A Jump-Diffusion Model for Option Pricing

S. G. Kou Department of Industrial Engineering and Operations Research, 312 Mudd Building, Columbia University, New York, New York 10027 [email protected]

rownian motion and normal distribution have been widely used in the Black–Scholes Boption-pricing framework to model the return of assets. However, two puzzles emerge from many empirical investigations: the leptokurtic feature that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and an empirical phenomenon called “volatility smile” in option markets. To incorporate both of them and to strike a balance between reality and tractability, this paper proposes, for the purpose of option pricing, a double exponential jump-diffusion model. In particular, the model is simple enough to produce analytical solutions for a variety of option-pricing problems, including call and put options, interest rate derivatives, and path- dependent options. Equilibrium analysis and a psychological interpretation of the model are also presented. (Contingent Claims; High Peak; Heavy Tails; Interest Rate Models; Rational Expectations; Overre- action and Underreaction)

1. Introduction fractal Brownian motion, and stable processes; see, for Despite the success of the Black–Scholes model based example, Mandelbrot (1963), Rogers (1997), Samorod- on Brownian motion and normal distribution, two nitsky and Taqqu (1994); (b) generalized hyperbolic empirical phenomena have received much attention models, including log t model and log hyperbolic recently: (1) the asymmetric leptokurtic features—in model; see, for example, Barndorff-Nielsen and other words, the return distribution is skewed to the Shephard (2001), Blattberg and Gonedes (1974); left, and has a higher peak and two heavier tails than (c) time-changed Brownian motions; see, for exam- those of the normal distribution, and (2) the volatility ple, Clark (1973), Madan and Seneta (1990), Madan smile. More precisely, if the Black–Scholes model is et al. (1998), and Heyde (2000). An immediate prob- correct, then the implied volatility should be constant. lem with these models is that it may be difficult to In reality, it is widely recognized that the implied obtain analytical solutions for option prices. More volatility curve resembles a “smile,” meaning it is a precisely, they might give some analytical formulae convex curve of the strike price. for standard European call and put options, but any Many studies have been conducted to mod- analytical solutions for interest rate derivatives and ify the Black–Scholes model to explain the two path-dependent options, such as perpetual American empirical phenomena. To incorporate the asymmet- options, barrier, and lookback options, are unlikely. ric leptokurtic features in asset pricing, a variety In a parallel development, different models are of models have been proposed:1 (a) chaos theory, also proposed to incorporate the “volatility smile” in option pricing. Popular ones include: (a) stochastic 1 Although most of the studies focus on the leptokurtic features volatility and ARCH models; see, for example, Hull under the physical measure, it is worth mentioning that the leptokurtic features under the risk-neutral measure(s) lead to the and White (1987), Engle (1995), Fouque et al. (2000); “volatility smiles” in option prices. (b) constant elasticity model (CEV) model; see,

Management Science © 2002 INFORMS 0025-1909/02/4808/1086$5.00 Vol. 48, No. 8, August 2002 pp. 1086–1101 1526-5501 electronic ISSN KOU A Jump-Diffusion Model for Option Pricing for example, Cox and Ross (1976), and Davydov The “volatility smiles” phenomenon is illustrated in and Linetsky (2001); (c) normal jump models pro- §5.3. The final section discusses some limitations of posed by Merton (1976); (d) affine stochastic-volatility the model. and affine jump-diffusion models; see, for example, Heston (1993), and Duffie et al. (2000); (e) models based on Lévy processes; see, for example, Geman 2. The Model et al. (2001) and references therein; (f) a numerical 2.1. The Model Formulation procedure called “implied binomial trees”; see, for The following dynamic is proposed to model the asset example, Derman and Kani (1994) and Dupire (1994). price, St, under the physical probability measure P: Aside from the problem that it might not be easy to find analytical solutions for option pricing, espe- dSt Nt = dt+ dWt+ d V − 1 (1) cially for path-dependent options (such as perpet- St− i ual American options, barrier, and lookback options), i=1 some of these models may not produce the asymmet- where Wt is a standard Brownian motion, Nt is ric leptokurtic feature (see §2.3). a Poisson process with rate , and Vi is a sequence The current paper proposes a new model with of independent identically distributed (i.i.d.) nonneg- the following properties: (a) It offers an explana- ative random variables such that Y = logV  has an tion for two empirical phenomena—the asymmetric asymmetric double exponential distribution2 3 with leptokurtic feature, and the volatility smile (see §§3 the density and 5.3). (b) It leads to analytical solutions to many −1y 2y option-pricing problems, including European call and fY y = p · 1e 1y≥0 + q · 2e 1y<0 put options (see §5); interest rate derivatives, such 1 > 1 2 > 0 as swaptions, caps, floors, and bond options (see §5.3 and Glasserman and Kou 1999); path-dependent where p q ≥ 0, p +q = 1, represent the probabilities of options, such as perpetual American options, bar- upward and downward jumps. In other words, rier, and lookback options (see §2.3 and Kou and +, with probability p Wang 2000, 2001). (c) It can be embedded into a ratio- logV  = Y =d (2) nal expectations equilibrium framework (see §4). (d) It −−, with probability q has a psychological interpretation (see §2.2). where + and − are exponential random variables The model is very simple. The logarithm of the with means 1/ and 1/ , respectively, and the nota- asset price is assumed to follow a Brownian motion 1 2 tion =d means equal in distribution. In the model, plus a compound Poisson process with jump sizes all sources of randomness, Nt, Wt, and Y s, are double exponentially distributed. Because of its sim- assumed to be independent, although this can be plicity, the parameters in the model can be easily relaxed, as will be suggested in §2.2. For notational interpreted, and the analytical solutions for option simplicity and in order to get analytical solutions pricing can be obtained. The explicit calculation is for various option-pricing problems, the drift  and made possible partly because of the memoryless the volatility  are assumed to be constants, and property of the double exponential distribution. the Brownian motion and jumps are assumed to be The paper is organized as follows. In §2, the model is proposed, is evaluated by four criteria, and is com- 2 pared with other alternative models. Section 3 studies If 1 = 2 and p = 1/2, then the double exponential distribution is the leptokurtic feature. A rational expectations equi- also called “the first law of Laplace” (proposed by Laplace in 1774), while the “second law of Laplace” is the normal density. librium justification of the model is given in §4. Some 3 Ramezani and Zeng (1999) independently propose the same jump- preliminary results, including the Hh functions, are diffusion model from an econometric viewpoint as a way of given in §5.1. Formulae for option-pricing problems, improving the empirical fit of Merton’s normal jump-diffusion including options on futures, are provided in §5.2. model to stock price data.

