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The copyright to this article is held by the Econometric Society, http://www.econometricsociety.org/. It may be downloaded, printed and reproduced only for personal or classroom use. Absolutely no downloading or copying may be done for, or on behalf of, any for‐profit commercial firm or other commercial purpose without the explicit permission of the Econometric Society. For this purpose, contact Claire Sashi, General Manager, at sashi@econometricsociety.org. Econometrica, Vol 68, No ŽNovember, 2000., 1343᎐1376 TRANSFORM ANALYSIS AND ASSET PRICING FOR AFFINE JUMP-DIFFUSIONS BY DARRELL DUFFIE, JUN PAN, AND KENNETH SINGLETON In the setting of ‘‘affine’’ jump-diffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems Example applications include fixed-income pricing models, with a role for intensity-based models of default, as well as a wide range of option-pricing applications An illustrative example examines the implications of stochastic volatility and jumps for option valuation This example highlights the impact on option ‘smirks’ of the joint distribution of jumps in volatility and jumps in the underlying asset price, through both jump amplitude as well as jump timing KEYWORDS: Affine jump diffusions, option pricing, stochastic volatility, Fourier transform INTRODUCTION IN VALUING FINANCIAL SECURITIES in an arbitrage-free environment, one inevitably faces a trade-off between the analytical and computational tractability of pricing and estimation, and the complexity of the probability model for the state vector X In light of this trade-off, academics and practitioners alike have found it convenient to impose sufficient structure on the conditional distribution of X to give closed- or nearly closed-form expressions for securities prices An assumption that has proved to be particularly fruitful in developing tractable, dynamic asset pricing models is that X follows an affine jump-diffusion Ž AJD , which is, roughly speaking, a jump-diffusion process for which the drift vector, ‘‘instantaneous’’ covariance matrix, and jump intensities all have affine dependence on the state vector Prominent among AJD models in the term-structure literature are the Gaussian and square-root diffusion models of Vasicek Ž1977 and Cox, Ingersoll, and Ross Ž1985 In the case of option pricing, there is a substantial literature building on the particular affine stochastic-volatility model for currency and equity prices proposed by Heston Ž1993 This paper synthesizes and significantly extends the literature on affine asset-pricing models by deriving a closed-form expression for an ‘‘extended transform’’ of an AJD process X, and then showing that this transform leads to analytically tractable pricing relations for a wide variety of valuation problems More precisely, fixing the current date t and a future payoff date T, suppose We are grateful for extensive discussions with Jun Liu; conversations with Jean Jacod, Monika Piazzesi, Philip Protter, and Ruth Williams; helpful suggestions by anonymous referees and the editor; and support from the Financial Research Initiative, The Stanford Program in Finance, and the Gifford Fong Associates Fund at the Graduate School of Business, Stanford University 1343 1344 D DUFFIE, J PAN, AND K SINGLETON that the stochastic ‘‘discount rate’’ RŽ X t , for computing present values of future cash flows, is an affine function of X t Also, consider the generalized terminal payoff function ă q ă X T e u X T of X T , where ă is scalar and the n elements of each of ¨ and u are scalars These scalars may be real, or more generally, complex We derive a closed-form expression for the transform Ž 1.1 ž ž Ht R Ž X , s ds Et exp y T s / / ă q ă X T e uи X T , where Et denotes expectation conditioned on the history of X up to t Then, using this transform, we show that the tractability offered by extant, specialized affine pricing models extends to the entire family of AJDs Additionally, by selectively choosing the payoff Ž ¨ q ¨ и X T e uи X T , we significantly extend the set of pricing problems Žsecurity payoffs that can be tractably addressed with X following an AJD To motivate the usefulness of our extended transform in theoretical and empirical analyses of affine models, we briefly outline three applications 1.1 Affine, Defaultable Term Structure Models There is a large literature on the term structure of default-free bond yields that presumes that the state vector underlying interest rate movements follows an AJD under risk-neutral probabilities Žsee, for example, Dai and Singleton Ž1999 and the references therein Assuming that the instantaneous riskless short-term rate rt is affine with respect to an n-dimensional AJD process X t Žthat is rt s q и X t Duffie and Kan Ž1996 show that the ŽT y t -period zero-coupon bond