Correlations of Bond Yields

8/8/2019 Correlations of Bond Yields

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Comments Welcome

Stochastic Volatilities and Correlations of

Bond Yields

Bing Han

Current Version: October 2004

Han is with the Fisher College of Business at the Ohio State University. I thank Martin Dierker,Mark Grinblatt, Jean Helwege, Jason Hsu, Jingzhi Huang, Andrew Karolyi, Francis Longstaff, MonikaPiazzesi, Pedro Santa-Clara, Kenneth Singleton and seminar participants at the Ohio State Universityfor helpful comments and suggestions. Financial support from the Dice Center for Financial Economicsat the Ohio State University is grateful acknowledged.

Corresponding Address: Bing Han, Department of Finance; The Ohio State University; 700D FisherHall; 2100 Neil Avenue; Columbus, OH 43210-1144. Phone: (614) 292-1875. E-mail: [email protected];


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Abstract

I develop an interest rate model with separate factors driving innovations in bondyields and their volatilities. My model features flexible and tractable affine structure

for the covariances of bond yields. Maximum likelihood estimation of the modelwith panel data on swaptions implies pricing errors that are almost always lowerthan half of the bid-ask spread. Further, market prices of interest rate caps do notdeviate significantly from their no-arbitrage values implied by the swaptions undermy model. This supports conjectures by Collin-Dufresne and Goldstein (2003), Daiand Singleton (2003), and Jagannathan, Kaplin and Sun (2003). I also extractmarkets view on the term structure of interest rate volatility and dynamics of bondcovariances.


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1 Introduction

The turmoil in the bond markets and increased interest rate volatility since the 1980s

have provided a boost for the rapid growth of hedging vehicles such as interest rate caps

and swaptions. As these interest-rate derivatives become more liquid, researchers startusing their prices to evaluate term-structure models. There is a rich cross-section of

swaptions and caps. Their prices are sensitive to both volatilities and correlations of the

bond yields and contain valuable information regarding markets view on the evolution

of the yield curve that is not available from interest rate data such as Treasury bonds,

Eurodollar futures or swap rates.

Recent studies find that it is challenging to explain market prices of swaptions and

caps under many popular term-structure models, even for models that by construction

fit the bond prices exactly. For example, Longstaff, Santa-Clara and Schwartz (2001)

calibrate string-market models to the swaption data. They find that short-dated and

long-dated swaptions tend to be priced inconsistently, and cap prices periodically deviate

significantly from the no-arbitrage values implied by the swaptions. Other studies that

document large and systematic pricing errors for swaptions and caps include Driessen,

Klaassen and Melenberg (2003) and Fan, Gupta and Ritchken (2001) for multi-factor

HJM models, and De Jong, Driessen and Pelsser (1999), as well as Hull and White (1999)

for the Libor and swap market models.

For the models studied in these papers, when model parameters are held constant, the

covariances of interest rates are either deterministic or depend at most on the underlying

interest rates. In the empirical work, models are recalibrated each date in order to capture

time variation in the covariances. While the recalibrated models yield significantly better

fit to the data compared to the case when model parameters are held constant (see, e.g.,

Driessen, Klaassen and Melenberg 2003), it is not enough to continuously recalibrate

simplistic models.

The affine framework of Duffie and Kan (1996) accommodates stochastic covariances

of bond yields, but may imply too strong restrictions on the covariances structure. Dai

and Singleton (2000) find that for the affine models to be admissible, there is an important

trade-off between flexibility in modeling the factor volatilities and correlations. In addi-

tion, the covariances of bond yields are affine functions of the state variables governing

bond yields in most affine models. Thus, risk factors that drive bond covariances gener-

ally can be hedged by a portfolio consisting solely of bonds. However, Collin-Dufresne

and Goldstein (2002) and Heidari and Wu (2001) show that interest-rate option market

exhibits risk factors unspanned by, or independent of, the underlying yield curve.

Empirically, Jagannathan, Kaplin and Sun (2003) find that multi-factor CIR models

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lead to pricing errors for caps and swaptions that are very large relative to the bid-

ask spread, although the fit to the swap rates are very good. They suggest that the

inability of the multi-factor CIR model to value caps and swaptions correctly may be

due to the restrictions it imposes on the nature of stochastic volatility, and it may be

necessary to consider models outside the affine class that are flexible in accommodatingstochastic volatility of more general forms. Collin-Dufresne and Goldstein (2003) and Dai

and Singleton (2003) conjecture that the ultimate resolution of the swaptions and caps

valuation puzzle may require time varying correlations and possibly factors affecting the

volatility of yields that do not affect bond prices.

In this paper, I develop a term structure model with properties in the conjectures

above. My model incorporates stochastic volatilities and correlations of bond yields.

The factors driving the covariances of bond yields are independent of the yield factors.

Empirically, I find that market prices of interest rate caps do not deviate significantly

from their no-arbitrage values implied from the swaptions under my model. This supportsthe conjecture by Collin-Dufresne and Goldstein (2003), Dai and Singleton (2003) and

Jagannathan, Kaplin and Sun (2003). I am also the first to extract markets view on the

term structure of interest rate volatility and dynamics of bond covariances from interest

rate derivatives. Such information is valuable for risk management and valuation of exotic

interest rate derivatives.

The rest of this paper is organized as follows. Section 2 introduces interest rate caps

and swaptions and the market convention for their valuations. In section 3, I develop the

model and derive closed-form pricing formulas for European swaptions and caps. The

main feature of the swaptions and caps data are discussed in section 4. Section 5 presents

the econometric method used to estimate the model and section 6 presents the empirical

results. Section 7 concludes the paper.

2 Swaptions and Interest Rate Caps

The swaptions studied in this paper are written on semi-annually settled interest rate

swaps. An interest rate swap can be viewed as an agreement to exchange a fixed rate

bond for a floating rate bond. Every six months till the maturity of the swap, one

counterparty receives a fixed annuity and makes a floating payment tied to the six-monthLibor rate to the other counterparty of the swap contract. The coupon rate on the fixed

leg of a swap, also known as the swap rate, is set so that the present value of the fixed

and floating legs are equal at the start of a swap.

Fix two dates T > . A European style by T (or into T ) receivers swaptionis a single option giving its holder the right, but not the obligation, to enter into a T

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year interest rate swap at date and receive semi-annual fixed payments at a pre-agreed

coupon rate between date and T. Let D(t, T) denote the time t price of a discount

Libor bond that matures at time T. Then at t < , the value of a forward swap that

starts at and matures at T with a coupon rate c is given by

V(t , , T, c) =c

2

2(T)i=1

D(t, i) + D(t, T) D(t, ) (1)

where i = +i2

years. The sum of the first two terms is the present value of the fixed leg

of the swap, and the third term is the present value of the floating leg. It follows that the

payoff to the holder of a European style by T receivers swaption at its maturity date

is given by

Max(V( , , T , c), 0) = Max c2

2(T)

i=1

D(, i) + D(, T)

1, 0 (2)

Thus, a swaption is an option on a portfolio of discount bonds.

A European style by T swaption is said to be at-the-money-forward when the coupon

rate c equals the corresponding forward swap rate F SR(0, , T )

F SR(0, , T ) = 2

D(0, ) D(0, T)2(T)

i=1 D(0, i)

(3)

There is a second type of swaption called payers swaption where the option holder has

the right to enter into a swap and pay fixed. For the at-the-money-forward swaptions, a by T payer swaption is worth the same as a by T receivers swaption.1

European type at-the-money-forward swaptions are actively traded in the over-the-

counter derivatives market, and quoted in terms of implied volatilities relative to the Black

(1976) model as applied to the corresponding forward swap rate. The market price for a

by T at-the-money-forward swaption is obtained by plugging the quoted Black implied

volatility into the following formula

(D(0, )

D(0, T))N

(

2

)

N(

2

) (4)where N() is the cumulative density function of a standard normal random variable.

1Standard no-arbitrage arguments show that the value of a forward swap must equal the value ofthe corresponding receivers swaption minus the value of the payers swaption. By equation (1) and thedefinition of forward swap rate (3), the date-0 value of a forward starting swap whose strike equals thecorresponding forward swap rate is zero.

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An interest rate cap provides insurance against the rate of interest on a floating rate

loan rising above the cap rate and gives its holder a series of European call options or

caplets on the underlying Libor rates. Each caplet has the same strike price as the others,

but a different expiration date. For example, a T-year cap on the six-month Libor rate

consists of 2T1 caplets:2

the first caplet matures in one year, and the last caplet maturesin T years. Let ti =i2

years, ai be the actual number of days between ti and ti+1 and Lidenote the six-month Libor rate applicable to period [ti, ti+1]. Then the cash flow on the

caplet maturing at ti+1 isai360

max(0, K Li). Alternatively, each caplet can be viewed asa put option on a Libor discount bond. An interest rate cap is thus a portfolio of options

on discount bonds.

For date t < < T, let F(t , , T ) denote the Libor forward rate at time t applicable to

the period from to T. Since Li = F(ti, ti, ti+1), the ith caplet is an option on the forward

rate F(t, ti, ti+1). Assuming that each forward Libor rates F(ti, ti, ti+1) is lognormal with

known volatility i, the Black model price of a cap with cap rate R is2T1i=1

ai360

D(0, ti+1)

F(0, ti, ti+1)N(di) RN(di i

ti)

(5)

where

di =ln(F(0, ti, ti+1)/R) +

2i ti/2

i

ti

The market convention is to quote the price of a cap in terms of an implied volatility ,

which is constant across caplets, so that the Black model price above at i = equals the

market price of the cap. A T-year cap is said to be at-the-money if the cap rate R equalsthe current T-year swap rate. Only at-the-money European-type interest rate caps and

swaptions are actively traded in the over-the-counter market. These are the data I use in

the empirical study.

