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Pricing Coupon-Bond Options and Swaptions in Affine Term

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					       Pricing Coupon-Bond Options and Swaptions in
                Affine Term Structure Models

                      Kenneth J. Singleton and Len Umantsev1

                              This draft : January 17, 2002




   1 Singleton is with the Graduate School of Business, Stanford University, Stanford, CA 94305 and
NBER, ken@future.stanford.edu. Umantsev is a Ph.D. student in the Department of Management
Science and Engineering, Stanford University, Stanford, CA 94305, uman@leland.stanford.edu. We
are grateful for financial support from the Gifford Fong Associates Fund, at the Graduate School
of Business, Stanford University.
                                         Abstract

This paper provides a numerically accurate and computationally fast approximation to the
prices of European options on coupon-bearing instruments that is applicable to the entire
family of affine term structure models. Exploiting the typical shapes of the conditional
distributions of the risk factors in affine diffusions, we show that one can reliably compute
the relevant probabilities needed for pricing options on coupon-bearing instruments by the
same Fourier inversion methods used in the pricing of options on zero-coupon bonds. We
apply our theoretical results to the pricing of options on coupon bonds and swaptions, and the
calculation of “expected exposures” on swap books. As an empirical illustration, we compute
the expected exposures implied by several affine term structure models fit to historical swap
yields.
1       Introduction
The affine family of term structure models (Duffie and Kan [1996], Dai and Singleton [2000])
is being increasingly widely used as a framework for pricing fixed-income derivatives. Among
the reasons for its attractiveness are: affine diffusions accommodate mean-reverting, corre-
lated factors and stochastic volatility, and zero-coupon bond prices are exponentials of affine
functions of the state (Duffie and Kan [1996]). Moreover, and of particular relevance for
option pricing, the conditional characteristic function (CCF) of an affine process is known
in closed form (Duffie, Pan, and Singleton [2000],Bakshi and Madan [2000]). The latter
observation, together with the fact that the optimal exercise boundary for an option on a
zero-coupon bond is a hyperplane, imply that the prices of zero-coupon bond options are
easily computed using Fourier inversion.1 Option pricing formulas for zero-coupon bonds
and the related pricing of caps/floors and “quanto” options are discussed in Chen and Scott
[1995], Chen [1996], Chacko and Das [1998], Nunes, Clewlow, and Hodges [1999], Duffie,
Pan, and Singleton [2000], and Bakshi and Madan [2000].
    While these results have substantially furthered our understanding of the pricing of cer-
tain fixed-income derivatives, they are largely silent on the pricing of such widely traded
options as European options on coupon bonds and swaptions. This paper fills this gap by
providing a numerically accurate and computationally fast approximation to the prices of
European options on coupon-bearing instruments that is applicable to the entire family of
affine term structure models. We proceed by approximating the (nonlinear) optimal exercise
boundary boundary for coupon-paying instruments with straight line segments that match
closely the segment of the boundary where the density of the affine state process Y is concen-
trated. With these approximations in hand, we can then compute the relevant probabilities
needed for pricing options on coupon-bearing instruments by the same Fourier inversion
methods used in the above literature on the pricing of options on zero-coupon bonds. The
key empirical observation underlying the computational success of our approach is that, for
model parameters chosen to fit historical term structure movements, the joint density of Y
typically overlaps a quite small segment of the nonlinear exercise boundary. Consequently,
for all of the examples we consider, the use of a single line segment to approximate the rel-
evant portion of the exercise boundary gives extremely accurate (and computationally very
fast) pricing of European options on coupon-bonds and swaps.2
    A different approach to approximating the prices of options on coupon bonds in affine
term structure models was proposed by Wei [1997] and subsequently extended by Munk
[1999]. Working in a one-factor setting (and thereby encompassing one-factor Vasicek [1977]
and Cox, Ingersoll, and Ross [1985] models), Wei showed that the price of a European option
on a coupon bond is approximately proportional to the price of an option on a zero-coupon
bond with maturity equal to the stochastic duration ( Cox, Ingersoll, and Ross [1979]) of the
    1
     This approach to the pricing of zero-coupon bond options is exactly analogous to the approach originally
set forth by Heston [1993] for pricing options on equities.
   2
     The focus on “affine” models gives several critical ingredients to the success of our approach: (1) knowl-
edge of the functional dependence of zero-coupon bond prices on Y , which in turn implies that (2) the
optimal exercise boundary is concave, and (3) knowledge of the CCF of Y . In principle, our approach could
be applied to non-affine models that implied sufficiently well-behaved exercise boundaries and knowledge of
the CCF of Y . However, we have not explored pricing in non-affine models.


                                                     1
coupon bond.3 Munk extends Wei’s Stochastic Duration approximation to the the general
case of multi-factor affine term structure models, and gives an analytic demonstration of why
the approximation works. In particular, he shows that for deep-in and deep-out of the money
calls the absolute pricing errors should be close to zero. However, both Wei and Munk show
through examples that their approach systematically over prices slightly in- and underprices
slightly out-of-the money calls, though by small amounts. In contrast, our approach, by
exploiting directly the known conditional characteristic function of affine diffusions, provides
relatively accurate option prices for coupon bonds over the entire range of exercise prices,
including those nearly at the money. This accuracy is achieved at the small cost of slightly
greater computational times relative to the use of the Stochastic Duration approximations.
    There is an even greater advantage, it seems, of our Affine approximation over the
Stochastic Duration approximation for the important case of pricing swaptions. A sig-
nificant portion of plain-vanilla swaps are priced “in arrears,” meaning that the payment
obligation of the floating-rate payer is determined by the LIBOR rate set at the beginning of
the payment period. In this case, pricing a swap between cash-flow dates is not equivalent
to pricing a coupon-bond between coupon dates; the value of a swap depends on the current
and lagged (as of the previous cash-flow date) values of the state vector. Whereas our Affine
approximation extends with minor modification to options written on swaps with LIBOR
set in arrears, as developed in the literature, the Stochastic Duration approximation does
not appear to be applicable.
    As an application of our approximation, we explore the sensitivity of “expected expo-
sures” on swaps to the specification of the market prices of risk or risk premiums in affine
term structure models. Following Sorensen and Bollier [1994], expected exposures is often
computed in practice by treating the credit risk as a portfolio of swaptions under restrictive
assumptions about the joint distribution of default times and treasury yields. Creditmetrics,
for instance, uses this approach under the additional assumption that the financial institu-
tion computing the risk is default free. To illustrate our approach, we provide an alternative
analytic calculation of the expected exposure on a newly issued swap that recognizes the
two-sided nature of the credit risk of an interest-rate swap and avoids the implicit restric-
tions in the Sorensen-Bollier approach on the process generating market and credit risks.
Additionally, we investigate the sensitivity of expected exposure to various specifications of
the market prices of risk in affine models. Building upon the recent work by Duffee [2001]
and Dai and Singleton [2001] on the forecasting properties of affine models we show, using
models estimated on time series of US$ Treasuries and LIBOR/Swap rates, that standard
formulations of the market prices of risk substantially understate the magnitudes of ex-
pected exposures. We hope that this analysis is of independent interest beyond our Affine
approximation that leads to analytic expected exposure calculations.
    The remainder of this paper is organized as follows. Section 2 sets up the pricing prob-
lem for general affine term structure models. Section 3 describes in detail our approach to
approximating the prices of options on coupon bonds for these models. Section 4 compares
   3
    For these one-factor models, the advantage of Wei’s approximation is largely computational, since exact
pricing formulas exist when zero-coupon bond prices are strictly monotonic functions of the (one-dimensional)
state (Jamshidian [1987]). The special cases of one-factor Gaussian and square-root diffusion models are
examined in Jamshidian [1989] and Longstaff [1993], respectively.