Management Science/Vol. 48, No. 8, August 2002 1087 KOU A Jump-Diffusion Model for Option Pricing one dimensional. These assumptions, however, can be have arbitrage opportunities, and thus are not self- easily dropped to develop a general theory. consistent (to give an example, it is shown by Rogers Solving the stochastic differential equation (1) gives 1997 that models using fractal Brownian motion may the dynamics of the asset price: lead to arbitrage opportunities). The double exponen- tial jump-diffusion model can be embedded in a ratio- 1 Nt nal expectations equilibrium setting; see §4. St = S0 exp  −  2 t + Wt V  (3) 2 i 2. A model should be able to capture some impor- i=1 tant empirical phenomena. The double exponential Note that EY  = p − q , VarY  = pq 1 + 1 2 +  p +     2 jump-diffusion model is able to reproduce the lep- 1 2 1 2 1 q , and tokurtic feature of the return distribution (see §3) 2 2 and the “volatility smile” observed in option prices EV  = EeY  (see §5.3). In addition, the empirical tests performed

2 1 in Ramezani and Zeng (1999) suggest that the double = q + p 1 > 1 2 > 0 (4) 2 + 1 1 − 1 exponential jump-diffusion model fits stock data bet- ter than the normal jump-diffusion model. Andersen The requirement  > 1 is needed to ensure that 1 et al. (1999) demonstrate empirically that, for the S&P EV  <  and ESt < ; it essentially means that 500 data from 1980–1996, the normal jump-diffusion the average upward jump cannot exceed 100%, which model has a much higher p-value (0.0152) than those is quite reasonable. of the stochastic volatility model (0.0008) and the There are two interesting properties of the dou- Black–Scholes model <10−5. Therefore, the combina- ble exponential distribution that are crucial for the tion of results in the two papers gives some empiri- model. First, it has the leptokurtic feature; see Johnson cal support of the double exponential jump-diffusion et al. (1995). As will be shown in §3, the leptokur- model.4 tic feature of the jump size distribution is inherited 3. A model must be simple enough to be amenable by the return distribution. Secondly, a unique fea- to computation. Like the Black–Scholes model, the ture (also inherited from the exponential distribution) double exponential jump-diffusion model not only of the double exponential distribution is the memo- yields closed-form solutions for standard call and ryless property. This special property explains why put options (see §5), but also leads to a variety the closed-form solutions for various option-pricing of closed-form solutions for path-dependent options, problems, including barrier, lookback, and perpet- such as barrier options, lookback options, and perpet- ual American options, are feasible under the dou- ual American options (see §2.3 and Kou and Wang ble exponential jump-diffusion model while it seems 2000, 2001), as well as interest rate derivatives (see impossible for many other models, including the nor- §5.3 and Glasserman and Kou 1999). mal jump-diffusion model (Merton 1976); see §2.3 for 4. A model must have some (economical, physical, details. psychological, etc.) interpretation. One motivation for the double exponential jump-diffusion model comes 2.2. Evaluating the Model from behavioral finance. It has been suggested from Because essentially all models are “wrong” and rough extensive empirical studies that markets tend to have approximations of reality, instead of arguing the “correctness” of the proposed model I shall evalu- 4 However, we should emphasize that empirical tests should not be ate and justify the double exponential jump-diffusion used as the only criterion to judge a model good or bad. Empirical model by four criteria. tests tend to favor models with more parameters. However, models 1. A model must be internally self-consistent. In with many parameters tend to make calibration more difficult (the calibration may involve high-dimensional numerical optimization the finance context, it means that a model must be with many local optima), and tend to have less tractability. This is arbitrage-free and can be embedded in an equilibrium a part of the reason why practitioners still like the simplicity of the setting. Note that some of the alternative models may Black–Scholes model.