price, Ž 1.2 ž ž Ht r ds E exp y T s / / Xt , is known in closed form, where expectations are computed under the riskneutral measure.2 Recently, considerable attention has been focused on extending these models to allow for the possibility of default in order to price corporate bonds and other credit-sensitive instruments To illustrate the new pricing issues that may arise with the possibility of default, suppose that, with respect to given risk-neutral probabilities, X is an AJD; the arrival of default is at a stochastic intensity t , and upon default the holder recovers a constant fraction w of face value Then, from results in Lando Ž1998., the initial price of a T-period zero-coupon bond is The entire class of affine term structure models is obtained as the special case of Ž1.1 found by setting RŽ X t s rt , u s 0, ă s 1, and ă s See, for example, Jarrow, Lando, and Turnbull Ž1997 and Duffie and Singleton Ž1999 AFFINE JUMP-DIFFUSIONS 1345 given under technical conditions by Ž 1.3 ž ž H0 Ž r q dt E exp y T t t // H0 q dt, qw T t where qt s Ew t expŽyH0t Ž ru q u du.x The first term in Ž1.3 is the value of a claim that pays contingent on survival to maturity T We may view qt as the price density of a claim that pays if default occurs in the ‘‘interval’’ Ž t, t q dt Thus the second term in Ž1.3 is the price of any proceeds from default before T These expectations are to be taken with respect to the given risk-neutral probabilities Both the first term of Ž1.3 and, for each t, the price density qt can be computed in closed form using our extended transform Specifically, assuming that both rt and t are affine with respect to X t , the first term in Ž1.3 is the special case of Ž1.1 obtained by letting RŽ X t s rt q t , u s 0, ¨ s 1, and ¨ s Similarly, qt is obtained as a special case of Ž1.1 by setting u s 0, RŽ X t s rt q t , and ă q ă и X t s t Thus, using our extended transform, the pricing of defaultable zero-coupon bonds with constant fractional recovery of par reduces to the computation of a one-dimensional integral of a known function Similar reasoning can be used to derive closed-form expressions for bond prices in environments for which the default arrival intensity is affine in X along with ‘‘gapping’’ risk associated with unpredictable transitions to different credit categories, as shown by Lando Ž1998 A different application of the extended transform is pursued by Piazzesi Ž1998., who extends the AJD model in order to treat term-structure models with releases of macroeconomic information and with central-bank interest-rate targeting She considers jumps at both random and at deterministic times, and allows for an intensity process and interest-rate process that have linearquadratic dependence on the underlying state vector, extending the basic results of this paper 1.2 Estimation of Affine Asset Pricing Models Another useful implication of Ž1.1 is that, by setting R s 0, ă s 1, and ă s 0, we obtain a closed-form expression for the conditional characteristic function of X T given X t , defined by Ž u, X t , t, T s EŽ e iuи X T < X t , for real u Because knowledge of is equivalent to knowledge of the joint conditional density function of X T , this result is useful in estimation and all other applications involving the transition densities of an AJD For instance, Singleton Ž2000 exploits knowledge of to derive maximum likelihood estimators for AJDs based on the conditional density f Žи < X t of X tq1 given X t , obtained by Fourier inversion of as Ž 1.4 f Ž X tq1 < X t s Ž 2 N Hޒ N eyi uи X tq 1 Ž u, X t , t , t q du 1346 D DUFFIE, J PAN, AND K SINGLETON Das Ž1998 exploits Ž1.4 for the specific case of a Poisson-Gaussian AJD to compute method-of-moments estimators of a model of interest rates Method-of-moments estimators can also be constructed directly in terms of the conditional characteristic function From the definition of , Ž 1.5 E e iuи X tq y Ž u, X t , t , t q < X t s 0, so any measurable function of X t is orthogonal to the ‘‘error’’ Ž e iuи X tq y Ž u, X t , t, t q Singleton Ž1999 uses this fact, together with the known functional form of , to construct generalized method-of-moments estimators of the parameters governing AJDs and, more generally, the parameters of asset pricing models in which the state follows an AJD These estimators are computationally tractable and, in some cases, achieve the same asymptotic efficiency as the maximum likelihood estimator Jiang and Knight Ž1999 and Chacko and Viceira Ž1999 propose related, characteristic-function based estimators of the stochastic volatility model of asset returns with volatility following a square-root diffusion.4 1.3 Affine Option-Pricing Models In an influential paper in the option-pricing literature, Heston Ž1993 showed that the risk-neutral exercise probabilities appearing in the call option-pricing