                                                     2
the properties of our Affine approximation to those of Stochastic Duration. Finally, sec-
tions 5 and 6 discuss analytic computation of swaption prices and expected exposures for
risk management of swap positions.


2         Affine Term-Structure Models and Option Pricing
We define the family of affine term structure models as those having the instantaneous short
rate rt modeled as an affine function of a multi-factor state process Yt ∈ Θ ⊂ RN following
an affine diffusion under both the physical (P) and risk-neutral (Q) measures. Under P,

                                        rt = δ0 + δY · Yt ,                                             (1)
                                      dYt = K(θ − Yt )dt + Σ                  St dWtP ,                 (2)

where WtP is a N-dimensional standard Brownian motion; the matrices K and Σ are N × N,
and δY and θ are N × 1 vectors; and the diagonal N × N matrix St has the instantaneous
factor variances along the diagonal,

                                        Sii,t = αi + βi · Yt , i = 1, . . . , N,                        (3)

for non-negative scalar αi and N × 1 vector βi . We assume, as in Duffee [2001], that the
N × 1 vector of “market prices of risk” is given by

                                               Λt =    St λ +           St− λYt ,
                                                                            ˜                           (4)

where λ is an N × 1 vector, λ is an N × N matrix, and St− is a diagonal matrix with ith
                            ˜
diagonal element defined as
                                                 1
                                Sii,t − =                , if inf (αi + βi · Yt ) > 0,                  (5)
                                            αi + βi · Yt      YT ∈Θ

and 0 otherwise. This specification assures that Y also follows an affine diffusion under Q,
driven by the Brownian motion WtQ = WtP + 0 Λu du.
                                                  t

     Following Dai and Singleton [2000], we classify uniquely each member of the family of
N-factor affine models into the subfamily AM (N) if there are M state variables driving
the factor variances Sii,t of all N state variables.4 The state space Θ ⊂ RN for an affine
term-structure model in AM (N) is a subset of the N-dimensional Euclidean space, bounded
by M hyperplanes imposing the non-negativity constraints on the diagonal elements of St ,
Sii,t ≥ 0. One can immediately see that for Gaussian models in A0 (N), Θ ≡ RN since the
conditional variances are constants; and for the full-rank models of AN (N), Θ √ RN , since
                                                                                  ≡ +
                                                           th
the model can be represented with the volatility of the i state variable being Yit .
     Yields in affine models are linear in Yt (by definition). Bond prices are hence exponentials
of affine functions of Yt :
                                                        T
                                  Bt (Yt ) = EQ [e−
                                   T
                                              t
                                                       t
                                                            ru du
                                                                    ] = eα(T −t)+β(T −t)·Yt ,           (6)
    4
        More precisely, the classification is done by the rank M of the matrix B = (β1 , . . . , βN ).

                                                              3
where the functions α(·) and β(·) are either known in closed form or can be obtained as
solutions of Ricatti ODEs. Since Runge-Kutta methods for such ODEs produce fast and
accurate solutions, we will not distinguish between conventional closed-form expressions and
solutions of such ODEs.
                                                                                     T
    Options written on zero-coupon bonds are easily priced in this setting. Letting Bt denote
price at date t of a zero-coupon bond that matures at date T , the price C(t, Yt ; S, T, K) a
call option with strike K and maturity S written on a zero-coupon bond with maturity T is
                                     S
     C(t, Yt ; S, T, K) = EQ [e−
                           t
                                    t
                                         ru du
                                                 (BS − K)+ ]
                                                   T

                                     T                  T                             S              S
                        = EQ [e−
                           t
                                    t
                                         ru du
                                                 ]EQ [1{BT >K} ] − KEQ [e−
                                                   t     S           t
                                                                                     t
                                                                                          ru du
                                                                                                  ]EQ [1{BT >K} ]
                                                                                                    t     S

                        = Bt P rt {BS > K} − KBt P rt {BS > K},
                           T    T   T          S    S   T
                                                                                                                    (7)
where P rt {X > K} is the conditional probability of the event {X > K}, based on the
          S

S−forward measure QS induced on Q by the price of a zero-coupon bond issued at date t and
                        t
                        S
maturing at time S, Bt . For the entire family of affine term structure models, these forward
probabilities are easily computed using the known conditional characteristic functions of
                          y
affine diffusions and Lev´ inversion (Bakshi and Madan [2000], Duffie, Pan, and Singleton
[2000]). That is, since {BS > K} ≡ {α(T − S) + β(T − S) · YS > ln K} and the characteristic
                           T

function of β(T − S) · YS conditional on Yt is known in closed form, two one-dimensional
Fourier transforms give the requisite probabilities under the two forward measures. This
approach cannot be used to price options on coupon bonds or swaptions, however. In the
next section we present very efficient approximations for prices of options on coupon-bearing
instruments in this affine setting.