1088 Management Science/Vol. 48, No. 8, August 2002 KOU A Jump-Diffusion Model for Option Pricing both overreaction and underreaction to various good or This makes it harder to persuade practitioners to bad news (see, for example, Fama 1998 and Barberis switch from the Black–Scholes model to more realis- et al. 1998, and references therein). One may interpret tic alternative models. The double exponential jump- the jump part of the model as the market response diffusion model attempts to improve the empirical to outside news. More precisely, in the absence of implications of the Black–Scholes model while still outside news the asset price simply follows a geo- retaining its analytical tractability. Below is a more metric Brownian motion. Good or bad news arrives detailed comparison between the proposed model according to a Poisson process, and the asset price and some popular alternative models. changes in response according to the jump size dis- (1) The CEV Model. Like the double exponential tribution. Because the double exponential distribu- jump-diffusion model, analytical solutions for path- 5 tion has both a high peak and heavy tails, it can be dependent options (see Davydov and Linetsky 2001) used to model both the overreaction (attributed to the and interest rate derivatives (e.g., Cox et al. 1985 and heavy tails) and underreaction (attributed to the high Andersen and Andersen 2000) are available under the peak) to outside news. Therefore, the double expo- CEV model. However, the CEV model does not have nential jump-diffusion model can be interpreted as an the leptokurtic feature. More precisely, the return dis- attempt to build a simple model, within the tradi- tribution in the CEV model has a thinner right tail tional random walk and efficient market framework, than that of the normal distribution. This undesirable to incorporate investors’ sentiment. feature also has a consequence in terms of the implied Incidently, as a by-product, the model also suggests volatility in option pricing. Under the CEV model the that the fact of markets having both overreaction and implied volatility can only be a monotone function of underreaction to outside news can lead to the lep- tokurtic feature of asset return distribution. the strike price. Therefore, if the implied volatility is a convex function (but not necessarily a decreasing 2.3. Comparison with Other Models function), as frequently observed in option markets, There are many alternative models that can satisfy at the CEV model is unable to reproduce the implied least some of the four criteria listed above. A main volatility curve. attraction of the double exponential jump-diffusion (2) The Normal Jump-Diffusion Model. Merton model is its simplicity, particularly its analytical tractabil- (1976) was the first to consider a jump-diffusion ity for path-dependent options and interest rate derivatives. model similar to (1) and (3). In Merton’s paper Unlike the original Black–Scholes model, many alter- Y s are normally distributed. Both the double expo- native models can only compute prices for stan- nential and normal jump-diffusion models can lead dard call and put options, and analytical solutions to the leptokurtic feature (although the kurtosis for other equity derivatives (such as path-dependent from the double exponential jump-diffusion model options) and some most liquid interest rate deriva- is significantly more pronounced), implied volatil- tives (such as swaption, caps, and floors) are unlikely. ity smile, and analytical solutions for call and put Even numerical methods for interest rate derivatives options, and interest rate derivatives (such as caps, and path-dependent options are not easy, as the con- floors, and swaptions; see Glasserman and Kou 1999). vergence rates of binomial trees and Monte Carlo The main difference between the double exponen- simulation for path-dependent options are typically tial jump-diffusion model and the normal jump- much slower than those for call and put options (for diffusion model is the analytical tractability for the a survey, see Boyle et al. 1997). path-dependent options. Here I provide some intuition to understand why 5 Interestingly enough, the double exponential distribution has been the double exponential jump-diffusion model can lead widely used in mathematical psychology literature, particularly in to closed-form solutions for path-dependent options, vision cognitive studies; see, for example, the list of papers on the web page of David Mumford at the computer vision group, Brown while the normal jump-diffusion model cannot. To University. price perpetual American options, barrier options,

Management Science/Vol. 48, No. 8, August 2002 1089 KOU A Jump-Diffusion Model for Option Pricing and lookback options for general jump-diffusion pro- if the risk-neutral return also has a power-type right cesses, it is crucial to study the first passage time of tail, then the call option price is also infinite: a jump-diffusion process to a flat boundary. When a E∗At− K+ ≥ E∗At− K = jump-diffusion process crosses a boundary sometimes it hits the boundary exactly and sometimes it incurs Therefore, the only relevant models with an “overshoot” over the boundary. t-distributed returns are models with discretely com- The overshoot presents several problems for option pounded returns. However, in models with discrete pricing. First, one needs to get the exact distribution compounding, closed-form solutions are in general of the overshoot. It is well known from stochastic impossible.6 renewal theory that this is only possible if the jump (4) Stochastic Volatility Models. The double expo- size Y has an exponential-type distribution, thanks to nential jump-diffusion model and the stochastic the special memoryless property of the exponential volatility model complement each other: The stochas- distribution. Secondly, one needs to know the depen- tic volatility model can incorporate dependent struc- dent structure between the overshoot and the first ture better, while the double exponential jump- passage time. The two random variables are condi- diffusion model has better analytical tractability, tionally independent, given that the overshoot is big- especially for path-dependent options and complex ger than 0, if the jump size Y has an exponential-type interest rate derivatives. One empirical phenomenon distribution, thanks to the memoryless property. This worth-mentioning is that the daily return distribu- conditionally independent structure seems to be very tion tends to have more kurtosis than the distribution special to the exponential-type distribution and does of monthly returns. As Das and Foresi (1996) point not hold for other distributions, such as the normal out, this is consistent with models with jumps, but distribution. inconsistent with stochastic volatility models. More Consequently, analytical solutions for the perpet- precisely, in stochastic volatility models (or essentially ual American, lookback, and barrier options can be any models in a pure diffusion setting) the kurtosis derived for the double exponential jump-diffusion decreases as the sampling frequency increases, while model. However, it seems impossible to get similar in jump models the instantaneous jumps are indepen- results for other jump-diffusion processes, including dent of the sampling frequency. the normal jump-diffusion model. (5) Affine Jump-Diffusion Models. Duffie et al. (3) Models Based on t-Distribution. The (2000) propose a very general class of affine jump- t-distribution is widely used in empirical studies diffusion models which can incorporate jumps, of asset pricing. One problem with t-distribution stochastic volatility, and jumps in volatility. Both nor- mal and double exponential jump-diffusion mod- (or other distributions with power-type tails) as a els can be viewed as special cases of their model. return distribution is that it cannot be used in models However, because of the special features of the with continuous compounding. More precisely, sup- exponential distribution, the double exponential pose that at time 0 the daily return distribution X has jump-diffusion model leads to analytical solutions a power-type right tail. Then in models with continu- for path-dependent options, which are difficult for ous compounding, the asset price tomorrow At is other affine jump-diffusion models (even numeri- given by At = A0eX . Since X has a power-type cal methods are not easy). Furthermore, the dou- right tail, it is clear that EeX  =. Consequently, ble exponential model is simpler than general affine EAt = EA0eX  = A0EeX  = 6 Another interesting point worth mentioning is that, for a sam- In other words, the asset price tomorrow has an infi- ple size of 5,000 (20-years-daily data), it may be very difficult to distinguish empirically the double exponential distribution from nite expectation. This paradox holds for t-distribution the power-type distributions, such as t-distribution (although it is with any degrees of freedom, as long as one considers quite easy to detect the differences between them and the normal models with continuous compounding. Furthermore, density).