formulas for bonds, currencies, and equities can be computed by Fourier inversion of the conditional characteristic function, which he showed is known in closed form for his particular affine, stochastic volatility model Building on this insight,5 a variety of option-pricing models have been developed for state vectors having at most a single jump type Žin the asset return., and whose behavior between jumps is that of a Gaussian or ‘‘square-root’’ diffusion.6 Knowing the extended transform Ž1.1 in closed-form, we can extend this option pricing literature to the case of general multi-dimensional AJD processes with much richer dynamic interrelations among the state variables and much richer jump distributions For example, we provide an analytically tractable method for pricing derivatives with payoffs at a future time T of the form Ž e bи X T y c q, where c is a constant strike price, bg ޒn, X is an AJD, and yq' maxŽ y, This leads directly to pricing formulas for plain-vanilla options on currencies and equities, quanto options Žsuch as an option on a common Liu, Pan, and Pedersen Ž2000 and Liu Ž1997 propose alternative estimation strategies that exploit the special structure of affine diffusion models Among the many recent papers examining option prices for the case of state variables following square-root diffusions are Bakshi, Cao, and Chen Ž2000., Bakshi and Madan Ž2000., Bates Ž1996., Bates Ž1997., Chen and Scott Ž1993., Chernov and Ghysels Ž1998., Pan Ž1998., Scott Ž1996., and Scott Ž1997., among others More precisely, the short-term interest rate has been assumed to be an affine function of independent square-root diffusions and, in the case of equity and currency option pricing, spot-market returns have been assumed to follow stochastic-volatility models in which volatility processes are independent ‘‘square-root’’ diffusions that may be correlated with the spot-market return shock AFFINE JUMP-DIFFUSIONS 1347 stock or bond struck in a different currency., options on zero-coupon bonds, caps, floors, chooser options, and other related derivatives Furthermore, we can price payoffs of the form Ž b и X T y c q and Ž e aи X T bи X T y c q, allowing us to price ‘‘slope-of-the-yield-curve’’ options and certain Asian options.7 In order to visualize our approach to option pricing, consider the price p at date of a call option with payoff Ž e dи X T y c q at date T, for given dg ޒn and strike c, where X is an n-dimensional AJD, with a short-term interest-rate process that is itself affine in X For any real number y and any a and b in ޒn, let Ga, b Ž y denote the price of a security that pays e aи X T at time T in the event that bи X T F y As the call option is in the money when ydи X T F yln c, and in that case pays e dи X T y ce 0и X T , we have the option priced at Ž 1.6 ps Gd ,y d Ž yln c y cG 0,y d Ž yln c Because it is an increasing function, Ga, b Žи can be treated as a measure Thus, it is enough to be able to compute the Fourier transform Ga, b Žи of Ga, b Žи., defined by Ga, b Ž z s qϱ Hyϱ e izy dGa, b Ž y , for then well-known Fourier-inversion methods can be used to compute terms of the form Ga, b Ž y in Ž1.6 There are many cases in which the Fourier transform Ga, b Žи of Ga, b Žи can be computed explicitly We extend the range of solutions for the transform Ga, b Žи from those already in the literature to include the entire class of AJDs by noting that Ga, b Ž z is given by Ž1.1., for the complex coefficient vector u s aq izb, with ă s and ¨ s This, because of the affine structure, implies under regularity conditions that Ž 1.7 Ga, b Ž z s e ␣ Ž0.q  Ž0.и X , where ␣ and  solve known, complex-valued ordinary differential equations ŽODEs with boundary conditions at T determined by z In some cases, these ODEs have explicit solutions These include independent square-root diffusion models for the short-rate process, as in Chen and Scott Ž1995., and the stochastic-volatility models of asset prices studied by Bates Ž1997 and Bakshi, Cao, and Chen Ž1997 Using our ODE-based approach, we derive other explicit examples, for instance, stochastic-volatility models with correlated jumps in both returns and volatility In other cases, one can easily solve the ODEs for ␣ and  numerically, even for high-dimensional applications In a complementary analysis of derivative security valuation, Bakshi and Madan Ž2000 show that knowledge of the special case of 1.1 with ă q ă XT s is sufficient to recover the prices of standard call options, but they not provide explicit guidance as to how to compute this transform Their applications to Asian and other options presumes that the state vector follows square-root or Heston-like stochastic-volatility models for which the relevant transforms had already been known in closed form 1348 D DUFFIE, J PAN, AND K SINGLETON Similar transform