3    Pricing Coupon-Bond Options in Affine Models
The difficulty of pricing coupon bond options is that the exercise region is defined implicitly
and, therefore, its probability is often difficult to compute. To illustrate the nature of the
problem, let
                                                                 (T −t)/∆
                       Vt = V (t, Yt ; T, ∆, c) = c/∆                       Bt −i∆ + Bt
                                                                             T        T
                                                                                                                    (8)
                                                                   i=0

be the price of a coupon bond with coupon rate c and cashflow frequency ∆. (Most coupon
bonds have semi-annual coupons, swaps have either annual or semi-annual cashflows, and
US$-denominated caps and floors typically have quarterly cashflows.) The price of a Euro-
pean option on this bond with strike K and maturity S is given by
                                                                         + 
                                                                       (T −S)/∆
                                                    S
       C(t, Yt ; S, K, T, ∆, c) =   EQ
                                     t
                                          e−      t
                                                        ru du   c/∆              BS −i∆ + BS − K  
                                                                                   T        T
                                                                                                                    (9)
                                                                         i=0
                                           (T −S)/∆
                               = c/∆                        Bt −i∆ P rt −i∆ {VS > K}
                                                             T        T

                                           i=0
                               +    Bt P rt {VS
                                     T    T
                                                            > K} − KBt P rt {VS > K}.
                                                                     S    S



                                                            4
More generally, if Vt = V (t, Yt ; {ci }n , {Ti }n ) is the price of a fixed-income instrument with
                                         i=1      i=1
certain cashflows c1 , c2 , . . . , cn payable at dates T1 , T2 , . . . , Tn , then the option price is given
by
                                                    n
       C(t, Yt ; S, {ci }n , {Ti }n ) =
                         i=1      i=1                      ci Bt i P rt i {VS > K} − KBt P rt {VS > K}.
                                                               T      T                S    S
                                                                                                                        (10)
                                                   i=0



                        00



                                                                                    00
                        00                                                          0




                        00
                   Y2




                        00




                        00




                        00




                             0
                                 0         0        0          0            0            0      0        0
                                                                       Y1

Figure 1: Exercise boundaries for 5-year a.t.m. Calls on 30-year 10% coupon and zero-coupon
bonds implied by an A2 (2) affine term structure model.

    The exercise region of this call option is the subset of Θ with
                                     n                                          n
           {VS > K} ≡                          T
                                           ci BS i (YS )   >K      ≡                ci eα(Ti −S) eβ(Ti −S)·YS > K   ,   (11)
                                     i=1                                    i=1

where we are assuming that there are n remaining cashflows after the expiration date of the
option. If all the future cashflows ci are positive, then this exercise boundary is a concave
surface. Further, if the bond has only one payment after the option maturity date (e.g. we
are dealing with a zero-coupon bond option), n = 1 and the concave boundary becomes a
hyperplane:

      {VS > K} ≡ c1 BS 1 (YS ) > K
                     T
                                                        ≡ β(T1 − S) · YS > ln(K/c1 ) − α(Ti − S) .                      (12)

Figure 1 illustrates these observations by plotting exercise boundaries for five-year at-the-
money (a.t.m.) calls on thirty-year 10% coupon and discount bonds implied by the two-factor

                                                                   5
square-root model (an A2 (2) model), with parameter values taken from Duffie and Singleton
[1997] and the state variables evaluated at their long-run means. In the case of zero-coupon
bonds, as noted above, the probabilities P rt {b · YS < a} are easily computed using one-
                                              τ

                y
dimensional Lev´ inversion.
        00
                                                                         00



        00                                                               00




                                                                         00
        00



                                                                         00
   Y2




        00




                                                                    Y2
                                                                         00
        00


                                                                         00

        00

                                                                              0


             0
                 0   00   0   0   0   0    0      0    0   0    0
                                                                     00
                                      Y1                                          0       00   0    0     0    0      0   0   0   0   0
                                                                                                               Y1

Figure 2: Bivariate density of a two-factor square-root model for Yt on the exercise set (left
panel) and our affine approximation to the exercise boundary in the relevant region (right
panel) for a 5-year a.t.m. Call on a 30-year 10% coupon bond.

    In order to price options on coupon bonds, we need to compute the probabilities of the
         τ    n       αi +βi YS
form P rt     i=1 ci e          > K for which these inversion methods are not directly available.
Our strategy for circumventing this problem is to approximate the optimal exercise boundary
using straight line segments that lend themselves, in affine models, to calculations based on
Fourier inversion. This approach is particularly appealing for affine models, because we have
found, for a wide variety of multi-factor affine term structure models fit to U.S. treasury and
swap yields, that the conditional density of Y is non-zero on a fairly small segment of the
exercise boundary. This observation is illustrated in the left panel of Figure 2 where the bi-
variate density of our illustrative two-factor square-root model for YT is shown along with the
exercise set. We therefore can approximate the boundary with a straight line that matches
closely the segment of the boundary where the conditional density of YS is concentrated.
The more concentrated is the density of YS , the shorter is this line and the more accurate
is the approximation. This basic Affine approximation is illustrated on the right panel of
Figure 2.
    For this case of a single line segment, our Affine approximation reduces the problem of
computing the exercise probability
                                           n
                                     τ
                                  P rt           ci eαi +βi ·YS ≤ K                   ,    τ = T1 , . . . , Tn , S,                       (13)
                                           i=1

to computing the n + 1 probabilities P rt {b · YS ≤ a} (one for each cashflow date after the
                                        τ

option expiration date and one for the strike payment), where the straight line b · Y = a

                                                                    6
approximates the concave boundary of the exercise set. The latter probabilities are exactly
of the form we know how to solve from the literature on pricing options on zero-coupon
bonds. For our two-factor square-root example, the probability of exercise of the option,
                                                   S
computed exactly under the forward measures P r0 , is 0.4678. Using the one-segment Affine
approximation, we obtain 0.4685. The corresponding prices of coupon bond options are
0.094360 (exact) and 0.094495 (approximate).
    This approximation strategy is easily extended to use several segments of straight lines to
approximate the exercise boundary in the relevant region and, hence, to obtain an even more
accurate approximation. The potentially better approximation to the exercise set is achieved
at the expense of potentially slower execution speed: for an instrument with n coupons
remaining after the option expiration one would need to compute n + 1 probabilities for
every segment. As before, this involves the computation of multiple one-dimensional Fourier
inversions.
    One convenient algorithm for choosing the straight-line segments underlying our Affine
approximation is as follows. Letting Y (i) denote the i entry of the vector Y and taking t = 0
to be the pricing date of the option and t = S to be its expiration date (so S =five years in
our example), we proceed in four steps:
                                                                                               (2)
   1. If n = 2, find the interval on the real line on which the univariate density of YS is
                                                                               (2)
      (approximately) concentrated. For our illustration, the density of Y5yr is negligible
      outside of [0.017, 0.025]. If n > 2, one must find 2n−1 vertices of a “cube” of dimension
      n − 1.

   2. Find the exercise boundary at the endpoints of this interval (or cube). In our example,
      the exercise boundary passes through the points Y5yr = (0.31, 0.017) and (0.22, 0.025).