1090 Management Science/Vol. 48, No. 8, August 2002 KOU A Jump-Diffusion Model for Option Pricing jump-diffusion models: It has fewer parameters that in Figure 1 along with the normal density with the makes calibration easier. The double exponential same mean and variance. The parameters are t = jump-diffusion model attempts to strike a balance 1 day = 1/250 year,  = 20% per year,  = 15% per between reality and tractability. year, = 10 per year, p = 030, 1/1 = 2%, and 1/2 = (6) Models Based on Lévy Processes (the pro- 4%. In this case, EY  =−22%, and SDY  = 447%. In cesses with independent and stationary increments). other words, there are about 10 jumps per year with Although the double exponential jump-diffusion the average jump size −22%, and the jump volatil- model is a special case of Lévy processes, because of ity 447%. The jump parameters used here seem to the special features of the exponential distribution it be quite reasonable, if not conservative, for the U.S. has analytical tractability for path-dependent options stocks. and interest rate derivatives, which are difficult for The leptokurtic feature is quite evident. The peak other Lévy processes. of the density g is about 31, whereas that of the nor- mal density is about 25. The density g has heav- 3. Leptokurtic Feature ier tails than the normal density, especially for the Using (3), the return over a time interval t is given left tail, which could reach well below −10% while by: the normal density is basically confined within −6%. Additional numerical plots suggest that the feature of St St+ t = − 1 having a higher peak and heavier tails becomes more St St pronounced if either 1/ (the jump size expectations) i 1 or (the jump rate) increases. = exp  −  2 t + Wt+ t − Wt 2 Nt+t 4. Equilibrium for General + Y − 1 i Jump-Diffusion Models i=Nt+1 Consider a typical rational expectations economy where the summation over an empty set is taken to (Lucas 1978) in which a representative investor be zero. If the time interval t is small, as in the case tries to solve a utility maximization problem of daily observations, the return can be approximated  max E' Uct tdt), where Uct t is the util- in distribution, ignoring the terms with orders higher c 0 ity function of the consumption process ct. There is than t and using the expansion ex ≈ 1 + x + x2/2, by an exogenous endowment process, denoted by *t, St √ available to the investor. Also given to the investor is ≈ t + Z t + B · Y (5) St an opportunity to invest in a security (with a finite where Z and B are standard normal and Bernoulli liquidation date T0, although T0 can be very large) random variables, respectively, with PB = 1 = t which pays no dividends. If *t is Markovian, it can and PB = 0 = 1 − t, and Y is given by (2). be shown (see, for example, pp. 484–485 in Stokey The density7 g of the right-hand side of (5), being and Lucas 1989) that, under mild conditions, the ratio- an approximation for the return St/St, is plotted nal expectations equilibrium price (also called the

7 The density with 1 − t x − t p q gx = √ $ √ Eg G = t + − t  t  t 1 2 2 2 22t/2 x − t −  1t 1 1 p q + t p e 1 e−x−t1 % √ Var G =  2t + pq + + + t 1 g 2 2  t 1 2 1 2 22t/2 2 + q e 2 ex−t2 p q 2 + − t1 − t   x − t +  2 t 1 2 × % − √ 2  t where $· is the standard normal density function.

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Figure 1

30 30 25 28 20 26 15 24 10 5 22

−0.1 −0.05 0.05 0.1 −0.01 −0.005 0.005 0.01

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

−0.1 −0.08 −0.06 −0.04 −0.02 0.02 0.04 0.06 0.08 0.1 Notes. The first panel compares the overall shapes of the density g and the normal density with the same mean and variance, the second one details the shapes around the peak area, and the last two show the left and right tails. The dotted line is used for the normal density, and the solid line is used for the model.

“shadow” price) of the security, pt, must satisfy the Although it is intuitively clear that, generally Euler equation speaking, the asset price pt should follow a similar EU *T TpT
 jump-diffusion process as that of the dividend pro- pt = c t ∀ T ∈ 't T ) (6) cess *t, a careful study of the connection between U *t t 0 c the two is needed. This is because pt and *t where Uc is the partial derivative of U with respect may not have similar jump dynamics (see the remark to c. At this price pt, the investor will never change after Corollary 1). Furthermore, deriving explicitly the his/her current holdings to invest in (either long change of parameters from *t to pt also provides or short) the security, even though he/she is given some valuable information about the risk premiums the opportunity to do so. Instead, in equilibrium the embedded in jump-diffusion models. investor finds it optimal to just consume the exoge- The work in this section builds upon and extends nous endowment; i.e., ct = *t for all t ≥ 0. the previous work by Naik and Lee (1990), in which In this section I shall derive explicitly the implica- the special case that Vi has a lognormal distribution tions of the Euler equation (6) when the endowment is investigated. Another difference is that Naik and process *t follows a general jump-diffusion process Lee (1990) require that the asset pays continuous div- under the physical measure P: idends and there is no outside endowment process, d*t Nt while here the asset pays no dividends and there is an =  dt +  dW t+ d V − 1 (7) *t− 1 1 1 i outside endowment process. Consequently, the pric- i=1 ing formulae are different even in the case of lognor- mal jumps. where the Vi ≥ 0 are any independent identically dis- tributed, nonnegative random variables. In addition, For simplicity, as in Naik and Lee (1990), I shall all three sources of randomness, the Poisson process only consider the utility function of the special forms Nt the standard Brownian motion W t, and the Uc t= e−-t c. if 0 <.<1 and Uc t= e−-t logc if 1 . jump sizes V are assumed to be independent. . = 0, where ->0 (although most of the results below