Although swaptions and interest rate caps are traded as separate products, they are

linked by no-arbitrage relations through the correlation structure of bond yields. The

relative valuation of caps and swaptions is a decreasing function of bond correlations. It

is important to note that the relative valuation of swaptions and caps can only be judged

under a term-structure model, and not by a simple comparison of their Black implied

volatilities quoted in the market.3

2Note that although the cashflow of this caplet is paid at time ti+1, the applicable Libor rate Li isdetermined at ti. For this reason, the cashflow for the first caplet maturing in six months is non-stochasticand thus omitted by market convention.

3The reason is that the Black implied volatilities apply to different underlying interest rates thatare assumed to be lognormally distributed: forward swap rates in the case of swaptions and forwardLibor rates in the case of interest rate caps. However, each forward swap rate is approximately a linearcombination of the underlying forward Libor rates. Thus, forward swap rates and forward Libor rates

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3 The Valuation Framework

With empirical work in mind, I now develop a term structure model with flexible and

intuitive specifications for the stochastic volatilities and correlations of bond yields. One

notable feature of my model is that bond covariances are affine in a set of volatilitystate variables. My model also captures the empirical evidences of unspanned stochastic

volatility (USV) documented in Collin-Dufresne and Goldstein (2002) and Heidari and

Wu (2001). In my model, there are separate factors driving the innovations in bond yields

and covariances of bond yields. Thus, my modelling framework is reminiscent of the large

literature on stochastic volatility (e.g., Heston 1993) that specifies the joint dynamics of

a traded asset and its volatility.

3.1 Model

Assume all bonds are traded. I directly model the risk-neutral dynamics of bond prices.

The risk-neutral drifts of bond prices are determined by the no-arbitrage condition that the

expected rate of return on all bonds equals the spot rate under the risk-neutral measure.

The focus of my model is bond covariances. I show in section 3.2 that once bond prices

are given, prices of swaptions and interest rate caps are determined only by the dynamics

of bond covariances.

By Girsanovs theorem, the instantaneous bond covariances are invariant with respect

to an equivalent change of probability measure. Thus, I can utilize information contained

in the historical estimates of bond covariances. It is well known that most of the observed

variation in historical bond prices are explained by a few common factors (e.g., Litterman

and Scheinkman 1991, and Dai and Singleton 2000). The explanatory power of these

factors are stable and the factor loadings show a persistent pattern. However, Bliss

(1997) and Perignon and Villa (2003) find significant time-variation in the variances of

the common factors.

Motivated by these findings, I assume that the yield curve is driven by N common

factors with time-invariant weights but possibly stochastic volatilies. The risk-neutral

dynamics of the discount bond prices are expressed as

dD(t, T)D(t, T)

= rtdt Nk=1

Bk(T t)

k(t) dZk(t) (6)

where rt is the instantaneous short rate. For each k = 1, , N, dZk is a Brownian motionterm representing shocks to the kth factor driving the yield curve, and k is the variance

can not be simultaneously lognormally distributed.

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of the kth yield factor. Without loss of generality, the yield factors are orthogonal to each

other. Bk(T t) describes the loadings of the bond with maturity T on the kth yieldfactor at time t. It is a deterministic function of the time-to-maturity T t only. Thisensures that the term structure dynamics under my model is time-homogeneous.

It follows from equation (6) and the Itos lemma that the date t instantaneous covari-ances of log-bond prices for a set of bonds with maturity 1, , n can be written as aproduct of three matrices

Bt Diag(t) Bt

where Bt is a n by N matrix whose (i, k)th element is Bk(i t), and Diag(t) is a N byN diagonal matrix whose diagonal elements are the instantaneous variances of the yield

factors k(t), k = 1, , N. The covariance matrix above is guaranteed to be positivesemi-definite. Further, the covariance between any two discount bonds is linear in the

variances of the N yield factors 1(t), , N(t).There are two sources for the time-variation in bond covariances under my model.

One is predictable: as time pass by, the time-to-maturities of bonds decrease, and hence

their loadings on the yield factors change correspondingly. Another source of movement

in the covariances of bond yields is the stochastic volatilities of the yield factors, which I

model next.

Consider the general case when K of the N yield factors (K N) display stochasticvariances, and the remaining N K factors have constant variances. Let IK denote theindex of yield factors with stochastic variances. For i / IK, denote the variance of the ithfactor by a positive constant i. For each i

IK, I model the variance of the ith factor

as a square root process:

di(t) = i(i i(t))dt + i

i(t) dWit, i IK (7)

Model parameters i and i are respectively the mean reversion speed and the long run

mean for the variance of the ith yield factor.

Ball and Torous (1999), Chen and Scott (2001), and Heidari and Wu (2001) find that

innovations in interest rate levels are almost uncorrelated correlated with innovations in

the volatility of interest rates. Motivated by this finding, I assume that each Brownian

motion dWit driving the variances of the yield factors is uncorrelated with the dZs inequation (6). This assumption implies that bond innovations are not contemporaneously

affected by volatility innovations. Therefore, bonds cannot be used to hedge volatility risk

instantaneously and thus my model exhibits unspanned stochastic volatility in the sense

Collin-Dufresne and Goldstein (2002). It also implies that dynamics for the variances of

the yield factors are the same under the risk neutral measure and all forward measures

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(see Goldstein, 2000).

In the rest of the paper, for a pair of integers K N, I denote by GAN,K the modelwith risk-neutral evolution of bond prices given by equations (6) and (7). Under the

GAN,K, there are N factors that drive innovations in the yield curve, the first K of which

display unspanned stochastic volatility, and the others have constant volatility.4 Theconditional covariances of bond yields are affine in the instantaneous variances of the first

K yield factors.

Risk-neutral bond prices dynamics are enough for deriving prices of swaptions and

interest rate caps (see section 3.2). For the purpose of carrying out a maximum likelihood

estimation of my model, I also need the dynamics of the volatility state variables under

the physical measure. The volatility risk premium can be modeled as i

i (see, e.g.,

Heston (1993) for an equilibrium justification based on the representative agents utility).

Under this assumption, i also follows an affine process under the physical measure:

di(t) = i(i i(t))dt + i

i(t) dWit (8)

where i = ii, i = iiii , and dWit is a standard Brownian motion under the physicalmeasure.

3.2 Model Valuation of Swaptions and Caps

Here I derive closed-form formulas for the prices of European swaptions and caps. My

models tractability greatly facilitates its econometric estimation with panel data of swap-

tions to be carried out later in this study.

Interest rate derivatives can be valued by taking expectation of their discounted payoffs

under the spot risk-neutral measure. Unlike the equity derivatives, one can not assume

a constant short rate, since the entire term structure is stochastic. In general, the short

rate process is non-Markovian under the HJM type models, and thus difficult to work

with.5 As the payoffs of swaptions and interest rate caps are homogeneous of degree 1 in

4As in principal component analysis, I label the factors so that for i < j, the ith factor has higherunconditional variance than the j factor. Theoretically, for a given pair (N, K), IK = {1, , K} is onlyone of the possible choices for the index of yield factors which display stochastic volatility. Empirically,

this choice fits the swaptions and interest caps data the best.5It can be shown that under my GAN,K model, the risk-neutral dynamics for the short rate r is

rt = y(0, t) +

t0

c(t s)V(s)ts

0

c()d

ds +

t0

c(t s)

V(s) dZQs

where B(T t) = [B1(T t) BN(T t)]; V(t) is N by N diagonal matrix whose ith diagonal elementis i(t); dZ

Qt is a N dimensional Brownian motion stacking dZ

Qk (t), k = 1, , N; c is a row vector of

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discount bond prices, it turns out that it is more convenient to use the forward risk-neutral

measure defined by Jamshidian (1987) to value swaptions and caps.

Let D(t , , T ) denote the date-t price of a forward contract to buy at date a bond

that matures at T. In the absence of arbitrage, the forward bond prices are related to the

discount bond prices by D(t , , T ) = D(t, T)/D(t, ). The forward risk-neutral measurecorresponding to date and denoted by Q uses the Libor discount bond that matures

at as the numeraire asset. By definition, each forward bond price D(t , , T ), T > ,

follows a martingale under Q. Thus, the dynamics of the forward bond prices under the

corresponding forward risk-neutral measure are determined by their covariances.

Below, I first show that given the initial bond prices, price of an European swap-

tion is determined by the covariances of a set of discrete-tenor forward bonds under the

corresponding forward risk-neutral measure. Then I explain how to operationalize the

modeling assumption in section 3.1 for the covariances of forward bonds. Finally, I derive

closed-form formulas for the prices of swaptions and caps.It is straight-forward to show that the payoff of a by T at-the-money-forward re-

ceivers swaption, previously given in equation (2) (with coupon rate c being equal to the

corresponding forward swap rate F SR(0, , T ) given in equation (3)), can be rewritten in

terms of the forward bond prices as

Max

A() 1, 0

(9)

where

A(t) =2(T)j=1

jSj(t), Sj(t) = D(t , , j)

D(0, , j), j = + j

2

j =jD(0, , j)2(T)

k=1 kD(0, , k)(10)

j =Kc

2, for j = 1, . . . , 2(T ) 1, and 2(T) = (1 + c

2)W

W =1

1 + (T )cNote that the js are positive constants and j j = 1.