analysis provides a price for an option with a payoff of the form Ž dи X T y c q, again for the general AJD setting For this case, we provide ˜a, b, d Žи., an equally tractable method for computing the Fourier transform of G ˜a, b, d Ž y is the price of a security that pays e dи X T aи XT at T in the event where G that bи X T F y This transform is again of the form 1.1., now with ă s a Given ˜a, b, d Ž y and the option price pЈ as this transform, we can invert to obtain G Ž 1.8 ˜a,y a, Žyln c y cG0, ya Žyln c pЈ s G As shown in Section 3, these results can be used to price slope-of-the-yield-curve options and certain Asian options Our motivation for studying the general AJD setting is largely empirical The AJD model takes the elements of the drift vector, ‘‘instantaneous’’ covariance matrix, and jump measure of X to be affine functions of X This allows for conditional variances that depend on all of the state variables Žunlike the Gaussian model., and for a variety of patterns of cross-correlations among the elements of the state vector Žunlike the case of independent square-root diffusions Dai and Singleton Ž1999., for instance, found that both time-varying conditional variances and negatively correlated state variables were essential ingredients to explaining the historical behavior of term structures of U.S interest rates Furthermore, for the case of equity options, Bates Ž1997 and Bakshi, Cao, and Chen Ž1997 found that their affine stochastic-volatility models did not fully explain historical changes in the volatility smiles implied by S& P 500 index options Within the affine family of models, one potential explanation for their findings is that they unnecessarily restricted the correlations between the state variables driving returns and volatility Using the classification scheme for affine models found in Dai and Singleton Ž1999., one may nest these previous stochastic-volatility specifications within an AJD model with the same number of state variables that allows for potentially much richer correlation among the return and volatility factors The empirical studies of Bates Ž1997 and Bakshi, Cao, and Chen Ž1997 also motivate, in part, our focus on multivariate jump processes They concluded that their stochastic-volatility models Žwith jumps in spot-market returns only not allow for a degree of volatility of volatility sufficient to explain the substantial ‘‘smirk’’ in the implied volatilities of index option prices Both papers conjectured that jumps in volatility, as well as in returns, may be necessary to explain option-volatility smirks Our AJD setting allows for correlated jumps in both volatility and price Jumps may be correlated because their amplitudes are drawn from correlated distributions, or because of correlation in the jump times ŽThe jump times may be simultaneous, or have correlated stochastic arrival intensities In order to illustrate our approach, we provide an example of the pricing of plain-vanilla calls on the S& P 500 index A cross-section of option prices for a given day are used to calibrate AJDs with simultaneous jumps in both returns AFFINE JUMP-DIFFUSIONS 1349 and volatility Then we compare the implied-volatility smiles to those observed in the market on the chosen day In this manner we provide some preliminary evidence on the potential role of jumps in volatility for resolving the volatility puzzles identified by Bates Ž1997 and Bakshi, Cao, and Chen Ž1997 The remainder of this paper is organized as follows Section reviews the class of affine jump-diffusions, and shows how to compute some relevant transforms, and how to invert them Section presents our basic option-pricing results The example of the pricing of plain-vanilla calls on the S& P 500 index is presented in Section Additional appendices provide various technical results and extensions TRANSFORM ANALYSIS FOR AJD STATE-VECTORS This section presents the AJD state-process model and the basic-transform calculations that will later be useful in option pricing 2.1 The Affine Jump-Diffusion We fix a probability space Ž ⍀ , F , P and an information filtration Ž Ft , and suppose that X is a Markov process in some state space D ; ޒn, solving the stochastic differential equation Ž 2.1 dX t s Ž X t dtq Ž X t dWt q dZt , where W is an Ž Ft -standard Brownian motion in ޒn ; : D ª ޒn, : D ª ޒn=n, and Z is a pure jump process whose jumps have a fixed probability distribution on ޒn and arrive with intensity Ä Ž X t :t G 04 , for some : D ª w0, ϱ To be precise, we suppose that X is a Markov process whose transition semi-group has an infinitesimal generator D of the Levy ´ type, defined at a bounded C function f : D ª ޒ, with bounded first and second derivatives, by Ž 2.2 Df Ž x sfx Ž x Ž x q q Ž x tr f x x Ž x Ž x Ž x i H ޒw f Ž xq z y f Ž x x d Ž z n Intuitively, this means that, conditional