1092 Management Science/Vol. 48, No. 8, August 2002 KOU A Jump-Diffusion Model for Option Pricing hold for more general utility functions). Under these Proof. See Appendix A.
types of utility functions, the rational expectations Given the endowment process *t, it must be equilibrium price of (6) becomes decided what stochastic processes are suitable for the asset price St to satisfy the equilibrium requirement Ee−-T *T.−1pT
 (8) or (11). I now postulate a special jump-diffusion pt = t  (8) e−-t*t.−1 form for St,

Assumption. The discount rate - should be large dSt = dt+  4dW t+ 1 − 42 dW t enough so that St− 1 2 Nt 5 1 .−1 2 + d Vi − 1 Vi = Vi (13) ->−1 − .1 + 1 1 − .2 − . + /1 2 i=1

a a a where W2t is a Brownian motion independent of where the notation /1 means /1 1= E'V − 1). W1t. In other words, the same Poisson process As will be seen in Proposition 1, this assumption affects both the endowment *t and the asset price guarantees that in equilibrium the term structure of St, and the jump sizes are related through a power interest rates is positive. function, where the power 5 ∈ −  is an arbitrary constant. The diffusion coefficients and the Brownian .−1 Proposition 1. Suppose /1 < . (1) Letting motion part of *t and St, though, are totally differ- Bt T be the price of a zero coupon bond with maturity ent. It remains to determine what constraints should T , the yield r1=−1/T − tlogBt T is a constant be imposed on this model so that the jump-diffusion independent of T , model can be embedded in the rational expectations equilibrium requirement (8) or (11). 1 2 .+5−1 .−1 r = - + 1 − .1 − 1 1 − .2 − . Theorem 1. Suppose / <  and / < . The 2 1 1 model (13) satisfies the equilibrium requirement (11) if and − / .−1 > 0 (9) 1 only if

(2) Let Zt 1= ertU *t t = er−-t*t.−1. Then .+5−1 .−1 c  = r + 141 − . − /1 − /1  Zt is a martingale under P, 1 2 = - + 1 − . 1 − 1 2 − . + 14 dZt 2 =− / .−1 dt +  .− 1dW t Zt− 1 1 1 − / .+5−1 (14) 1 Nt .−1 If (14) is satisfied, then under P∗, + d Vi − 1  (10) i=1 dSt = rdt− ∗E∗V5 − 1dt Using Zt, one can define a new probability measure P∗: St− i ∗ ∗ dP /dP 1= Zt/Z0 Under P , the Euler Equation (8) Nt ∗ 5 holds if and only if the asset price satisfies + dW t+ d Vi − 1  (15) i=1

−rT−t ∗ ∗ ∗ St = e E ST
t ∀ T ∈ 't T0) (11) Here, under P , W t is a new Brownian motion, Nt is ∗ .−1 a new Poisson process with jump rate = EVi  = Furthermore, the rational expectations equilibrium price of .−1 /1 + 1, and {Vi are independent identically dis- a (possibly path-dependent) European option, with the pay- tributed random variables with a new density under P∗: off 3S T at the maturity T , is given by ∗ 1 .−1 fx = x fVx (16) −rT−t ∗ V / .−1 + 1 3S t = e E 3S T
t ∀ t ∈ '0 T) (12) 1

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Proof. See Appendix A.
see Abramowitz and Stegun (1972, p. 691). The Hh The following corollary gives a condition under function can be viewed as a generalization of the which all three dynamics, *t and St under P and cumulative normal distribution function. St under P∗, have the same jump-diffusion form, The integral in (18) can be evaluated very fast by which is very convenient for analytical calculation. many software packages (for example, Mathematica).8 Corollary 1. Suppose the family  of distributions In addition, of the jump size V for the endowment process *t satisfies √ −n/2 −x2/2 that, for any real numbers a ∈ '0 1 and b ∈ − , Hhnx = 2 8e 1 1 1 1 1 3 1 b a−1 F  n+ x2 F  n+1 x2 V ∈  and const · x f x ∈  (17) 1 1 2 2 2 2 1 1 2 2 2 V × √ −x 1 ;1 + 1 n 2;1+ n 2 2 a−1 −1 2 where the normalizing constant, const, is /1 + 1 (provided that / a−1 < ). Then the jump sizes for the 1 where 1F1 is the confluent hypergeometric function. asset price St under P and the jump sizes for St under A three-term recursion is also available for the Hh the rational expectations risk-neutral measure P∗ all belong function (see pp. 299–300 and p. 691 of Abramowitz to the same family  . and Stegun 1972): Proof. Immediately follows from (7), (13), and

(16).
nHhnx = Hhn−2x − xHhn−1x n ≥ 1 (19) Condition (17) essentially requires that the jump size distribution belongs to the exponential family. Therefore, one can compute all Hhnx n ≥ 1, by It is satisfied if logV  has a normal distribution or using the normal density function and normal dis- a double exponential distribution. However, the log tribution function. The Hh function is illustrated in power-type distributions, such as log t-distribution, Figure 2. do not satisfy (17).

5.2. European Call and Put Options 5. Option Pricing Introduce the following notation: For any given prob- In this section I will compute the rational expectations ability P, define equilibrium option-pricing formula (12) explicitly for the European call and put options. For notational sim- <   p   9a T1= PZT  ≥ a plicity, I will drop ∗ in the risk-neutral notation, i.e., 1 2 write  instead of ∗, etc. To compute (12), one has to 1 1 where Zt = t + Wt+ Nt Y Y has a dou- study the distribution of the sum of the double expo- i=1 i ble exponential distribution with density f y ∼ p · nential random variables and normal random vari- Y  −1y y2 ables. Fortunately, this distribution can be obtained 1e 1y≥0 + q · 2e 1y<0, and Nt is a Poisson in closed form in terms of the Hh function, a special process with rate . The pricing formula of the call function of mathematical physics. option will be expressed in terms of < , which in turn can be derived as a sum of Hh functions. An 5.1. Hh Functions explicit formula for < will be given in Theorem B.1 in For every n ≥ 0, the Hh function is a nonincreasing Appendix B. function defined by:

  1 n −t2/2 8 A short code (about seven lines) can be downloaded from Hhnx = Hhn−1ydy = t − x e dt ≥ 0 n! the author’s Web page. Note that the integrand in (18) has x x √ n = 0 1 2  (18) a maxima at x + x2 + 4n/2; therefore, to numerically com- √ √ pute the integral, one should split the integral more around the −x2/2 Hh−1x = e = 28$x Hh0x = 28%−x9 maxima.