The no-arbitrage price of a contingent claim, which settles at time , is given by first

taking the expectation of its payoff under the forward risk-neutral measure, and then

multiplying it by D(0, ) (see, e.g., Lemma 13.2.3 of Musiela and Rutkowski, 1997). It

functions defined to be the derivative of B: c() = B()

; and y(0, t) = log D(0, t)/t.

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follows that the date-0 price of a by T at-the-money-forward receivers swaption is:

P(, T) = D(0, ) EQMax

A() 1, 0

(11)

where Q is the forward risk-neutral measure. Thus, the valuation of swaption is reduced

to computing the expectation under the forward risk-neutral measure of an arithmetic

sum of a set of random variables Sj(), for j = 1, . . . , 2(T ). Since each Sj(t) is justa constant multiple of D(t , , j), its drift under Q

is zero. Therefore, the covariances of

{D(t , , j)}, j = + j2 , j = 1, . . . , 2(T ), determines their joint distributions underthe forward risk-neutral measure Q, which in turns determines the price of a by T

European swaption by equation (11).

Note that to value swaptions and caps, I need the covariances of forward bonds with

fixed maturities, but it is more convienient to first model the covariances of bonds with

fixed time to maturity (e.g., multiples of six-months).6 Let t be the date t instantaneous

covariance matrix of changes in the logarithm of the six-month forward Libor bond prices{D(t, t + ti, t + ti+1)}19i=0, where ti = i2 years. Let H be the corresponding unconditionalcovariance matrix. Since H is a semi-definite symmetric matrix, it can be decomposed as

H = U0U

, where 0 is a diagonal matrix whose diagonal elements are the eigenvalues

of H, and columns of U are the corresponding eigenvectors.7 The conditional covariance

matrix under the GAN,K model is given by

t = UtU (12)

where t

is a diagonal matrix whose first N main diagonal elements are the conditional

variances of the N yield factors. Among them, the first K yield factors have stochastic

variances which I model as CIR processes as in equation (7). The remaining NK factorshave constant variances. All other elements of t are zero.

The covariances of forward bonds with fixed maturities can be obtained from t in

two steps. First,

log(D(t , , j)) = log(D(t , , 1)) + log(D(t, 1, 2)) + + log(D(t, j1, j))6Note that the swap rates, swaptions and caps data are all refreshed each date, in the sense that they

have constant time-to-maturities rather than fixed maturity. The longest total maturity of the swaptions

in my dataset is ten years. Therefore I consider bonds with time-to-maturity no more than ten years.7I normalize U to have unit length for each column. The ith column of U and the ith main diagonal

element of 0 are the weights and the unconditional variances of the ith common factor driving the yieldcurve. Further, a rotation of these factors (obtained by multiplying a orthogonal matrix) does not changethe factor variances (is). Since all structural parameters in my model are for the dynamics of the is,they are properly identified.

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Second, every six months from date 0 till , j ts are multiples of six-month. On thesedates, the instantaneous covariances of the bonds on the right hand side of last equation

can be read off from t = {cij(t)}ij. On other dates, I linearly interpolate the covariancesto preserve the continuity of the covariances as functions of time-to-maturity. 8 More

precisely, fix integer i, j such that 1 i < j < 20; at any time t ti =i

2 , let k be theinteger such that tk t < tk+1. Then I assume

Cov(D(t, ti, ti+1), D(t, tj, tj+1)) = (12(ttk)) cik,jk(t)+2(ttk) cik1,jk1(t) (13)

Now I continue with the valuation of an European by T swaption given by equa-

tion (11). Based on the assumption that the volatility state variables are instantaneously

uncorrelated with innovations in the yield curve, I can use an argument similar to Hull

and White (1987) to show that each Sj() is lognormal conditional on average values of

k (k = 1,

, N) between date 0 and . A(), an arithmetic sum of

{Sj()

}j, is not con-

ditionally lognormal. However, the corresponding geometric sum, G(t) =

2(T)j=1 S

jj (t),

is. Upon replacing the difference of A() and G() by the mean of the difference,9 the

price of an by T swaption approximately equals:

P(, T) = D(0, ) EQMax

G() g, 0

(14)

where g = 1 + EQ

[G() A]. I apply the law of iterative expectation and derive thefollowing approximate closed-form formula for prices of at-the-money-forward European

swaptions (see the appendix for details):

Proposition 1 Under the GAN,K model, the price of a by T European at-the-money-

forward swaption is given approximately by:

P(, T) = D(0, )

2N

1

2

1)

(15)

=1

21l=0

AlUDiag

l1U

A

l + Al+1UDiagl2U

A

l+1 AlUDiagl2U

A

l

(16)

where N() is the cumulative density function of a standard normal random variable. Theweights j, j = 1,

, 2(T

), as given by (10), depend on , T as well as the current

8I experiment with alternative interpolating schemes using functions that are exponentially decayingin time-to-maturity, and find that the interpolation scheme has a negligible influence on the prices ofswaptions and caps.

9This approximation technique has been applied to price Asian options and basket options. It is knownto be very accurate (see, e.g., Vorst (1992)). In my case, simulations show that the typical approximationerror is less than 0.1%, even for maturity as long as = 10, and for values of the volatility state variablesthat generate bond yield volatilities that are twice as high as those observed in the data.

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prices of discount bonds. U is the matrix of eigenvectors of the unconditional covariance

matrix of changes in the logarithm of the six-month forward Libor bonds. For each non-

negative integer l 21, Diagl1 and Diagl2 are 20-by-20 matrices whose entries are zeroexcept the first N diagonal elements

Diagl1(i, i) =1

2i +

eil/2 ei(l+1)/2i

(i(0) i), i = 1, , K

Diagl1(i, i) =1

2i, i = K+ 1, , N

Diagl2(i, i) =1

4i + e

il/2

2

2i e

i2

i 2

2ie

i2

(i(0) i), i = 1, , K

Diagl2(i, i) =1

4i, i = K+ 1, , N

For i = 1,

, K, i(0) denotes the spot variance of the ith common yield factor. For each

l, Al is a 2(T )-by-20 matrix. The 2 l + 1 to 2T l column of Al forms a lowertriangular matrix whose elements on and below diagonal are 1. All other elements of Alare zero.

The valuation of an interest rate cap is easier. The price of a cap is the sum of the

prices of its constituent caplets. Each caplet can be viewed as a put options on the

corresponding forward bond.

Proposition 2 The price of an at-the-money cap on the six-month Libor rate with time

to maturity T is:

2T1i=1

D(0, ti)N(d2i) (1 + 12

R)D(0, ti+1)N(d1i) (17)

d1i =log(D(0, ti, ti+1)(1 +

12

R)) + 12

2i ti2i ti

and d2i = d1i

2i ti

where ti =i2

year, the cap rate R equals the T year swap rate andi2 is the average expected

variance over [0, ti] of changes in the logarithm of forward bond price D(t, ti, ti+1), given

by equation (16) with = ti and T

= 1

2.

3.3 Relation to Other Term-Structure Models

I now discuss how my model relates to models in previous and contemporaneous studies.

First, the dynamics of discount bond prices under my model as given by equation (6) can

be derived from the affine model of Duffie and Kan (1996). To see this, consider an affine

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model with N factors driving the short rate:

rt = 0 + 1Xt

The risk-neutral dynamics of the risk factors Xt are given by

dXt = ( Xt)dt +

S(t)dZQt (18)

where and are N by N matrices, and S is a diagonal matrix with components

Sii(t) = i + iX(t)

The discount bond prices are given by

D(t, T) = eA(Tt)B(Tt)X(t) (19)

where A() and B() can be solved from a system of ODEs with A(0) = 0, B(0) = 0.

By equation (18), equation (19) and the Itos lemma, the risk-neutral dynamics for the

discount bond prices under the affine model are

dD(t, T)

D(t, T)= rtdt B(T t)

S(t) dZQt (20)

which is of the same shape as bond prices dynamics under my model.

Just like under the affine model, the covariances of bond yields in my model are

affine in the volatility state variables. The difference is that in my model, innovations inbond yields are not contemporaneously affected by volatility innovations. In contrast, the

stochastic volatility factors in the affine framework typically enter into bond prices, and

can be represented as linear combinations of bond yields. In other words, the volatility

factors in the traditional affine framework drive both the cross-sectional differences in bond

yields and changes in conditional volatility. My model and affine models with unspanned

stochastic volatility both break this dual role of stochastic volatility.10 Thus, my model can

be viewed as a reduced-form representation of affine model with unspanned stochastic

volatility.

My model takes the current term structure of interest rates as given, and focus onunderstanding what drives stochastic covariances of bond yields and the valuation of

10See Collin-Dufresne and Goldstein (2002) for examples of affine models that display unspannedstochastic volatility. Collin-Dufresne, Goldstein and Jones (2004) compare traditional affine models toan affine model with unspanned stochastic volatility. They show that the latter generates much morerealistic volatility estimates without sacrificing fit to the cross-sectional bond prices.