on the path of X, the jump times of Z are the jump times of a Poisson process with time-varying intensity Ä Ž X s :0 F s F t , and that the size of the jump of Z at a jump time T is independent of Ä X s :0 F s - T and has the probability distribution The filtration Ž Ft s Ä Ft :t G 04 is assumed to satisfy the usual conditions, and X is assumed to be Markov relative to Ž Ft For technical details, see for example, Ethier and Kurtz Ž1986 The generator D is defined by the property that Ä f Ž X t y H0t D f Ž X s ds:t G 04 is a martingale for any f in its domain See Ethier and Kurtz Ž1986 for details 1350 D DUFFIE, J PAN, AND K SINGLETON For notational convenience, we assume that X is ‘‘known’’ Žhas a trivial distribution Appendices provide additional technical details, as well as generalizations to multiple jump types with different arrival intensities, and to timedependent Ž , , , We impose an ‘‘affine’’ structure on , i , and , in that all of these functions are assumed to be affine on D In order for X to be well defined, there are joint restrictions on Ž D, , , , , as discussed in Duffie and Kan Ž1996 and Dai and Singleton Ž1999 The case of one-dimensional nonnegative affine processes, generalized as in Appendix B to the case of general Levy ´ jump measures, corresponds to the case of continuous branching processes with immigration ŽCBI processes For this case, Kawazu and Watanabe Ž1971 provide conditions Žin the converse part of their Theorem 1.1 on , , , and for existence, and show that the generator of the process is affine Žin the above sense if and only if the Laplace transform of the transition distribution of the process is of the exponential-affine form.10 2.2 Transforms First, we show that the Fourier transform of X t and of certain related random variables is known in closed form up to the solution of an ordinary differential equation ŽODE Then, we show how the distribution of X t and the prices of options can be recovered by inverting this transform We fix an affine discount-rate function R: D ª ޒ The affine dependence of , i , , and R are determined by coefficients Ž K, H, l, defined by: Ž x s K q K x, for Ks Ž K , K g ޒn = ޒn=n Ž Ž x Ž x i i j s Ž H0 i j q Ž H1 i j и x, for H s Ž H0 , H1 g ޒn=n = ޒn=n=n Ž x s l q l и x, for l s Ž l , l g ޒ = ޒn RŽ x s q и x, for s Ž , g ޒ = ޒn For c g ރn, the set of n-tuples of complex numbers, we let Ž c s H ޒn expŽ c и z d Ž z whenever the integral is well defined This ‘‘jump transform’’ determines the jump-size distribution The ‘‘coefficients’’ Ž K, H, l, of X completely determine its distribution, given an initial condition X Ž0 A ‘‘characteristic’’ s Ž K, H, l, , captures both the distribution of X as well as the effects of any discounting, and determines a transform : ރn = D = ޒq= ޒqª ރof X T conditional on Ft , when well defined at t F T, by ⅷ ⅷ ⅷ ⅷ Ž 2.3 ž ž Ht R Ž X ds Ž u, X t , t , T s E exp y T s / / e uи X T Ft , Independently of our work, Filipovic ´ Ž1999 applies these results regarding CBI processes to fully characterize all affine term structure models in which the short rate is, under an equivalent martingale measure, a one-dimensional nonnegative Markov process Extending the work of Brown and Schaefer Ž1993., Filipovic ´ shows that it is necessary and sufficient for an affine term structure model in this setting that the underlying short rate process is, risk-neutrally, a CBI process 10 AFFINE JUMP-DIFFUSIONS 1351 where E denotes expectation under the distribution of X determined by Here, differs from the familiar Žconditional characteristic function of the distribution of X T because of the discounting at rate RŽ X t The key to our applications is that, under technical regularity conditions given in Proposition below, Ž 2.4 Ž u, x, t , T s e ␣ Ž t q  Ž t и x , where  and ␣ satisfy the complex-valued ODEs11 Ž 2.5 ˙Ž t s y K 1i  Ž t y Ž 2.6 ␣ ˙ Ž t s 0 y K0 и  Ž t y 2 i  Ž t H1  Ž t y l 1Ž Ž  Ž t y , i  Ž t H0  Ž t y l Ž Ž  Ž t y , with boundary conditions  ŽT s u and ␣ ŽT s The ODE Ž2.5 ᎐ Ž2.6 is easily conjectured from an application of Ito’s Formula to the candidate form Ž2.4 of In order to apply our results, we would need to compute solutions ␣ and  to these ODEs In some applications, as for example in Section 4, explicit solutions can be found In other cases, solutions would be found numerically, for example by Runge-Kutta This suggests a practical advantage of choosing a jump distribution with an explicitly known or easily computed jump transform The following technical conditions will justify this method of calculating the transform DEFINITION: A characteristic Ž K, H, l, , is well-behaă ed at u, T g ރn = w0, ϱ if Ž2.5 ᎐ Ž2.6 are solved uniquely by  and ␣ ; and if Ži E žH / žH / T Ž ii E < ␥ t < dt - ϱ, where ␥ t s ⌿t Ž Ž  Ž t y Ž X t , T Ž iii 1r2 i - ϱ, where t s ⌿t  Ž t Ž X t , and t и t dt E Ž