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Figure 2 The Hh Function for n = 1 3 5 with the Steepest Curve for n = 5 and the Flattest Curve for n = 1

80

60

40

20

−4 −2 2 4

Theorem 2. From (12), the price of a European call The result (20) resembles the Black-Scholes for- option 9 is given by mula for a call option under the geometric Brown- ian motion model, with < taking the place of %. The 1 3 0 = S0< r +  2 − /  ˜ p ˜ ˜ ˜ 9 proof of Theorem 2 is similar to that of Theorem 3 in c 2 1 2 Kou and Wang (2001), hence is omitted. logK/S0 T Now I consider the problem of pricing options on futures contracts. Assume, for now, that the term 1 structure of interest rate is flat, and r is a constant. − Ke−rT · < r −  2 − /  p   9 2 1 2 Then the futures price, Ft T∗, with delivery date T ∗, ∗ ∗ ∗ rT∗−t is given by Ft T  = E ST 
t = e St. logK/S0 T (20) Corollary 2. The price of the European call option on where a futures contract is given by

p 1 p˜ = · ˜1 = 1 − 1 ∗ 1 + /  − 1 3c F D F0 T  T  1 p q 1 ˜ =  + 1 ˜ = / + 1 / = 1 + 2 − 1 = D · F0 T∗<  2 − /  ˜ p ˜ ˜ ˜ 9 2 2  − 1  + 1 2 1 2 1 2 The price of the corresponding put option, 3 0, can be K 1 p log T − K< −  2 − /  obtained by the put-call parity: F0 T∗ 2 K 3 0 − 3 0 = e−rTE∗K − ST+ − ST− K+ p   9 log T p c 1 2 F0 T∗ = e−rTE∗K − ST = Ke−rT − S0 where D = e−rT. The put option can be priced according to

9 the put-call parity: To give a numerical example, if 1 = 10 2 = 5 = 1 p= 04  = 016 r = 5% S0 = 100 K = 98 T = 05 then (20) yields the call ∗ ∗ price 9.14732. Although in the pricing formula < involves infinite 3p F D F0 T  T  − 3c F D F0 T  T  series, my experience suggests that numerically only the first 10 to −rT ∗ 15 terms in the series are needed for most applications. = e K − F0 T 

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∗ Proof. Since FT T ∗ − K+ = erT −TST − Figure 3 Midmarket and Model-Implied Volatilities for Japanese Ke−rT∗−T+, we have LIBOR Caplets in May 1998 E'e−rTFT T ∗ − K+) 1 = erT∗−T S0< r +  2 − /  ˜ p ˜ ˜ ˜ 9 2 1 2 logKe−rT∗−T/S0 T 1 − Ke−rT∗−Te−rT< r −  2 − /  p 2 −rT∗−T 1 29 logKe /S0 T 1 = e−rT F0 T∗< r +  2 − /  ˜ p ˜ ˜ ˜ 9 2 1 2 logK/F 0 T∗ + rT T Notes. The parameters used in the fitted model are: for the two-year caplet 1 = 372 = 18p= 004= 14 = 021; and for the nine-year caplet 1 1 = 232 = 18p= 009= 02 = 009. − K< r −  2 − /  p   9 2 1 2 logK/F 0 T∗ + rT T 5.3. The “Volatility Smile” To illustrate that the model can produce “implied from which the conclusion follows by noting that volatility smile,” I consider a real data set used first Pr + T + WT + NT Y ≥ a + rT = PT + i=1 i in Andersen and Andreasen (2000) for two-year and WT+ NT Y ≥ a.
i=1 i nine-year caplets in the Japanese LIBOR market as of Using the fact that for every t ≥ 0, Zt converges in late May 1998. Figure 3 shows both observed implied distribution to t + Wt as both  →and  → 1 2 volatility curves and calibrated implied volatility , one easily gets the following corollary. curves derived by using the futures option formula Corollary 3. (1) As the jump size gets smaller and in Corollary 2, with the discount parameter D being smaller, the pricing formulae in Theorem 2 and Corollary the corresponding bond prices and the underlying 2 degenerate to the Black–Scholes formula and Black’s asset being the LIBOR rate. For details of calibration futures option formula. More precisely, as both 1 → and the theoretical justification of using the futures and 2 →while all other parameters remain fixed, option formula for caplets; see Glasserman and Kou  −rT  (1999). 3c0 → S0%b  − Ke %b  + − I should emphasize that this example is not meant −rT ∗   3c F 0 → e F 0 T %b+ F − K%b− F to be an empirical test of the model; it only serves as where an illustration to show that the model can produce a logS0/K + r ± 2/2T close fit even to a very sharp volatility skew. b 1= √ ±  T logF0 T∗/K ±  2T/2  6. Limitations of the Model b± F 1= √   T There are several limitations of the model. First, (2) If the jump rate is zero, i.e., = 0, then the pricing one disadvantage of the model is that the pricing formulae again degenerate to the Black–Scholes and Black’s formulae, although analytical, appear quite compli- futures option formulae, respectively. cated. This perhaps is not a major problem because