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interest rate derivatives. This aspect of my model is similar to the HJM (1992) framework

or its generalization such as the string or random field framework (see, e.g., Goldstein 2000,

and Santa-Clara and Sornette, 2001). If a model fits the bond prices with errors, such

errors will get carried over to the interest rate derivatives and confound the inferences

about the factors driving the bond covariances.My model is closely related to the string-market model of Longstaff, Santa-Clara

and Schwartz (2001) who also directly model the risk-neutral dynamics of discount bond

prices. The two models share the property that the instantaneous covariance matrix of

bond yields and the unconditional historical covariance matrix are diagonalized by the

same matrix. The difference is that Longstaff, Santa-Clara and Schwartz (2001) consider

only constant covariances while I explicitly model the dynamics of the risk factors driving

the stochastic bond covariances.

Two contemporaneous theoretical papers contain models similar to mine. First, Kim-

mel (2004) develops a class of random field models in which the volatility of forward ratesdepends on a set of latent variables, not the level or shape of the forward curve. The latent

volatility variables themselves follow general diffusion process. He focuses on conditions

necessary for the existence and uniqueness of the forward rate process, characterizing

the behavior of the latent variable random field model and deriving theoretic results for

derivative pricing. For tractability of empirical work, I choose bond prices rather than

forward rates as model fundamentals, and focus on affine process for the state variables

that drive the stochastic covariances of bond yields. These state variables also have eco-

nomic meaning in my model: they are the instantaneous variance of common factors that

drive the yield curve innovations. Second, my model is closely related to the generalized

affine framework proposed by Collin-Dufresne and Goldstein (2003), which also allows

unspanned stochastic volatility but maintains the tractability of the affine models.

4 Data

A panel dataset of interest rates and implied volatility for at-the-money-forward European

type swaptions is used to estimate my model. The interest rates include six-month and

one-year Libor rates as well as two, three, four, five, seven, ten and fifteen-year swap rates.

The total maturity of the swaptions I use is no greater than ten years and there are 34such swaptions. The maturities of the swaptions range from six months to five years and

the tenors of the underlying swaps are between one and seven years. Implied volatilities

for Libor interest-rate caps of maturity two, three, four, five, seven and ten years are used

in the study of relative valuation of swaptions and caps. All data are collected from the

Bloomberg system and represent the average of best bid and ask among many large swap

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and swap derivatives brokers. There are 220 weekly observations for each series, sampled

every Friday from January 24, 1997 to April 6, 2001.

Discount bond prices (with maturity that are multiples of six-months) in the Libor-

swap market are needed in order to compute the market prices and model prices for

swaptions and caps. I apply a least-square cubic spline approximation to the Libor andswap rates to get the par yield curve, and then bootstrap the discount bond prices from

the par yield curve.11 The unconditional covariance matrix of forward bond yields is

computed using the discount bond prices between January 17, 1992 and January 17,

1997. Its eigenvector matrix U is used in forming instantaneous covariances as specified

in (12).

Table 1 reports the mean and standard deviation of swaption implied volatilities.

On average, the implied volatility is humped as a function of swaption maturity, with a

maximum at two years. On high volatility dates, the swaption implied volatility tends

to decrease monotonically with option maturity. Consistent with mean reversion in theinterest rates, the implied volatility of swaption is usually monotonically decreasing as a

function of the tenor of the underlying swap. Further, the standard deviation of swaption

implied volatility decreases with option maturity as well as the tenor of the underlying

swap. For example, although the implied volatilities of 0.5 into 1 and 5 into 5 swaption

have about the same sample mean, the first is almost three times as volatile as the latter.

Figure 1 confirms that there is a fair amount of time-series variation in the swaption

implied volatilities, especially for short-dated swaptions. My sample period includes the

Asian crisis, the Russian moratorium and the Long Term Capital Management crisis,

the crash of technology stocks, as well as quiet periods of low interest rate volatility.These different volatility environments help us to estimate the dynamics of bond-yield

covariances.

Figure 1 also shows that implied volatilities for short-dated and long-dated swaptions

do not always track each other closely. For example, the sample correlation between the

implied volatility for the 0.5 into 1 swaption and that for the 5 into 5 swaption is only

0.51. A principal component analysis also suggests there are at least two factors driving

the innovations in the swaption implied volatilities.

11The least-squares cubic spline approximation fits the swap rates very well, with average absolute

fitted error about 0.76 basis points. Interpolation schemes that exactly fit observed Libor-swap ratestend to lead to unreasonably high estimates for the volatilities of long term bonds and somewhat ruggedcorrelations. An alternative method to generate reasonable estimates for the bond covariances is to putsome smoothness condition on the shape of the forward rate curve (see, e.g., Driessen, Klaassen andMelenberg 2003).

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5 Econometric Method

The econometric exercise in my paper parallels many previous studies that evaluate the

performance of short-rate based term structure models and infer dynamics of the risk

factors driving the short rate using bond prices or swap rates. The focus of my studyis the stochastic covariances of bond yields. I infer dynamics of the risk factors driving

bond covariances and obtain conditional estimates of bond covariances from the swaption

implied volatilities data.12

I adopt the maximum likelihood approach used by Chen and Scott (1993), Pearson

and Sun (1994), Duffie and Singleton (1997), Jagannathan, Kaplin and Sun (2003) and

others. The volatility state variables are unobservable. Thus, they need to be filtered

at the same time as the model is estimated. For a given set of model parameters, the

volatility state variables are inverted from the prices of swaptions that are assumed to

be priced exactly. The percent pricing errors of the other swaptions are assumed to be

i.i.d and jointly normal N(0, ). The estimated parameters maximize the sum of thelog-likelihoods of the exactly fitted swaptions and the percent pricing errors of the other

swaptions.

The estimated parameters for the volatility dynamics under the GAN,K model maxi-

mize a log-likelihood function function log L = log L1 + log L2, where

log L1 =T1t=1

Ki=1

lnf(i,t+1|i,t) T

t=2

ln(J,t)

log L2 = (34

K)(T

1)

2 ln(2) T

1

2 ln|| 1

2

Tt=2

e

t1et

where et is a vector of the percent pricing errors of the non-exactly-fitted swaptions,

J,t is the determinant of the matrix of first order partial derivatives of K exactly-fitted

swaptions with respect to each of the Kvolatility factors, and f(i,t+1|i,t) is the transitiondensity of the ith volatility factor (assumed to follow independent CIR process). Omitting

the subscript i,

f(t+1|t) = 2ceuw

w

u

q/2Iq(2

uw)

withc =

2

2(1 e(Tt)) , u = cte(Tt), w = cT, q =

2

2 1,

and Iq is a qth order modified Bessel function of the first kind. Since the data are sampled

12Collin-Dufresne, Goldstein and Jones (2004) find that interest rate volatility can not be extractedfrom the cross-section of bond prices.

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weekly, the above formula is applied with T t = 1/52 year.To implement the maximum likelihood approach above, I have to choose the swaptions

that are to be fitted exactly. Each swaption provides information about the covariances

of a segment of the yield curve and there are no good reasons to choose one swaption

over another to be fitted exactly.13 For all results reported in this paper, I invert theK volatility state variables under the GAN,K model by assuming that K portfolios of

swaptions (corresponding to the first K principal components of the swaption implied

volatilities) are fitted exactly, and measurement errors apply to other principal compo-

nents.14 This approach not only circumvents the arbitrariness of standard approach of

fitting specific instruments exactly, it also approximately orthogonalizes the matrix of

measurement errors.

6 Empirical Results

The empirical results presented below answer, among other things, the following questions:

1) how many yield factors and how many covariance factors are needed to explain the

swaptions data? 2) can my model explain the relative valuation of swaptions and interest

rate caps? 3) how do the covariances of bond yields implied from the swaptions compare

to the historical estimates based on bond prices alone?

6.1 Number of Factors Underlying the Swaptions Data

Recall that my GAN,K model have N yield factors, and the first K yield factors have

stochastic volatilities. When K = 0, my model specifies a constant covariance matrix

for bonds with fixed time-to-maturity. There are only N parameters under the GAN,0model, which are the constant variances of the N yield factors. They can be estimated

via the General Methods of Moments approach. The pricing error of each swaption

provides a moment condition. The test statistic for over-identifying moment restrictions

is asymptotically distributed as 234N.

The constant covariance model could not fit the swaptions data well. Even when there

13For my model with only one volatility factor, I find that the parameter estimates and the impliedvolatility state variable can differ significantly depending on the choice of the swaption to be fitted exactly.This reflects the fact that my model with only one volatility factor is unable to fit simultaneously bothshort-maturity options on short-tenor swaps and long-maturity options on long-tenor swaps. My modelwith multiple factors driving the volatilities and correlations of bond yields fits the data much better andthe estimation results are not sensitive to the choices of swaptions to be fitted exactly.

14Collin-Dufresne, Goldstein and Jones (2004) has applied a similar approach in estimating variousaffine model using bond prices.

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are 4 factors driving the innovations in the yield curve, the mean absolute pricing error

is 9.86%, and the average RMSE for the swaptions is 10.21%. The constant covariance

model tends to underprice swaptions. The test statistics for over-identifying moment

restrictions is 129.18. Since the 99% critical value for 230 is 50.89, I can reject the

GAN,0 model. Constant covariance model with fewer yield factors performs even worse.Increasing the number of yield factors to N > 4 does not significantly improve model fit.