   3. Fit a straight line b · Y = a through these two points. If n > 2, a hyperplane b · Y = a
      can be fit through 2n−1 points by the method of least squares.

   4. Compute the price of the option using the probabilities P r0 i {b · YS ≤ a} under the
                                                                 T

      relevant forward measures.

One can easily assess the quality of the resulting approximation by replacing one of the
end-points of each interval with the mid-point of the segment and then computing the prob-
abilities of the sets bounded by straight lines fit through the mid-points of the intervals.
If the obtained price is substantially different from the original price, more than one line
segment is necessary.
    To understand, more formally, why our approximation scheme is so accurate consider the
nonlinear dependence of the forward par rate rt (S; T ) (the coupon on a T-year bond that
makes it trade at par S years forward) on the state Y :5

                                                eα0 +β0 Yt − eαn +βn Yt
                                 rt (S; T ) =              αi +βi Yt
                                                                        .
                                                        ie
  5
    In the case of swaptions examined subsequently, we would focus on the forward swap rate which can
similarly be represented as a forward par rate.



                                                    7
We show in the Appendix A, following Umantsev [2001], that these rates can be efficiently
approximated by linear functions of the state vector
                                 rt (S; T ) = aS(S; T ) + bS(S; T )Yt + t ,                                     (14)
where the error term t can be expressed as a difference of two strictly positive terms each
on the order of r × T 2 × var[r], where r and var[r] are the average value and variance of
                 ¯                         ¯
the underlying rate. Not surprisingly, the error is greater for volatile rates and long swap or
bond maturities.
    Empirically, the bounds on t are typically small for the coupon-bond option and swap-
tions examples we have examined. (See our subsequent discussion of swaptions for a detailed
example.) Accordingly, the exercise regions of the options written on such rates can be reli-
ably replaced with hyperplanes. Computation of the exercise probabilities for these options
under the forward measures is no more complicated that that for the options on forward LI-
BOR rates (caps and floors) and on discount bonds. For our two-factor square-root example,
                                                                                             S
the probability of exercise of the option, computed exactly under the forward measures P rt ,
is 0.4678. Using the one-segment Affine approximation, we obtain 0.4685. The corresponding
prices of coupon bond options are 0.094360 (exact) and 0.094495 (approximate).


4    Affine Approximation Versus Stochastic Duration Ap-
     proximation for Coupon-Bond Options
It is instructive to compare the pricing errors of our affine approximation to those obtained
by the Wei-Munk approximation based on stochastic duration. Briefly, the latter approach
approximates the initial problem of computing the coupon bond option price with a much
simpler problem of computing a zero-coupon bond option price. The stochastic duration of a
coupon bond is defined as the maturity T ∗ of a zero-coupon bond that has the same volatility
of instantaneous return as that of the instantaneous return on the coupon bond. Munk [1999]
shows that under certain conditions the solution T ∗ exists and is unique. Furthermore, the
price of a call option on a coupon bond can be reasonably approximated as the multiple
            T∗
V (t, Yt )/Bt of the price of a call option on a T ∗ maturity zero-coupon bond. This scaling
adjusts for the fact that the prices of the underlying bonds at time t are different. Munk shows
that the error from using this approximation for any choice of T > 0 can be represented as
                    S                             +     V (t, Yt ) Q −    S                    Bt T         +
         EQ
          t   e−   t
                        su du
                                V (S, YS ) − K        −      T
                                                                  Et e   t
                                                                              ru du    T
                                                                                      BS   −            K
                                                          Bt                                 V (t, Yt )
                                              +                         +
               S         V (S, YS )  K                 V (t, Yt )  K
     =   Bt EQ
          T
             t                T
                                    − T           −         T
                                                                  − T
                            BS       BS                  Bt        BS
When K is large relative to V (t, Yt ) (out-of-the-money calls), both ()+ terms are zero with
probability close to 1. The pricing error is therefore close to zero. When K is small (deep-
in-the-money calls), both ()+ terms are positive with probability close to 1 and the error is
close to
           S  V (S, YS )  K     V (t, Yt )   K             S  V (S, YS )    T V (t, Yt )
    Bt EQ
       T
         t         T
                         − T −       T
                                           + T = Bt EQT
                                                         t         T
                                                                         − Bt      T
                                                                                         =0
                 BS       BS      Bt        BS                   BS             Bt

                                                        8
(the last equality holds because prices deflated by Bt are QT -martingales, by definition of
                                                     T

the forward measure). The approximation error can therefore only be significantly different
from zero when the probability of only one of two ()+ terms being positive is relatively high.
This happens for the intermediate values of K (near-the-money calls).6
                                                               3
                                                        x 10
                                                 2.5

                                                                                               Affine Approximation + Fourier Transform
                                                   2
                                                                                               Stochastic Duration
                                                                                               Monte Carlo, 10000 paths
                                                 1.5
                Absolute Approximation Errors




                                                   1


                                                 0.5


                                                   0


                                                -0. 5


                                                  -1


                                                -1. 5


                                                  -2


                                                -2. 5
                                                    0.2            0.4   0.6   0.8     1          1.2     1.4     1.6      1.8     2

                                                                               Moneyness (Strike/Forward)


Figure 3: Absolute Approximation Errors vs. Moneyness: 5-yr. Calls on a 30-yr. 10% bond.

    The absolute pricing errors for these two approximation schemes – our Affine approxi-
mation and the Stochastic-Duration approximation – are illustrated in Figure 3. For our
illustrative affine model, both “Affine Approximation” and “Stochastic Duration” are very
precise: all absolute errors are on the order of 5 × 10−3 . However, the Affine Approxi-
mation does not manifest the systematic over-pricing (under-pricing) of slightly in- (out-)
of-the money calls, but rather shows uniformly small absolute pricing errors for all degrees
of moneyness. Also, given that the error of the stochastic duration approximation changes
sign around-the-money, one can infer that the error is very close to zero for at-the-money
options. Unfortunately, we know of no theoretical result that can help us pinpoint where
exactly the error is changing its sign or, alternatively, help us estimate how large the error
is for a.t.m. options.
    Figure 4 displays the corresponding relative pricing errors. Deep out-of-the money calls
have very small prices and so small pricing errors can show up as large errors relative to
market prices. This is indeed the case for the Stochastic Duration approximation for which
relative errors become substantial for options that are twenty percent or more out-of-the
money on a forward basis. In contrast, the relative pricing errors are again very small for
   6
    There is no presumption that using the maturity T ∗ gives an optimal approximation in the sense of
minimizing the approximation error over all choices of T for the maturity of the zero-coupon bond used in
the approximation.