1096 Management Science/Vol. 48, No. 8, August 2002 KOU A Jump-Diffusion Model for Option Pricing the Hh function can be computed easily, and what Appendix A: Derivation of the Rational appears to be lengthy to human eyes might make lit- Expectations tle difference in terms of computer programming, as Proof of Proposition 1. (1) Since BT T= 1, Equation (8) yields long as it is a closed-form solution. Secondly, a more serious criticism is the difficul- E*T.−1
 Bt T = e−-T−t t  (A1) ties with hedging. Due to the jump part, the mar- *t.−1 ket is incomplete, and the conventional riskless hedg- Using the facts that ing arguments are not applicable here. However, it *T  .−1 1 should be pointed out that the riskless hedging is 2 = exp .− 1 1 − 1 T − t really a special property of continuous-time Brownian *t 2 NT motion, and it does not hold for most of the alterna- .−1 + 1.− 1W1T− W1t Vi tive models. Even within the Brownian motion frame- i=Nt+1 work, the riskless hedging is impossible if one wants NT  ' T − t)j E V.−1 = e− T −t / .−1 + 1j i j! 1 to do it in discrete time. For some suggestions about i=Nt+1 j=0 hedging with jump risk, see, for example, Grünewald = exp / .−1T − t and Trautmann (1996), Merton (1976), and Naik and 1 Lee (1990). Equation (A1) yields Finally, just like all models based on Lévy processes, 1 Bt T = exp −T − t - − .− 1  −  2 one empirical observation that the double exponential 1 2 1 jump-diffusion model cannot incorporate is the pos- 1 −  2.− 12 − / .−1 sible dependence structure among asset returns (the 2 1 1 so-called “volatility clustering effect”), simply because from which (9) follows. the model assumes independent increments. Here a (2) Note that (A1) implies possible way to incorporate the dependence is to use −rT−t some other point process, Nt with dependent incre- e = EUc *T  T /Uc *t t
t  (A2) ments, to replace the Poisson process Nt, while still retaining the independence between the Brownian which shows that Zt is a martingale under P. Furthermore, (7) and (9) lead to motion, the jump sizes, and Nt . The modified model no longer has independent increments, yet is simple .−1 r−-t 1 2 Zt = *0 e exp .− 1 1 − 1 t enough to produce closed-form solutions, at least for 2 Nt standard call and put options. However, it seems dif- .−1 + 1.− 1W1t Vi ficult to get analytical solutions for path-dependent i=1 1 options by using the new process Nt instead = *0.−1 exp −  2.− 12 − / .−1 t of Nt. 2 1 1 Nt .−1 + 1.− 1W1t Vi i=1

Acknowledgments from which (10) follows. Now by (8) and (A2), This paper was previously titled “A Jump Diffusion Model for Option Pricing with Three Properties: Leptokurtic Feature, Volatil- EUc *T T3S T
t  −rT ZT 3S t = = e E 3S T
t ity Smile, and Analytical Tractability.” The author is grateful to Uc *t t Zt an anonymous referee who made many helpful suggestions. The −rT ∗ = e E 3S T
t 
author also thanks many people who offered insight into this work, including Mark Broadie, Peter Carr, Gregory Chow, Savas Dayanik, Proof of Theorem 1. The Girsanovtheorem for jump-diffusion ∗  Paul Glasserman, Lars Hansen, Chris Heyde, Vadim Linetsky, Mike processes (see Björk et al. 1997) tells us that under P W1t 1= ∗ Staunton, and Hui Wang. This research is supported in part by NSF W1t − 1. − 1t is a new Brownian motion, and under P the ∗ .−1 .−1 grants DMI-9908106 and DMS-0074637. jump rate of Nt is = EVi  = /1 +1 and Vi has a new

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∗ .−1 .−1 density f x = 1// +1x fx Therefore, the dynamics of where P and Q are given by V 1 V n k n k St is given by n−1 i−k n−i n − k − 1 n 1 2 i n−i Nt Pn k = · p q dSt 5 i − k i  +   +  = dt+ 4dW t+ 1 − 42 dW t+  V − 1 i=k 1 2 1 2 St− 1 2 i i=1 1 ≤ k ≤ n − 1  =  +  4.− 1 dt + 4dW t+ 1 − 42 dW t n−1 n−i i−k 1 1 2 n − k − 1 n   Q = · 1 2 pn−iqi Nt n k i − k i 1 + 2 1 + 2 +  V5 − 1  i=k i n n i=1 1 ≤ k ≤ n − 1 Pn n = p Qn n = q Because and 0 is defined to be one. Here + and − are i.i.d. exponential random 0 i i  ∗ 5 5 1 .−1 E V  = x x f xdx variables with rates 1 and 2, respectively. i .−1 V 0 / + 1 1 As a key step in deriving closed-form solutions for call and 1 / .+5−1 + 1 = EV.+5−1 = 1 put options, this proposition indicates that the sum of i.i.d. double .−1 .−1 /1 + 1 /1 + 1 exponential random variables can be written, in distribution, as a (randomly) mixed gamma random variable.10 To prove Proposition we have ∗E∗V5 − 1 = / .+5−1 − / .−1. Therefore, i 1 1 B.1, the following lemma is needed. dSt =  +  4.− 1 + / − /  dt Lemma B.1. St− 1 .+5−1 .−1 n m − ∗E∗V5 − 1dt+ 4dWt+ 1 − 42 dW t + − i 1 2 i − j i=1 j=1 Nt +  V5 − 1    i  k  i=1   with prob.   i   i=1  Hence, to satisfy the rational equilibrium requirement St =   n−k m n − k + m − 1   1 2 ·  −rT−t ∗   +  +  e E ST
t  we must have  + 14. − 1 + /.+5−1 −  1 2 1 2 m − 1    /  = r from which (14) follows. If (14) is satisfied, under the   .−1 d k = 1  n ∗ =  (B3) measure P , the dynamics of St is given by  l     −  with prob.  dSt  i  = rdt− ∗E∗V5 − 1dt+ 4dWt+ 1 − 42 dW t  i=1      i 1 2  n m−l n − l + m − 1  St−  1 2   ·   1+2 1+2 n − 1  Nt   5 l = 1  m +  Vi − 1 i=1 Proof. Introduce the random variables An m = n  − i=1 i from which (15) follows.
m ˜ j=1 j . Then Appendix B: Derivation of the < Function An − 1 m− 1 + + / +   An m =d 2 1 2 − An − 1 m− 1 −  1/1 + 2 B.1. Decomposition of the Sum of Double Exponential Random Variables An m − 1  / +   =d 2 1 2 The memoryless property of exponential random variables yields An − 1 m 1/1 + 2 + −− + >− =d + and + −− + <− =−d − thus leading to the conclusion that via (B1). Now imagine a plane with the horizontal axis repre- senting the number of + and the vertical axis representing the + with probability  / +   i + − − =d 2 1 2 (B1) number of −. Suppose we have a random walk on the integer −− with probability  / +   j 1 1 2 lattice points of the plane. Starting from any point n m, n m ≥ 1, because the probabilities of the events + >− and + <− are the random walk goes either one step to the left with probability 1/1 + 2 or one step down with probability 2/1 + 2, and 2/1 + 2 and 1/1 + 2, respectively. The following proposi- tion extends (B1).