Allowing stochastic covariances greatly improves models fitting performance for swap-

tions. Table 2 shows that under the GA4,1 model, the overall mean absolute swaption

pricing error is reduced to 3.28%, and the average (median) RMSE for the swaptions

is 3.97% (3.53%). The swaption RMSE under GA4,1 is always smaller than that under

GA4,0. The difference is especially big in the months leading up to the Long Term Capital

Management (LTCM) crisis in the fall of 1998. Between January 1998 to the end of Au-

gust 1998, the swaption RMSE under the constant covariance GA4,0 model raises steadily

from around 10% to over 30%, but it stays around 2.5% under the GA4,1 model.However, the GA4,1 model, with only one factor driving the stochastic covariances

of bond yields, has difficulty fitting simultaneously both short and long dated swaptions,

especially during the LTCM crisis. As can be seen from Figure 1, the implied volatility for

the 0.5 into 1 swaption jumps from just above 10% before the LTCM crisis to more than

24% during the crisis, while the implied volatility for the 5 into 1 swaption only increases

from about 13% to 17%. In order to generate this feature of the data using models with

one volatility factor, the mean reversion speed 1 must be high. A large 1 implies fast

reversion to the mean. Thus, a volatility shock is expected to die out quickly, which

implies much smaller changes in the implied volatilities for the long-dated swaptions than

short-dated swaptions. Yet a high 1 would produce large swaption pricing errors during

normal market when implied volatilities for short dated and long dated swaptions tend

to move by about the same amount.

My model with multiple covariance factors solves the problem above. The mean

absolute pricing error for the swaptions is 2.17% under the GA4,2 model. The median

swaption RMSE is 2.61% and the maximum is 8.00%. The GA4,3 model performs even

better. It implies a mean absolute pricing error of 2.06%. Panel A of Figure 2 shows that

the swaption RMSE under GA4,3 is lower than 4% except during the LTCM crisis. On

161 of the 220 weeks, the RMSE is lower than 3% (about half of the bid-ask spread).15

In addition to having multiple factors driving the bond covariances, it is also important

to have a sufficient number of yield factors. For example, although GA2,2 has more

15Note that the bid-ask spread for the swaption implied volatility is usually 1 volatility point (i.e., 1%)during my sample period. For a typical implied volatility of 16% for at-the-money-forward swaption, thistranslates into a bid-ask spread of about 6% bid-ask spread for the swaption price, since price of swaptionat-the-money-forward is almost linear in its Black implied volatility.

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volatility factors than GA4,1, it often produces larger pricing errors. The GA4,2 model

significantly dominates both GA4,1 and GA2,2, as formal statistical tests presented below

also shows.

The swaption pricing errors obtained in the previous studies are substantially higher

than ours, even when the models are frequently recalibrated in these studies in order tofit the data better. For example, De Jong, Driessen and Pelsser (1999) find that for a

variety of Libor and swap market model specifications, the mean absolute pricing error

ranges from 0.87 to 2.28 volatility points, or about 5% to 14% assuming an average im-

plied volatility of 16%. Driessen, Klaassen and Melenberg (2003) report average absolute

swaption pricing error between 7.23% and 9.50% for several HJM type models. Jagan-

nathan, Kaplin and Sun (2003) report swaption pricing errors that are larger than the

bid-ask spread. For example, under the two-factor CIR model, the mean absolute pricing

error is about 8% for 3-month into 2-year swaption, and 31 .39% for 2-year into 5-year

swaption.

16

The best prior fitting performance for the swaptions data comes from the four-factor

string-market model of Longstaff, Santa-Clara and Schwartz (2001). The median swaption

RMSE is 3.10%, but it becomes over 15% during the LTCM crisis. Just like my GA4,0model, their model assumes constant covariances of bond yields (with constant time-to-

maturities). The difference is that they recalibrate their model each date which implicitly

allows four factors to drive the covariances of bond yields. In contrast, I explicitly model

the dynamics of the risk factors that drive the covariances of bond yields. As a result,

my GA4,2 model, with only two factors driving the covariances of bond yields, is able

to fit the swaption data better, both under normal market condition and during market

crises.17

Table 3 reports the Diebold and Mariano (1995) test statistics for pairwise comparisons

of alternative specifications of my model. Each model specification implies a time series

of mean squared swaption pricing errors SS E(t) corresponding to the optimized model

parameters: SS E(t) = 134

34i=1

2i,t, where i,t is the percent pricing error for the ith

swaption on date t. Consider two models that give rise to {SS E1(t)}Tt=1 and {SS E2(t)}Tt=1respectively (T = 220 in my case). The null hypothesis that the two models have the

same pricing accuracy is equivalent to the null hypothesis that the population mean ()

16

Umantsev (2001) shows that the three-factor affine model A1(3) of Dai and Singleton (2000) fitsthe swaption volatilities better than the multi-factor CIR model, especially when the swaptions data areincluded in the estimation procedure. However, the pricing errors for swaptions with long option or swapmaturity are still quite large.

17As reported earlier, over the entire sample of 220 weekly observations, the median swaption RMSEunder the GA4,2 model is 2.61% and the maximum is 8.00%. The sample in Longstaff, Santa-Clara andSchwartz (2001) is the first 128 weeks of mine. Over this subsample, the median RMSE of my GA4,2model is 2.43%.

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of the difference in pricing errors (dt) is zero, where dt = SS E1(t) SS E2(t). Dieboldand Mariano (1995) show that if{dt}Tt=1 is covariance stationary and short memory, thenthe asymptotic distribution of the sample mean of pricing error difference d = 1

T

Tt=1 dt

is normally distributed as T(

d ) N(0, 2fd(0))

where fd(0) is the spectral density of the pricing error difference at frequency:

fd(0) =1

2

=

rd() where rd() = E[(dt )(dt )]

The formula for fd(0) shows correction for serial correlation in dt. Diebold and Mariano

(1995) obtain a consistent estimate of 2fd(0) by the sum of available autocovariances up

to a certain truncation lag q:18

2 fd(0) =q

=q

rd(), where rd() = 1T

Tt=||+1

(dt d)(dt|| d)

Thus, under the null hypothesis of equal pricing accuracy across the two models, the

following statistic

S =

T d

2 fd(0)

is distributed asymptotically as standard normal N(0, 1).Table 3 confirms my earlier results on the comparison of model fitting performance.

Models with more yield factors and volatility factors usually fit the swaptions data signifi-

cantly better. The largest improvement comes from increasing the number of yield factors

from one to two, or from two to three, and from increasing the number of volatility factors

from one to two. In each pair of such comparisons, models with more yield or volatility

factors imply at least 1% lower swaption RMSE on at least 200 weeks out of a total of 220

weeks. These cases also give the largest Diebold and Mariano (1995) statistics. Overall,

the GA4,3 model with four yield factors and three separate volatility factors best fit the

swaptions data. The GA4,2 model and the GA3,3 model also provides excellent fit to the

swaptions data.

Results in Table 3 are also consistent with Heidari and Wu (2001) who find by princi-pal component analysis that one needs three volatility factors that are independent of the

underlying yield curve to capture the movement of the swaption implied volatility surface.

However, the principal component analysis is unconditional, and completely ignores the

18In Table 3, lag order q = 52. I verify that my results are robust to alternative lag order q such as 5,10, 26, and 102.

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rich information contained in the cross-section of swaptions that my stochastic-covariance

model tries to capture. Through rigorous modeling and econometric estimation, I obtain

estimates of conditional covariances of bond yields and infer information about the dy-

namics of the factors that drive bond covariances. I can also address issues such as the

relative valuation of swaptions and interest rate caps. These are impossible to achievewith principal component analysis.

6.2 Relative Valuation of Swaptions and Caps

The relative valuation of swaptions and interest rate caps crucially depends on the co-

variances of bond yields, and thus provides a diagnostic test for my model. This is an

out-of-sample test in the sense that I first estimate my model with the swaptions data,

then compare the model implied no-arbitrage values for the interest rate caps given by

equation (17) to their market prices. The differences are referred to as cap pricing errors.Table 4 reports the mean and mean absolute pricing errors of interest rate caps. The

constant-covariance model, even with as many as four yield factors, still implies large

mean absolute pricing error (10.36%). Just like the case for the swaptions, introducing

stochastic covariances substantially improves model fit to the caps. The mean absolute

cap pricing error is 5.41% under GA4,1, 4.88% under GA4,2, and 4.54% under the GA4,3model. The average pricing error is 0.62% under the GA4,2 model and only 0.03% under

the GA4,3 model. Only the 2-year cap shows some systematic overvaluation by the GA4,2model. The GA4,3 model, which is favored by the statistical tests based on models fit

to the swaptions, also implies the smallest pricing errors for the interest rate caps. Its

improvement over the GA4,2 model is mainly reducing the pricing error of the 2-year cap.

Panel B of Figure 2 shows that differences between the market prices of interest rate

caps and their no-arbitrage values implied from the swaptions according to my GA4,3model are typically lower than 6% (about the bid-ask spread). There are only two periods

when the cap RMSE is higher than 6%. The first occurs during the LTCM crisis. For

example, the cap RMSE jumps to 19.9% on August 28, 1998 and 30.4% on September

4, 1998, right after The Times headlined Meriwether fund plummets by 44%. For the

next 4 weeks, the average cap pricing error stays below 6% until it shoots up again to

25.8% on October 2, 1998. During that week, many banks revealed huge exposure to or

investment in the hedge-fund which led to a 3.5 billion bail out of LTCM.

The second period of relatively large cap pricing errors starts at the beginning of

August 1999, when, according to my GA4,3 model, the no-arbitrage prices of individual

interest rate caps implied from the swaptions are lower than the market prices of caps by

about 8%. Supporting the view that caps are overvalued relative to swaptions during this

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period, various news reports indicate that proprietary desks are short cap volatility and

long swaption volatility.19 The cap pricing errors revert to around 3% (about half of the

bid-ask spread) by the end of 1999.