                                                                                           9
                                                   1

                                                                Affine Approximation + Fourier Transform
                                                 0.8
                                                                Stochastic Duration
                                                                Monte Carlo, 10000 paths
                                                 0.6



                Relative Approximation Errors
                                                 0.4


                                                 0.2


                                                   0


                                                -0. 2


                                                -0. 4


                                                -0. 6


                                                -0. 8


                                                  -1
                                                    0.2   0.4        0.6     0.8      1       1.2     1.4   1.6   1.8   2

                                                                              Moneyness (Strike/Forward)


Figure 4: Relative Approximation Errors vs. Moneyness: 5-yr. Calls on a 30-yr. 10% bond.

the Affine approximation scheme.
    For comparison, we also display the absolute and relative pricing errors based on a Monte-
Carlo analysis of an Euler discretized model with a step size of one day (= 1/250 year) from
time 0 until the maturity of the option, normal random shocks, and 10,000 paths. The
Affine approximation clearly dominates due to residual noise in this particular Monte Carlo
scheme.
    In terms of computational complexity, stochastic duration requires computing only one
zero-coupon bond option price or, equivalently, computing two probabilities of the form
P rt b · YS < a – one for the stochastic duration of the coupon bond τ = T ∗ and the
   τ

other one for the option maturity time τ = S. On the other hand, Affine approximation
requires computing n + 1 probabilities for all cashflow dates up to the bond maturity T . If
T is considerably larger than T ∗ (as is the case with most long-maturity bonds), and the
distribution function takes longer to compute for large T ’s, the Affine approximation will
be slower than the Stochastic Duration approximation. Therefore, the choice among these
approximations involves a speed–accuracy tradeoff. In the case of our illustrative 30-year 10%
bond, its stochastic duration is 12.0071 years. Thus, the Stochastic Duration approximation
gives a coupon bond option price of 0.094537, compared to the exact price of 0.094360 and
our Affine approximate price of 0.094495.
    Table 1 provides some information about the speed/accuracy tradeoff by providing the
computational times of the approximations for this illustrative coupon-bond option.7 The
  7
   The bond prices and conditional characteristic functions were solved for using a variable-step Runge-
Kutta method, coded in C (called from MATLAB), even though exact solutions were available for the CIR
models. The reason for this is to make the times comparable to the ”intermediate” AM (N ), 0 < M < N


                                                                                      10
Affine approximation is substantially faster than both Monte-Carlo and numerical integration
schemes, and somewhat slower than the Stochastic Duration approximation. Both of these
approximations have the potential disadvantage relative to Monte-Carlo methods of being
roughly additive in the sense that the number of calculations required goes up proportionately
with the number of options being valued. In contrast, the same Monte-Carlo paths can be
used to value a variety of options on the same underlying.

                              Method                   Time, sec.
                              Monte-Carlo, 10000 paths     12.73
                              Affine Approximation            1.42
                              Stochastic Duration           0.24

Table 1: Computational Times, in seconds: 5yr. call on a 10% 30yr. bond in the A2 (2)
model.

    Though we have focused on Treasury bonds, our pricing approach extends immediately
to the case of defaultable corporate or sovereign bonds and to defaultable options (the writer
of the option might default) written on defaultable bonds. Specifically, following Duffie and
Singleton [1999] and assuming that recovery in the event of default is expressed as a fraction
of market value, we can replace the riskless rate r in the valuation of the coupon bond (in
                        T
the determination of BS in (8) by the default-adjusted discount rate appropriate for pricing
the underlying defaultable bond. Similarly, if the writer of the option might default, then
(10) becomes
                                        n
     C(t, Yt ; S, {ci}n , {Ti}n )
                      i=1     i=1   =         ci Bt i P rt i {VS > K} − KBt P rt {VS > K},
                                                 ˆ OT    T                OS   S
                                                                                                     (15)
                                        i=0

        OT
where Bt is the value of a defaultable zero-coupon bond obtained by discounting a promised
payoff of $1 by the default-adjusted discount rate appropriate for the writer of the option,
and the forward default probabilities are redefined accordingly.
    We can also easily extend our approach to pricing to quantos options in which the under-
lying coupon bond is issued in a different currency than the currency of the option payoff.8
Letting Ft denote the spot exchange rate– expressed as units of the option-payoff currency
per unit of the currency in which the coupon bond is issued– two versions of such quantos
options are: (Q1) the payoff on a European option at maturity date S is CS = FS (VS − K)+ ,
                                                                                   ˆ    ˆ
                ˆ denominated in the currency of the underlying bond; and (Q2) the payoff
with the strike K
is CS = (FS VS − K)+ , with the strike K denominated in the payoff currency and hatted vari-
            ˆ
ables denoting foreign currency-denominated prices. In case (Q1), pricing is accomplished
by applying the Affine approximation to price the foreign bond option.
models, where no closed-form solutions for the coefficients on the state exist. All computations were done
on a Ultra-60 SUN Sparc in MATLAB 6.1.
   8
     See Duffie, Pan, and Singleton [2000] for a discussion of the pricing of quantos options on zero-coupon
bonds in an affine diffusion setting.




                                                    11
           ˆ
   Letting VS denote the foreign coupon bond price, the value of the (Q2) quanto is deter-
mined similarly to the value of a coupon bond option as in (10):
                           S                                    +
        C0 = EQ e−         0
                               ru du       ˆ
                                           VS FS − K
                               S
             − KEQ e−          0   ru du
                                           1{VS FS >K} =
                                             ˆ

                     (T −S)/∆
                                                           Ti                                         T
             = c/∆                 F0 B0 −i∆ P r {VS FS > K} + F0 B0 P r {VS FS > K} −
                                      ˆT     ˆ    ˆ               ˆT ˆ    ˆ
                      i=S/∆

             −   KB0 P r S {VS FS
                   S        ˆ               > K},                                                                        (16)

where the exercise region is
                      n                                                        n
                                                                                                          ˆ
    {VS FS > K} ≡
     ˆ                        ˆT
                           ci BS i (YS )F (YS ) > K                     ≡                ˆ
                                                                                     ci eα(Ti −S) e(ψ+β(Ti −S))·YS > K   (17)
                     i=1                                                       i=1

Note that in this case, for the first n + 1 terms, the Affine approximation is applied using
the forward measures for foreign-exchange denominated assets QTi (hence the notation P r).
                                                             ˆ                       ˆ


