construction the relative prices
of caps and swaptions.
( can be used to calibrate cap prices perfectly and ( can be used to
calibrate swaption prices perfectly and independently.
With truly stochastic correlation and volatility, need both caps and
swaptions to estimate (or calibrate) model.
– In forward bond model, swaption are more sensitive to changes in
correlations (( ) than are caps . In fact, caps are only functions of
volatility ( .
– In bond model, caps are more sensitive to correlation () than are
swaptions.
4.8 Optimal Portfolio Choice and Preferred Habitat
Consider Dynamic portfolio choice when agents can invest in continuum of assets (Heaney and Cheng (1984), Bjork et al. (2000))
Consider general affine model under the historical measure:
2 , 2 2 2 2 2 0 2 (177)
2
the stochastic volatility state variable has -measure dynamics given
by:
2 2 2 6 2 ?2
(178)
where ? is a Brownian motion, and ? 2 0 2 for all maturities .
Risk-premia are given by 0 2 0 2
71
, 2
2 2
The correlation structure is specified as: 0 2 0 2 22
In this model, no bond is redundant.
Consider an agent who can invest in a continuum of zero-coupon
bonds with
maturities consumption:
to maximize his expected utility of terminal
) !
)
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-9
held.
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 $ +20 +2 where, we introduce the notation
$ , ;$ +2 ,+2 and $ + 72
;+
2$ 2(181)
Impose that trading strategies be admissible (i.e., such that SDE for
is
well-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:
1 ) (182)
5 has the following solution:
.
) 1 (183)
where the functions solve the system of ODE (s.t. ):
. ,
where: ($ . +
6
(184)
. (185)
. ($ +
+
$ $ $ A and A , )
73
(186)
The optimal trading strategy solves:
($ 5 s.t. $ The actual portfolio holding $& + $++ are:
+
#
.
$&+ + , .
.
/
(187)
where # is the Lagrange multiplier associated with the portfolio constraint given by:
, ' # ' ' (188)
In contrast to finite dimensional model we obtain a unique optimal
portfolio choice.
Agents invest more in assets with higher mean return or lower correlation with overall portfolio (Merton (1973)).
In contrast to Merton (1973), preferred habitat for maturity
translates in over (under) investment in preferred habitat bond depending
on 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 are
predictable and driven by a single factor which is a portfolio of forward rates.
This is easily captured in generalized affine framework.
74
Consider, for example, the dynamics:
2
2 , 2'2 2 2 0 2
2
Risk-premia are driven by innovations of all forward rates:
' 4 4' 60 2
(189)
(190)
The instantaneous sharpe ratio (or risk-premium) on bonds are given
by , '2.
Consistent with CP, each forward rate contributes information to predictability regressions since:
Corr ' 6
66 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 +
. +
A+ A+ '
6 + ,+
& 6 + '
The optimal trading strategy is thus given by
/
/
! / / /
/
/
where # # # ' is the Lagrange multiplier associated with
the 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 changes
in 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 stochastic correlation and volatility changes, can fit relative pricing of
caps and swaptions by construction. Both instruments are required to calibrate/estimate the model.
Investigate optimal bond portfolio choice for agent with preferred
habitat 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 maturity depending on whether more (less) risk-averse than log.
– Can accommodate time varying sharpe ratios consistent with predictability regressions of Cochrane and Piazzesi (2002). In that
case, optimal bond portfolio holdings are time varying due to
hedging demand.
References
[1] A. Carverhill. When is the short rate markovian. Mathematical Finance, 4 (Oct):305–312, 1994.
[2] O. Cheyette. Markov representation of the Heath-Jarrow-Morton
model. BARRA Inc. working paper, 1995.
[3] P. Collin-Dufresne and R. S. Goldstein. Generalizing the affine
framework to HJM and random field models. Carnegie Mellon University Working Paper, 2002.
77
[4] Q. Dai and K. J. Singleton. Specification analysis of affine term
structure models. Journal of Finance, 55:1943–1978, 2000.
[5] F. de Jong and P. Santa-Clara. The dynamics of the forward interest
rate curve: A formulation with state variables. JFQA, 34:131–157,
1999.
[6] D. Duffie, J. Pan, and K. Singleton. Transform analysis and option
pricing for affine jump-diffusions. Econometrica, 68:1343–1376,
2000.
[7] Dybvig and Huang. Non-negative wealth, absence of arbitrage, and
feasible 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 term
structure: A probabilistic approach. Banque de France Working Paper, 1993.
[10] A. Frachot and J.-P. Lesne. Econometrics of the linear gaussian
model 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 term
structure of interest rates: A new methodology for contingent claims
evaluation. Econometrica, 60:77–105, 1992.
[13] S. L. Heston. A closed form solution for options with stochastic
volatility. Review of financial studies, 6:327–343, 1993.
[14] J. Hull. Options, Futures and Other Derivative Securities. Prentice
Hall, 2000.
78
[15] F. Jamshidian. Contingent claim evaluation in the gaussian interest
rate model. Research in Finance, 9:131–170, 1991.
[16] A. Jeffrey. Single factor Heath-Jarrow-Morton term structure models
based on markov spot interest rate dynamics. Journal of Financial
and Quantitative Analysis, 30n04:619–642, 1995.
[17] N. E. Karoui and J. Rochet. A pricing formula for options on couponbonds. Cahier de Recherche du GREMAQ-CRES,no. 8925, 1989.
[18] T. C. Langetieg. A multivariate model of the term structure. Journal
of Finance, 35:71–97, 1980.
[19] P. Ritchken and L. Sankarasubramaniam. Volatility structure of forward rates and the dynamics of the term structure. MF, 5:55–72,
1995.
[20] D. Williams. Probability with Martingales. Cambridge University
Press, 1991.
79