Under my GA4,3 model, the market prices of interest rate caps do not deviate signif-

icantly from their no-arbitrage values implied from the swaptions. In contrast, using afour-factor constant-covariance model, Longstaff, Santa-Clara and Schwartz (2001) find

periodically large and somewhat bimodally distributed pricing errors for interest rate

caps. The average cap pricing error is 10.38%, and the mean absolute cap pricing error is

14.19%. Note that their model and my model both have four factors driving the innova-

tions in the yield curve. The difference is that my model also has separate factors driving

the covariances of bond yields. My model seems to realistically capture the dynamics

of bond covariances and largely eliminates the large and systematic mispricing between

swaptions and caps found by Longstaff, Santa-Clara and Schwartz (2001) and several

other studies.

20

As an additional test of goodness-of-fit for my stochastic covariance model, I regress

changes in the market prices of interest rate derivatives on the changes in the fitted prices

under the GA4,3 model. The regression is run individually for each swaption and interest

rate cap. If my model describes the data well, then the estimated intercept should be

very close to 0, and the slope coefficient to be indistinguishable from 1, as well as a high

R2 for the regression. This is what I find. None of the intercept is even close to being

statistically different from 0. The average estimated slope coefficient is not significantly

different from 1. The average R2 is 0.92 (0.78) across all the regressions corresponding to

the swaptions (interest rate caps).

6.3 The Volatility Factors and Implied Covariances

Through the model estimation, I also extract markets view about the dynamics of the

volatility factors and obtain implied bond covariances. Here I compare the implied volatil-

ities and correlations of bond yields to their historical estimates (both unconditional and

conditional). All results below are based on the estimated GA4,3 model.

19However, this position is risky. On August 5, 1999, Dow Jones Newswire reports that two major USinvestment banks lose millions of dollars when they had to unwind big swaption positions and cover shortcap positions. Interestingly, this correlation play on the forward rates was reported to be put back onjust three weeks later.

20For example, the smallest mean absolute pricing error for caplets among all the Libor and swapmarket models tested in De Jong, Driessen and Pelsser (1999) is about 20%. Driessen, Klaassen andMelenberg (2003) find that the average absolute pricing errors for several multi-factor HJM models areall bigger than 10%. Jagannathan, Kaplin and Sun(2003) find smallest mean absolute pricing error forcaps under one, two, or three-factor CIR models are still much larger than the bid-ask spreads.

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Recall that instantaneous covariances of bond yields under the GA4,3 model are linear

in the variances of four yield factors. Plots of the factor loadings obtained from the

principal component analysis confirm that the first three yield factors approximately

represent the level, slope and curvature of the term structure. The fourth factor only

affects the short end of the yield curve (less than 2-year maturity).The first volatility factor, or the variance of the level yield factor, is quite persistent.

On the other hand, the second and the third volatility factor, or the variance of the slope

and the curvature yield factor mean revert very fast. The half-life, defined as log2

,

where is the mean reversion speed, measures the time when a mean-reverting variable is

expected to reach a value halfway between the current level and its long-run mean. Based

on the model parameter estimates presented in Table 5 Panel A, the half life of the first

volatility factor is about 6 years under the physical measure, while it is only about 3 to

4 months for the second and the third volatility factor.

Several implications follow from the weights of the yield factors and the mean reversionspeeds of their stochastic volatility. First, the long-dated swaptions are mainly influenced

by the first volatility factor. Second, swaptions of different maturities and tenors are about

equally sensitive to innovations in the first volatility factor. Therefore, the first volatility

factor proxies for the average level of the surface of swaption implied volatilities. Third,

the second volatility factor proxies for the spread between the implied volatilities of short

and long dated swaptions of the same tenor.21 Finally, only short-dated swaptions on

one-year and two-year swaps are sensitive to the third volatility factor. I have confirmed

these implications with numerical analysis.

Panel B of Table 5 reports the sample correlations between the yield factors and thevolatility factors under the GA4,3 model. It strongly supports my models assumption

that shocks to the yield factors are uncorrelated with their stochastic volatilities. Panel C

of Table 5 shows that the time series average of the implied variances of the yield factors

are very close to the unconditional variances of the yield factors. Figure 3 shows that

the average implied covariances of bond yields are also very close to their unconditional

historical estimates.22

Finally, Figure 4 compares the implied covariances of forward bonds to the conditional

covariances obtained from the Dynamic Conditional Correlation (DCC) model, a new class

of multivariate GARCH models proposed by Engle (2002). He shows that this model

provides a very good approximation to a variety of time varying correlation process, yet it

21The sample correlations between implied 2 and the difference between the Black implied volatilitiesfor 0.5 into T and 5 into T year swaptions are all about 0.95, for T = 1, . . . , 5.

22The only noticeable difference occurs for the correlations of bonds with maturity below 3 years wherethe implied correlations are lower than the historical correlations. But most of the differences are smallerthan 0.05. The maximum difference is 0.08.

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still has the flexibility of univariate GARCH model. I apply this model to changes in the

logarithm of six-month forward bonds. Conditional covariances are estimated following a

two-step procedure as described in detail by Engle and Sheppard (2001).

Several interesting results can be observed from Figure 4. The instantaneous volatility

of six-month forward rate implied from the swaptions closely tracks the GARCH(1,1)estimate. For example, in the case of six-month forward rate with two year maturity

plotted in the top panel of Figure 4, the correlation between the two conditional volatility

estimates is 0.62. There is no systematic difference between the two conditional volatility

estimate. The bottom panel in Figure 4 plots estimates for the correlation between the

two and five year maturity six-month forward rates. The correlation implied from the

swaptions is more volatile than that estimated from the bond prices under the DCC

model. The sample correlation between the two correlation series is only 0.15. During

periods of market crises, the implied correlation drops precipitously but this is not reflected

in the DCC estimate. The comparison of implied correlations and historical correlationsdeserves further investigation in future studies.

7 Conclusion

Recent studies find that it is difficult for many popular term-structure models to explain

the valuation of swaptions and interest rate caps, although these models can price the

underlying bonds very well or even fit them exactly by design. A potential limitation of

these models is that the covariances of bond yields are either deterministic, or can be com-

pletely hedged by the underlying bonds. Collin-Dufresne and Goldstein (2003), Dai andSingleton (2003) and Jagannathan, Kaplin and Sun (2003) conjecture that the resolution

of the swaptions and caps valuation puzzle may require models that can accommodate

more general forms of stochastic volatility and time varying correlations, such as having

volatility factors that do not affect bond prices.

In this paper, I develop and evaluate a term-structure model that has separate factors

driving innovations in bond yields and their stochastic volatilities and correlations. The

covariances of bond yields are affine in a set of volatility state variables that are not

spanned by bonds. Empirically, my model explains the cross-section and the time-series

variation of the swaption implied volatilities very well, and reconciles the relative valuationof swaptions and interest rate caps. These results support the conjectures by Collin-

Dufresne and Goldstein (2003), Dai and Singleton (2003) and Jagannathan, Kaplin and

Sun (2003).

The modelling of factors driving stochastic covariances of bond yields and the econo-

metric estimation using interest rate derivative prices in my study are parallel to a large

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literature that studies short-rate based term-structure models using bond prices. More

empirical research is needed to better understand the rich dynamics of interest rates

volatilities and correlations as well as the valuation of interest rate derivatives, and to

evaluate the class of models proposed in this paper as well as in Collin-Dufresne and

Goldstein (2003) and Kimmel (2004).

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Table 1: Descriptive Statistics of European Swaption Volatilities

This table presents descriptive statistics for the mid-market Black implied volatilities for 34 at-the-money-forward European swaptions analyzed in the paper. The data are Friday close collected from theBloomberg, from January 24, 1997 to April 6, 2001. Expiration refers to the number of years till optionexpiration and Tenor refers to the maturity of the underlying swap. The last column reports the firstorder serial correlation of each swaption implied volatility series. The swaption implied volatilities areannualized and expressed in percentage points.