5     Analytic Swaption Prices
The value of a (settled in arrears) swap today (date t) that matures at date Tn is given by
                                                     n                                 T
                                                                                     Bt it
                                       Vt = c               T
                                                           Bt i     +    T
                                                                        Bt n   −      T
                                                                                              ,                          (18)
                                                    i=it                           BTiit−1
                                                                                          t


where the Ti are the cashflow dates and it is the index of the next cashflow date at time t.
The last term in (18) appears, because the LIBOR floating side of the contract is settled in
arrears using the LIBOR rate at the preceding cashflow date. An important consequence
of this settlement convention is that Vt depends not only on the current state, but also on
the value of the state on the previous cashflow date. Only on cashflow dates, when the last
term simplifies to unity, does the direct parallel between a swap and a coupon bond emerge.
Consequently, the Stochastic Duration approximation cannot be applied (at least, not as
developed in the literature) for the pricing of swaptions. The Affine approximation can still
be used successfully, however, as we now demonstrate.
    On cashflow dates, the floating side of swap is at par so the swaption price is equal to
the price of a call of the same maturity and strike of one written on a coupon bond with
maturity and coupon rate equal to those of the swap. Specifically, letting T = Tn − S, at
the inception of a “T-in-S” swaption– the right to enter into a T -period swap at some future
date S– the swaption price is
                                                                  n                               +
                                             S
                            EQ
                             t
                                       −
                                       e    t
                                                 ru du
                                                         · c            BS i + BS n − 1
                                                                         T      T
                                                                                                      ,                  (19)
                                                                i=iS


                                                                  12
where rt is being set to the discount rate implicit in the pricing of swaps. We could easily
extend our valuation approach to the case where counterparties in the swaption contract had
different ratings than those (say AA) underlying the pricing of generic swaps by introducing a
different discount rate for pricing swaptions versus pricing swaps. Our pricing of both swaps
and swaptions recognizes the two-sided nature of the credit risk of swaps. However, following
Duffie and Singleton [1997], we are assuming that the counterparties have symmetric credit
risks. As shown by Duffie and Huang [1996], asymmetry of credit quality has very little
effect on the pricing of at-market interest rate swaps. Within this framework, the pricing of
newly issued swaptions proceeds as in the case of a coupon-bond option.
    At times between cashflow dates, the swaption value is given by
                                                           
                                  n                  TiS
                       S                           B      +
           EQ e− t ru du · c
             t                       BS i + BS n − TS
                                      T      T
                                                     i
                                                            =                           (20)
                                i=iS              BTi S−1
                                                      S
                                                                                     
               n                                                  Ti
                                                                           1
                  Bt i QTi {VS > 0} + Bt n QTn {VS > 0} − EQ e− t ru du Ti 1{VS >0} 
                   T                      T                         S
        = c              t                   t               t
             i=iS                                                       BTi S−1
                                                                                                                S

The first two terms on the right-hand-side of (20) have exact counterparts in the coupon-bond
option pricing formulas, so we already know how to compute them efficiently. Therefore,
if we can accurately approximate the third term, then we will have extended our Affine
approximation to the case of swaptions. This is easily accomplished, because this term can
also be represented as a probability of {VS > 0}, though under a different measure than we
have heretofore examined.
    The expectation we are interested in can be written as
                                      
                  Ti                                Ti
                           1                                −α(1)−β(1)·YTi −1
         EQ e− t ru du TS 1{VS >0}  = EQ e− t ru du e
                    S                                 S
           t                                 t
                                                                          S   1{VS >0} , (21)
                        BTi t−1
                                        S

                                                    T              α(1)+β(1)·Y
where we have used the fact that BTiSt−1 = e       Ti −1
                                                     S   . Central to pricing coupon-bond
                                    S
options are the equivalent Martingale measure Qt induced by a “money-market account”
and the S-forward measure QS induced on Qt by the price of Bt :
                             t
                                                               S

                                                                                                      S
                                S        1               S                    dQS   e− t ru du
                             EQ [∗]
                              t        = S EQ [e−       t
                                                             rτ dτ
                                                                     · ∗], or     =            .
                                        Bt t                                  dQ      Bt S


For pricing swaptions between cashflow dates we extend this idea to let Q(τu ,τv ,u,v) denote
the following equivalent measure:
                                                                        τu
                                            dQ(τu ,τv ,u,v)    et            u·Y d +v·Yτv
                                                            = Q           τu
                                                                               u·Y d +v·Yτv
                                                                                                  ,                                  (22)
                                               dQ            Et [e       t                    ]
                       (S,0,−δ,0)
so that QS = Qt
         t                          . Using this notation, we have
                                         
         Ti                                                  Ti
                         1                                           ru du −α(1)−β(1)·YTiS −1             (TiS ,TiS −1 ,−δ,−β(1))
EQ e−                           1{VS >0}  = EQ e−
           S                                                   S
 t
         t     ru du
                        Ti                     t
                                                             t           e                            P rt                          {VS > 0},
                       BTi S−1
                          S
                                                                                                                                     (23)

                                                                   13
where the first term on the right-hand side of (23) is known in closed-form (Duffie, Pan, and
Singleton [2000]). Finally, note that the swaption exercise region is
                       n
                             α(Ti −S)+β(Ti −S)·YS          α(Tn −S)+β(Tn −S)·YS       eα(TiS −S)+β(TiS −S)·YS
 {VS > 0} ≡       c          e                           +e                       −       α(1)+β(1)·YTi
                                                                                                                >0
                                                                                        e             S −1
                      i=iS
                   n−iS +1
                                    ηj ·YS −ˆj ·YTi
                                            η
            ≡                    νj e             S −1   >1 ,
                      j=1

for some νj ∈ R, ηj , ηj ∈ RN , j = 1, . . . , n − iS + 1, and the Affine approximation can be used
                      ˆ
to approximate exercise regions of the form {VS > 0} as {b · YS + ˆ · YTiS −1 < a}, just like in
                                                                         b
the case of coupon bonds discussed in Section 3.
    As argued in Section 3, the reliability of our Affine approximation scheme depends on
the approximation error in (14) being small empirically. The variances of forward rates,
var[r], are on the order of 10−4 . The maturities of swaps for the actively traded European
swaptions, which are the subject of this study, are not larger than 10 years. Combining these
                                                 ¯
observations, and using a (high) average r of 10%, the upper bound for the approximation
                                2
error for such swaptions is rT var[r] = .1×(10/2)2 ×10−4 ≈ 2.5 basis points. The accuracy of
                            ¯
the reported mid-market swap rates is ±1bps., and the swap (bid-ask) spreads have declined
from about 10bps. in the early 90’s to about 2 − 3bps. now. Also, the majority of the
swaptions studied here are written on swaps of 3 to 7 year maturity (the corresponding
upper bound for a *-into-4 forward swap rate is less than .5bps.). Based on these facts and
noting that the realized error is likely to be smaller than the upper bound, we conclude that
the approximation is very accurate.


