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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 References Bakshi, G. and D. Madan (2000). Spanning and Derivative-Security Valuation. Journal of Financial Economics 55, 205–238. Chacko, G. and S. Das (1998). Pricing Average Interest Rate Options: A General Ap- proach. Working Paper, Harvard Business School, Harvard University. Chen, L. (1996). Stochastic Mean and Stochastic Volatility – A Three-Factor Model of the Term Structure of Interest Rates and Its Application to the Pricing of Interest Rate Derivatives. 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