Expiration Tenor Mean Median Standard Min Max SerialDeviation Correlation

0.5 1 14.77 13.85 3.29 9.70 27.00 0.931 1 16.12 15.45 2.79 11.80 26.00 0.952 1 16.85 16.50 2.11 13.10 23.20 0.943 1 16.66 16.40 1.87 13.00 21.80 0.934 1 16.35 16.10 1.69 13.00 20.70 0.925 1 16.05 15.90 1.50 12.90 20.25 0.93

0.5 2 15.50 14.55 3.02 10.40 27.00 0.941 2 16.10 15.50 2.48 12.20 24.00 0.942 2 16.37 16.05 1.95 13.00 22.50 0.933 2 16.12 15.90 1.70 12.90 21.00 0.934 2 15.82 15.73 1.51 12.80 20.10 0.925 2 15.51 15.50 1.37 12.70 19.75 0.92

0.5 3 15.48 14.58 2.89 10.60 27.00 0.931 3 15.87 15.30 2.28 12.20 23.00 0.942 3 16.04 15.70 1.81 12.90 21.50 0.933 3 15.79 15.70 1.57 12.80 20.20 0.934 3 15.50 15.50 1.41 12.70 19.85 0.92

5 3 15.18 15.20 1.29 12.60 19.50 0.91

0.5 4 15.42 14.60 2.73 10.80 26.50 0.931 4 15.65 15.30 2.12 12.10 22.50 0.932 4 15.74 15.55 1.67 12.80 20.30 0.933 4 15.50 15.40 1.48 12.70 19.95 0.934 4 15.20 15.20 1.33 12.60 19.50 0.915 4 14.85 14.90 1.22 12.40 19.05 0.91

0.5 5 15.36 14.60 2.60 11.00 26.50 0.921 5 15.40 15.10 2.00 11.30 21.50 0.932 5 15.46 15.30 1.57 12.70 19.80 0.923 5 15.22 15.18 1.40 12.60 19.55 0.92

4 5 14.89 14.90 1.27 12.50 19.10 0.915 5 14.50 14.53 1.15 12.10 18.55 0.90

0.5 7 15.18 14.50 2.45 11.00 26.00 0.921 7 15.12 14.80 1.80 12.00 20.50 0.922 7 15.07 14.90 1.49 12.40 19.45 0.923 7 14.82 14.70 1.35 12.30 19.20 0.91

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Table 2: Swaption Pricing Errors

This table reports the mean absolute value of the percent pricing errors of 34 at-the-money-forwardEuropean swaptions under various model specifications. For the model corresponding to the columnlabelled by (N, K), there are N factors driving the evolution of yield curve, and the first K yield factorshave unspanned stochastic volatility. All models are estimated using a panel dataset consisting of 220weekly observations on 34 swaptions from January 24, 1997 to April 6, 2001. The percent pricing error fora swaption is the difference between its fitted model price and its market price expressed as a percentageof market price.

Expiration Tenor Model(4,0) (4,1) (2,2) (4,2) (3,3) (4,3)

0.5 1 13.36 7.78 12.45 4.23 3.43 3.311 1 10.56 4.22 7.46 3.32 2.74 2.842 1 9.78 3.13 3.57 3.08 3.14 3.143 1 9.63 3.14 3.39 2.93 2.91 2.794 1 9.58 3.84 4.11 3.37 3.15 3.185 1 9.44 4.48 3.80 3.03 2.94 3.01

0.5 2 13.19 5.16 2.69 2.12 3.28 1.851 2 10.76 2.41 3.34 2.59 3.15 2.572 2 9.71 1.50 2.48 1.50 1.49 1.613 2 9.44 1.77 1.49 1.41 1.36 1.304 2 9.11 2.42 1.39 1.47 1.19 1.305 2 9.22 3.18 1.18 1.27 1.35 1.31

0.5 3 13.26 4.20 6.65 2.85 2.64 2.281 3 10.62 1.70 4.91 2.17 1.87 1.992 3 9.33 1.56 2.12 0.99 1.13 1.003 3 8.97 1.97 0.91 1.17 1.08 1.024 3 8.82 2.59 0.79 1.05 0.95 1.005 3 9.08 3.24 1.26 1.36 1.46 1.35

0.5 4 12.55 3.66 5.51 2.50 2.13 2.251 4 10.11 1.76 3.46 1.55 1.43 1.512 4 8.84 2.28 0.94 1.20 1.43 1.183 4 8.58 2.70 1.02 1.33 1.32 1.294 4 8.59 3.13 1.15 1.29 1.33 1.285 4 9.18 3.78 2.22 2.26 2.23 2.14

0.5 5 11.71 3.72 3.49 2.84 2.86 2.931 5 9.97 2.31 2.15 1.76 1.81 1.762 5 8.54 3.06 1.27 1.44 1.55 1.47

3 5 8.42 3.47 1.48 1.45 1.54 1.514 5 8.66 3.92 1.89 2.01 2.10 1.975 5 9.76 4.37 3.52 3.42 3.15 3.13

0.5 7 11.04 4.26 4.69 4.52 4.54 4.651 7 9.03 2.99 2.72 2.38 2.37 2.402 7 8.24 3.80 2.16 1.81 1.86 1.863 7 8.28 4.15 2.00 1.97 2.06 1.96

Average 9.86 3.28 3.05 2.17 2.15 2.0626


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Table 3: Tests of Alternative Model Specifications

This table reports pairwise comparisons of pricing accuracy of alternative specifications of my model. Allmodels are estimated using a panel dataset consisting of 220 weekly observations on 34 swaptions fromJanuary 24, 1997 to April 6, 2001. For the model labelled by (N, K), there are N factors driving theevolution of the term structure of bond yields. The first K factors have unspanned stochastic volatility,and the other NK factors have constant volatility. The DM column reports the Diebold and Mariano(1995) statistic that tests the null hypothesis that the models in the first two columns have the samepricing accuracy for swaptions. Under the null hypothesis, the Diebold and Mariano statistic is distributedasymptotically as standard normal N(0, 1). Column four reports the number of weeks (out of a totalof 220 weeks) that Model 1 implies a higher root-mean-squared percent pricing error (RMSE) for thecross-section of swaptions, and column five reports the average reduction in swaption RMSE (reportedin percent) implied by Model 2 during these weeks. Similarly, the sixth column reports the number ofweeks that Model 2 implies a higher swaption RMSE, and the last column reports the average reductionin swaption RMSE (reported in percent) implied by Model 1 during these weeks.

Model 1 Model 2 DM RM SE 2 < RMSE1 RMSE2 > RMSE1Statistic Frequency Mean Difference Frequency Mean Difference

(1,1) (2,1) 6.46 211 1.88 9 0.47

(2,1) (3,1) 9.59 213 1.29 7 0.31

(3,1) (4,1) 0.26 119 0.17 101 0.16

(2,1) (2,2) 2.87 218 1.02 2 0.11

(2,2) (3,2) 6.68 211 1.36 9 0.48

(3,2) (4,2) 2.18 178 0.19 42 0.08

(3,1) (3,2) 3.61 200 1.17 20 0.21

(3,2) (3,3) 3.55 166 0.32 54 0.21

(3,3) (4,3) 2.50 161 0.16 59 0.10

(4,0) (4,1) 2.38 209 7.17 11 0.62

(4,1) (4,2) 2.83 214 1.21 6 0.18

(4,2) (4,3) 2.10 149 0.31 71 0.21

(4,3) (4,4) 0.52 127 0.21 93 0.20

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Table 4: Interest Rate Caps Pricing Errors

This table reports the average and mean absolute pricing errors for the interest rate caps of maturitytwo, three, four, five, seven and ten years under various model specifications. For the model labelled by(N, K), there are N factors driving the evolution of the term structure of bond yields. The first K factorshave unspanned stochastic volatility, and the other N K factors have constant volatility. The pricingerror for an interest rate cap is defined as the difference between its no-arbitrage model price impliedfrom the swaptions and its market price expressed as a percentage of the market price. The model price is

obtained from equation (17) using the model parameters (and implied volatility state variables) estimatedfrom the swaptions data. The dataset consists of 220 weekly observations on each series from January24, 1997 to April 6, 2001.

Maturity Mean Percent Pricing Errors Mean Absolute Percent Errors(4,0) (4,1) (4,2) (3,3) (4,3) (4,0) (4,1) (4,2) (3,3) (4,3)

2 -3.04 0.21 6.07 -2.82 4.52 10.96 6.27 7.26 4.52 5.943 -3.71 -3.26 0.91 -4.57 0.04 10.45 5.12 4.72 5.25 4.254 -3.37 -3.10 0.04 -3.92 -0.47 10.67 5.06 4.58 5.12 4.365 -3.35 -3.92 -1.51 -4.43 -1.77 10.53 5.35 4.51 5.33 4.457 -1.69 -3.33 -1.51 -3.44 -1.64 10.16 5.44 4.34 4.86 4.35

10 0.81 -2.07 - 0.28 - 1.88 - 0.49 9.43 5.24 3.86 4.14 3.92

Average -2.39 -2.58 0.62 -3.51 0.03 10.36 5.41 4.88 4.87 4.54

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Table 5: Maximum Likelihood Estimation of the GA4,3 Model

This table reports the results of maximum likelihood estimation of the GA4,3 model. There are four factorsdriving the innovation of bond yields. The first three factors display unspanned stochastic volatility, andthe fourth factor has constant volatility. The model is estimated with a panel dataset of 220 weeklyobservations on 34 at-the-money-forward European swaptions from January 24, 1997 to April 6, 2001.Panel A reports the estimates for i, i and i (i = 1, 2, 3) that govern the risk-neutral dynamics ofthe volatility state variables. The is are the market prices of volatility risk. The standard errors ofparameter estimates reported in the parentheses are obtained by calculating the inverse of the informationmatrix based on the Hessian of the likelihood function. Panel B reports the sample correlations betweenthe yield factors and each volatility state variable, as well as the correlations between the implied volatilitystate variables. Panel C reports the sample mean of the implied variances of the first three yield factorsand their unconditional variances.

Panel A: Parameter Estimates

100 1 0.0509 0.0354 0.0081 -0.0582

(0.0016) (0.0091) (0.0033) (0.1811)2 1.9703 0.0041 0.0138 -0.1479

(0.0329) (0.0001) (0.0056) (0.4186)3 2.6144 0.0018 0.0079 -0.2007

(0.3945) (0.0003) (0.0032) (0.2936)4 - 0.0026 - -

- (0.0002) - -

Panel B: Correlations between Yield Factors and Volatility Factors

Factor Yield Factors Volatility Factors

z1 z2 z3 z4 1 2 3z1 1z2 -0.0000 1z3 -0.0000 0.0000 1z4 -0.0000 -0.0000 0.0000 11 -0.0302 0.0084 -0.0162 -0.0579 12 0.0028 -0.0362 0.0613 0.0714 0.0955 13 -0.0291 -0.0434 -0.0791 -0.1288 0.3488 0.1473 1

Panel C: Unconditional and Average Implied Factor Variance

1031 1032 1033Historical 0.4584 0.0617 0.0159

Avergae Implied 0.4428 0.0761 0.0178

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Figure 1: Time Series of Swaption Implied Volatilities

This figure plots the time series of implied volatilities for 4 at-the-money-forward European-style swap-

tions. The data are mid-market quotes of annualized implied volatilities measured in percentage. The

sample is weekly from January 24, 1997 to April 6, 2001.