6     Expected Exposures on Interest-Rate Swaps
As an illustration of these pricing results for swaptions, we explore the computation of
the “expected exposure” on a swap, a commonly used measure of credit risk by financial
institutions. Specifically, we will use our Affine approximation to compute the expected
exposures on a ten-year, at-the-money interest-rate swap initiated at time t = 0. The
expected exposure on a swap is defined as the expected positive part of the swap’s future
                               +
market value, E(T ) = EP [VT ], where VT is the value of the swap at time T . Given the
                           0
cashflow pattern of a swap, E0 = E10yr = 0, and ET > 0 for 0 < T < 10yr.
    Our objective is not only to illustrate the ease of computing expected exposures (and
swaption prices) using our approach, but also to explore the sensitivity of expected exposures
to alternative parameterizations of the market prices of risk in affine term structure models.
Expected exposures, as risk measures, need to be computed under the physical measure P,
whereas pricing is done under the various risk-neutral measures Q. It is the specification of
the market prices of risk (4) that determines the transformation between these measures. To
examine these issues, we proceed to compute expected exposures for the lifetime of the swap
in three 3-factor affine models (N = 3), estimated using the daily time series of US$ LIBOR
and swap rates over the sample period of January 1, 1992 to December 31, 1997 (1516 daily
observations). We considered “most flexible” affine models for Y with m elements of Y

                                                              14
driving the volatilities of all 3 state variables (m = 0, 1, 3), subject to the requirement that
the m volatility factors have non-negative support. When m = 0 (Model A0 (3)), Y follows
a correlated Gaussian diffusion, and when m = 3 (Model A3 (3)), Y follows a three-factor
CIR-style model with independent risk factors. The intermediate case of m = 1 (Model
A1 (3)) has one factor driving the volatilities of all three factors (two factors are Gaussian
conditional on the time-path of the other).
    The parameters of these models were fixed over the entire sample period and estimated
by the method of maximum likelihood. In this manner we use the entire sample period to
“pin down” the distributional properties of swap rates. Six-month LIBOR and the two-
and ten-year swaps were assumed to be priced perfectly by the three-factor model, and the
rates on the 1-year LIBOR and three-, four-, five-, and seven-year swaps were assumed to be
priced with serially-correlated autoregressive mean zero errors.9 The conditional densities
of the state used in constructing our likelihood functions, are known to be Gaussian and
non-central chi-square for the models with m = 0 and m = 3, respectively. For the case
of m = 1, the density is not known exactly in closed-form and we used the approximation
proposed by Duffie, Pedersen, and Singleton [2001]. All subsequent calculations of exposures
are done using state variables backed out from the model on the first date of the sample,
January 1, 1992.

                                    0.08                                                   A0 (3)
                                                                                           A1 (3)
                                                                                           A3 (3)
                                    0.07



                                    0.06
                Expected Exposure




                                    0.05



                                    0.04



                                    0.03



                                    0.02



                                    0.01



                                      0
                                           0   1   2   3     4        5    6       7   8        9   10
                                                           Calendar Time (years)


Figure 5: Expected Exposures on a 10yr interest-rate swap in three affine models under P
(solid lines) and Q (dotted lines).

   Figure 5 displays the results for all three models. We see that the exposures under
the P and Q measures are notably different, and that the Gaussian model exhibits the
   9
   See Chen and Scott [1993], Pearson and Sun [1994], and Duffie and Singleton [1997] for similar estimation
approaches for the case of CIR-style models.


                                                                 15
largest difference across measures. Indeed, at their peak (around 4 - 5 years out), the
exposures under the physical measure are approximately 400% larger than their risk-neutral
counterparts in the Gaussian model. Furthermore, there is the interesting pattern of the
differences between the physical and risk-neutral exposures becoming smaller as we increase
the number of factors m driving the stochastic volatilities of Y . One interpretation of this
pattern, suggested by the findings of Dai and Singleton [2001] and Duffee [2001], is that
affine models are not sufficiently flexible to simultaneously explain the predictability in excess
holding period returns and the degree of time-varying volatility in swap markets. As m is
increased, the likelihood function places more weight on matching the stochastic volatility
manifested in the historical data, at the expense of replicating the historical autocorrelations
or predictability of the risk factors and yields. Since expected exposures are determined, to
a large degree, by the degree of mean reversion in the state variables, the increased difficulty
of matching the predictability of returns as m increases evidently is manifested in Figure 5
in the form of smaller expected exposures under P. To the extent this explanation is correct,
then the large differences for the Gaussian Model A0 (3) more accurately represent history.
A complementary consideration is that the flexibility of the market price of risk specification
                 ˜
(4) induced by λ = 0 is reduced as m increases, because the requirement of “admissibility”
                                                            ˜
of an affine model imposes additional zero restrictions on λ. Again, this observation suggests
that the qualitative findings for Model A0 (3) are the more reliable.
    Pursuing the latter point about risk premiums, we compare in Figure 6 the expected
exposures within the Gaussian model for the cases where the market price of risk is given by
                                      ˜
(4) and the special case of (4) with λ set to zero. The latter case is the multivariate Vasicek
[1977] model (see Langetieg [1980]) in which the market prices of risk are proportional to
the constant factor volatilities. The expected exposures under Q are virtually on top of each
other for these two risk premium specifications. However, under P, the exposures computed
      ˜                                                            ˜
with λ = 0 are substantially larger than those computed with λ = 0 and, importantly, the
data support the former specification of risk premiums.




                                              16
                                   0.08                                                                            ~
                                                                                                             A0 (3)¸ = 0
                                                                                                                   ~
                                                                                                             A0 (3)¸ = 0
                                   0.07



                                   0.06
               Expected Exposure
                                   0.05



                                   0.04



                                   0.03



                                   0.02



                                   0.01



                                     0
                                          0      1         2         3     4           5         6      7      8      9      10
                                                                         Calendar Time (years)


Figure 6: Expected Exposures on a 10year interest-rate swap in three-factor A0 (3) (Gaussian)
            ˜         ˜
models with λ = 0 and λ = 0 under P (solid lines) and Q (dotted lines).

A      Bound on Approximation Errors
In this appendix we derive a bound on the approximation error in the linearization (14)
                                                       (Ti )
for the case of forward swap rates. Letting wt (i) = PtPt (Tj ) , a stochastic strictly decreasing
                                                                                                j
                                                            S
weight function on i = 1, . . . , n, the forward swap rate rt (S; T ) can be expressed as
                                                               n
                                                                                             1
                                               S
                                              rt (S; T )   =         ft (Ti )wt (i) =          (wt (n) − wt (0)) .
                                                               i=1
                                                                                             τ

Our objective is to show that the difference (wt (n) − wt (0)) is approximately linear in the
state, even though each weight wt (i) may have nonlinear dependence.
    To show this we start by computing the second order Taylor expansion of the weight
function wt (i):

            ∂wt (i)   Pt (Ti )bB(Ti ) Pt (Ti )                                 j   Pt (Tj )bB(Tj )
                    =                −                                                    2             = wt (i)(bB(Ti ) − bBt ),
             ∂Y           j Pt (Tj )       (                                   j Pt (Tj ))


where bBt = j bB(Tj )wt (j) is a weighted mean of the bond-pricing coefficients taken with
state-dependent weights wt (j). The Jacobian of bBt is given by

                                     ∂bBt                          ∂wt (j)
                                          =              bB(Tj )           =               wt (i)(bB(Ti ) − bBt )bB(Tj ) .
                                     ∂Y              j
                                                                    ∂Y             j


                                                                               17
This Jacobian can be interpreted as a weighted covariance function of the bond coefficients
taken with the same stochastic weights wt (i), so we denote it by vart [bB]. Similarly, the
Hessian of the weight function is given by
                ∂ 2 wt (i)
                           = wt (i)(bB(Ti ) − bBt )(bB(Ti ) − bBt ) − wt (i)vart [bB].
                 ∂Y Y
Combining these obserservations, the second-order expansion of the weight function around
the long-run mean of the state vector θ with Yt = Yt − θ is
                                                   ˆ

     wt (i) = w(i) + w(i)(bB(i) − bBt ) Yt + w(i)Yt (bB(i) − bBt )(bB(i) − bBt ) − vart [b] Yt
              ¯      ¯                  ˆ    ¯ ˆ                                            ˆ

                + o((bB(i) − bBt ) Yt )2 ,
                                   ˆ                                                                 (24)
       ¯
where w(i) is the weight function evaluated at θ.
   The terms of the form (bB(i) − bBt ) Yt have a natural interpretation in terms of forward
                                        ˆ
rates. Specifically, the forward rate from time u to time v > u is given by
                1                                              ¯           1
  ft (u, v) =        aB(u) − aB(v) + (bB(u) − bB(v)) Yt = ft (u, v) +          (bB(u) − bB(v)) Yt .
                                                                                               ˆ
              v−u                                                        v−u
Hence, these terms represent the deviations of the forward rates from their long-run means
                                                     ¯
scaled by the length of the interval; (ft (u, v) − ft (u, v))(v − u). In particular, the order of
the error in the second-order expansion (24) is smaller than var[ft (Ti , T )] × (T − Ti )2 , where
                                                                             ¯      ¯
 ¯                                                         B
T is some maturity between S and S + T , so long as b (·) is sufficiently close to its weighted
average value bB.
    The weights wt (i) at the ends of the interval [0, n] are anticorrelated: deviations (bB(i)−bB)
in the vector of first-order coefficients in (24) have opposite signs for i close to 0 and n. The
forward swap rate, being the difference of the anticorrelated first and last weights, is thus
itself highly variable. However, these weights enter the gradient of the forward swap rate
with opposite signs, sign(bB0 − bB) = −sign(bBn − bB), so rt (S; T ) is relatively stable.
    We can see this more formally by noting that
           rt (S; T ) = wt (0) − wt (n)
                      = w(0) − w(n) + bS(S; T ) Yt + Yt A(S; T )Yt + o(∆T 2 var[f ]),
                        ¯        ¯              ˆ    ˆ          ˆ                                    (25)
               ¯                  ¯
where ∆T = (T − S) ∨ (S + T − T ), the larger of the distances between the ends of the
                                      ¯
interval and the “weighted mid-point” T , A(S; T ) is the second-order coefficient matrix,

     w(0) (bB(0) − bB)(bB(0) − bB) − var[b] − w(n) (bB(n) − bB)(bB(n) − bB) − var[b]
     ¯                                        ¯                                                      ,

and bS(S; T ) is the linear coefficient that is used to approximate the forward swap rate,
                             bS(S; T ) = w(0)(bB(0) − bB) − w(n)(bB(n) − bB).
                                         ¯                  ¯
To see that the quadratic form in (25) is small, note that
                      ˆ        ˆ
                      Yt var[b]Yt =              wt (i)Yt (bB(i) − bB)(bB(i) − bB) Yt
                                                       ˆ                           ˆ
                                             i

                                    ≈            wt (i)(ft (Ti , T ) − ft (Ti , T ))2 (Ti − T )2 ,
                                                                 ¯     ¯        ¯           ¯
                                             i


                                                           18
which behaves like the (weighted) average variance of forward rates with maturities ranging
from 0 to ∆T multiplied by (Ti − T )2 . Hence the quadratic term reduces to
                                  ¯

                                  r(S; T ) × ∆T 2 × (η − ν),
                                  ¯

                                                ¯
the average value of the forward swap rate r (S; T ) times the squared half-length of the
                    2
interval [S, T ], ∆T , multiplied by the difference of two positive terms η and ν, each of
which is of order var[f ]. Therefore, the order of the approximation error, which consists of
the quadratic and o terms, is substantially smaller than r(S; T ) × var[f ] × ∆T 2 . We conclude
                                                         ¯
that the spot and forward swap rates can be approximated by a linear function of the state,
                             S
                            rt (S; T ) = aS(S; T ) + bS(S; T )Yt + t .

    Though we have this approxiation to be adequate for our purposes, if a finer approxima-
tion is desired, one can use a quadratic expansion. Noting that the coefficient matrices in
the definition of A(S; T ), (bB(0) − bB)(bB(0) − bB) and (bB(n) − bB)(bB(n) − bB) , are close to
each other, the quadratic term can be re-written as

       (w(0) − w(n))Yt (bB(0) − bB)(bB(0) − bB) − var[b] Yt = r¯ (S; T )Yt ΩS (S; T )Yt,
        ¯      ¯    ˆ                                    ˆ     S        ˆ            ˆ

where ΩS (S; T ) is a symmetric positively definite matrix. Thus, the exact forward swap
rate is always higher than the linear approximation, and the quadratic approximation to the
forward swap rate will be highly accurate.




                                               19
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