01/24/97 07/25/97 01/23/98 07/24/98 01/22/99 07/23/99 01/21/00 07/22/00 01/20/01

8

10

12

14

16

18

20

22

24

26

28

Date

Swaptio

nImpliedVolatility

0.5 into 1

5 into 1

0.5 into 5

5 into 5

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Figure 2: Time Series of Pricing Errors for Swaptions and Interest Rate Caps

This graph plots the time series of root mean squared pricing errors (RMSE) for swaptions and interest

rate caps under the GA4,3 model. The model is estimated via maximum likelihood using 34 at-the-

money-forward European swaptions. Panel A plots the RMSE of these 34 swaptions. Panel B plots the

RMSE for interest rate caps with maturity two, three, four, five, seven and ten years. For each interest

rate cap, the pricing error is calculated as the difference between its no-arbitrage value implied from the

swaptions according to the GA4,3 model and its market price, expressed as a percentage of the market

price. The dataset consists of 220 weekly observations on each series from January 24, 1997 to April 6,

2001.

01/24/97 11/24/97 09/24/98 07/25/99 05/24/00 03/24/01

0

2

4

6

8

Date

RootMeanSquaredPercentErr

or

Swaptions

01/24/97 11/24/97 09/24/98 07/25/99 05/24/00 03/24/010

10

20

30

40

Date

RootMe

anSquaredPercentError

Interest Rate Caps

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Figure 3: Average Implied Covariances versus Unconditional Covariances

This graph compares the time series average of the option-implied conditional covariances of the six-month

forward rates (up to 10 years in maturity) under the GA4,3 model to the their unconditional estimates.Panel A plots the average implied volatility and the sample standard deviation of the six-month forward

rates (both annualized). Panel B plots the difference between the average implied correlations and the

sample correlations based on weekly data from January 24, 1997 to April 6, 2001. The implied volatilities

and implied correlations are calculated according to the covariance structure specified in equation (12)

under the estimated GA4,3 model.

1 2 3 4 5 6 7 8 9 100.008

0.009

0.01

0.011

0.012

0.013

0.014

Maturity

Volatility of SixMonth Forward Rates

Average ImpliedUnconditional

1 23 4

5 67 8

9

1234

56789

0.1

0.05

0

0.05

0.1

Maturity

Difference Between Average Implied and Unconditional Correlations

Maturity

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Figure 4: Conditional Covariances: GARCH versus Option-Implied

This graph compares the conditional volatilities and correlations of the six-month Libor forward ratesimplied from the swaptions according to my GA4,3 model to those estimated from the historical bond

prices according to Engle (2002)s GARCH model with dynamic conditional correlations. Panel A illus-

trates two estimates of the volatility of the six-month forward rate with two-year maturity. Panel B plots

two estimates of the correlation between the two-year maturity and five-year maturity six-month forward

rates. In both panels, the horizonal line marks the unconditional estimate. The data sample is weekly

from January 24, 1997 to April 6, 2001.

01/24/97 11/24/97 09/24/98 07/25/99 05/24/00 03/24/010.005

0.01

0.015

0.02

Date

Volatility

GARCH(1,1)OptionImplied

01/24/97 11/24/97 09/24/98 07/25/99 05/24/00 03/24/010.4

0.5

0.6

0.7

0.8

0.9

1

Date

Correlation

Bivariate GARCH

OptionImplied

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Appendix

Proof of Proposition 1:

Under the forward risk-neutral measure Q, Sj(t) is a martingale, since it differs from

D(t , , j) only by a constant, for all j = 1, , 2(T). Moreover, the covariances among{d log D(t , , j)}j and {d log Sj}j are the same. Let S(t) be a vector of length 2(T )that stacks all Sj(t). Denote t as the instantaneous covariance matrix of d log S and Rtas the Cholesky decomposition of t so that RtR

t = . Using equations (12) and (13),t can be expressed explicitly in terms of U and k(t), k = 1, , n, as follows: for anytime t < = i/2, let integer m be such that m/2 t < (m + 1)/2, then

t = (1 2(t m/2)) A1UtUA1 + 2(t m/2) A2UtU

A

2

where A1 and A2 are both 2(T ) by 20 matrix. All elements ofA1 (resp. A2) are zero,except that the 2(T

) by 2(T

) submatrix consisting of i

m + 1th to i

m +jth

column (resp. from i mth to i +j 1 mth column) is a lower triangular matrix whoseelements on and below diagonal are 1. It follows each variance and covariance term in tis linear in the volatility state variables k(t), k = 1, , N.

Let dZt be 2(T ) dimension standard Brownian motions, then under Q,dS(t)

S(t)= RtdZt

By Itos lemma and definition of G, this implies that

d log G =

2(T)j=1

jd log Sj = 1

2

diag()dt +

RtdZt

where diag is an operator that takes the diagonal of a matrix. Note that if k(t) isdeterministic instead of following process (7), then G() is lognormal.

Now I calculate the expectation in equation (14) via the law of iterated expectationby first conditioning on a vector V which stacks the average values of k(t) over timeinterval [0, ], for k = 1, , n. Based on the assumption that the volatility state vari-ables are instantaneously uncorrelated with innovations in the yield curve, and followingan argument of Hull-White (1987) (see their lemma on P284-285), G() is lognormallydistributed conditional on V:

G() = exp( ) (21)where = 1

2

diag() and is a normal mean zero random variable whose variance

is (

). is the average covariance matrix of S over [0, ] corresponding to V. Tocalculate the expectation in (14), I use the following simple lemma:

Lemma 1 Let be a Gaussian random variable with zero mean and the variance 2. For

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any strictly positive numbers a and b, I have

E

max(ae0.52 b, 0)

= aN(h) bN(h )

where h = 1ln(a/b) + 0.5.

Note that EQ

[A] = 1 since each Sj is a martingale under Q and Sj(0) = 1. Hence

g = EQ

[G()]. Apply lemma 1 to G() as in (21), it follows that the price of a by Tat-the-money-forward swaption is approximately

P = D(0, ) EQ

V

EMax

G g, 0

|V

= D(0, ) EQ

V

N(1

2

) N(1

2

)

= D(0, )

2N

1

2

EQ

V[]

1)

Here I use the well known fact at-the-money option is almost linear in the Black volatilitywhich follows from the property of the normal cumulative density function N() and thefact that expectation operator is linear.

To finish the evaluation of the expectation above, I just need to compute the averageexpected value of the s from time 0 to and hence the average expected covariancematrix of the forward bonds involved in pricing by T swaption. This can be done withsimple but tedious algebra, using the conditional expectation of a mean reverting process:

E0[j(t)] = ejtj(0) + (1 ejt)j

Proof of Proposition 2: A cap on six-month Libor rate of final maturity T year withcap rate R consists of 2 T 1 caplets Ci, i = 1, . . . , 2T 1. Let i = i2 and D(t, i, i+1be the forward Libor bond price. Each caplet Ci protects the six-month Libor rate Liapplicable between time [i, i+1] year from rising above R. Its pays off

12

max(Li R, 0)per unit dollar of principal at time i+1. Since Li = 2(

1D(i,i,i+1)

1), the payoff of capletCi discounted to time i can be rewritten as

1

1 + 0.5Li

1

2max(Li R, 0) = (1 + 0.5R)max( 1

1 + 0.5R 1

1 + 0.5Li, 0)

= (1 + 0.5R)max(

1

1 + 0.5R D(i, i, i+1), 0)Hence each caplet Ci is a European put option on forward Libor bond D(t, i, i), witha strike price 1

1+0.5Rand maturing at i. A cap can be also viewed as a portfolio of put

options on forward Libor bond. I can price each caplet Ci using the corresponding forwardmeasure Qi. The key quantity is the average expected variance of each forward Liborbond (of fixed maturity), which can be calculated from equations (12), (7) and (13).

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References

[1] Ball, C. and W. Torous (1999), The Stochastic Volatility of Short-Term InterestRates: Some International Evidence, Journal of Finance, 54, no.6, 2339-2359.

[2] Black, F. (1976), The Pricing of Commodity Contracts, Journal of Financial Eco-nomics 3, 167 179

[3] Bliss, R., (1997), Movements in the Term Structure of Interest Rates, FederalReserve Bank of Atlanta Economic Review, Fourth Quarter, 1997, 16-33.

[4] Chen, R. and L. Scott (1993), Maximum Likelihood Estimation For a MultifactorEquilibrium Model of the Term Structure of Interest Rates, Journal of Fixed Income,3, 14-31.

[5] Chen, R. and L. Scott (2001), Stochastic Volatility and Jumps in Interest Rates:

An Empirical Analysis, Working Paper, Rutgers University.[6] Collin-Dufresne, P. and R. S. Goldstein (2002), Do Bonds Span the Fixed Income

Markets? Theory and Evidence fo

Documents

Correlations of Bond Yields