10 Proposition B.1. For every n ≥ 1, we have the following A result similar to the decomposition (B2) was first discovered decomposition by Shanthikumar (1985), although (B2) gives a more explicit cal- culation of P and Q . Furthermore, the proofs are totally dif- n k n k n k + with probability P k= 1 2  n ferent: A combinatorial approach is used here, while the proof in Y =d i=1 i n k (B2) i − k − with probability Q k= 1 2  n i=1 i=1 i n k Shanthikumar (1985) is based on the Laplace transform.

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the random walk stops once it reaches either the horizontal or ver- B.2. Results on Hh Functions tical axis. For any path from n m to k 0 1 ≤ k ≤ n, it must reach First of all, note that Hh x → 0, as x →, for n ≥−1; and n √ k 1 first before it makes a final move to k 0. Furthermore, all the Hh x →,asx →−, for n ≥ 1; and Hh x = 28%−x → √ n 0 paths going from n m to k 1 must have exactly n − k lefts and 28,asx →−. Also, for every n ≥−1, as x →, n−k+m−1 m − 1 downs, whence the total number of such paths is . m−1 1 Similarly, the total number of paths from n m to 0 l,1≤ l ≤ m, −x2/2 lim Hhnx/ e = 1 (B4) is n−l+m−1 . Thus, x→ xn+1 n−1   and as x →−,  k    with prob.   i   i=1  n   Hhnx = O x  (B5)    n−k  m n − k + m − 1   1 2   ·    +   +  m − 1   1 2 1 2  Here (B4) follows from Equations (19.14.3) and (19.8.1) in  k = 1  n  An m =d Abramowitz and Stegun (1972); and (B5) is clearly true for n =−1,  l    while for n ≥ 0 note that as x →−,  −  with prob.   i   i=1     n m−l  1 2  1 2 n − l + m − 1  n −t /2  ·  Hhnx = t − x e dt   +   +  n − 1  n! x  1 2 1 2    n  n  l = 1  m 2 2 2 2 ≤ t ne−t /2 dt + x ne−t /2 dt = O x n n! − n! − and the lemma is proven.
For option pricing it is important to evaluate the integral Proof of Proposition B.1. By the same analogy used in I c9. 5 *, Lemma B.1 to compute probability P ,1≤ k ≤ n, the probability n n k weight assigned to k + when we decompose n Y , it is equiv-  i=1 i i=1 i .x Inc9. 5 *1= e Hhn5x − * dx n ≥ 0 (B6) alent to consider the probability of the random walk ever reach c k 0 starting from the point i n − i 0 ≤ i ≤ n, with probability of starting from i n − i being n piqn−i. Note that the point k 0 for arbitrary constants . c, and 5. i can only be reached from points i n − i such that k ≤ i ≤ n − 1, Proposition11 B.2. (1) If 5>0 and . = 0, then for all n ≥−1, because the random walk can only go left or down, and stops once it reaches the horizontal axis. Therefore, for 1 ≤ k ≤ n − 1 e.c n 5 n−i I c9. 5 * =− Hh 5c − * (B3) leads to n . . i i=0 √ n+1 2 n−1 5 28 .* + . . P = Pgoing from i n− i to k 0 · Pstarting from i n− i + e 5 252 % −5c + * +  (B7) n k . 5 5 i=k n−1  i−k  n−i i + n− i− k − 1 n (2) If 5<0 and .<0, then for all n ≥−1 = 1 2 · piqn−i  +   +  n− i− 1 i i=k 1 2 1 2 e.c n 5 n−i i−k n−i n−1 Inc9. 5 * =− Hhi5c − * n − k − 1 n 1 2 i n−i . . = p q i=0 n − i − 1 i 1 + 2 1 + 2 √ i=k n+1 2 5 28 .* + . . 5 2 n−1 n − k − 1 n  i−k  n−i − e 25 % 5c − * −  (B8) = 1 2 piqn−i . 5 5 i − k i  +   +  i=k 1 2 1 2 Proof. Case 1. 5>0 and . = 0. Since, for any constant . and n .x Of course, Pn n = p . Similarly, we can compute Qn k: n ≥ 0 e Hhn5x − * → 0asx →thanks to (B4), integration by parts leads to n−1 Q = Pgoing from n−i i to 0 k·Pstarting from n−i i  n k 1 .x i=k In = Hhn5x − * de . c n−1 n−i i−k   n−i+i−k−1 n 1 5  = 1 2 · pn−iqi =− Hh 5c − *e.c + e.x Hh 5x − * dx  +  + n−i−1 n−i n n−1 i=k 1 2 1 2 . . c n−1 n−k−1 n  n−i  i−k = 1 2 pn−iqi i−k i  +  + 11 If 5>0 and . = 0, then for all n ≥ 0 I c9. 5 *= 1 Hh 5c − i=k 1 2 1 2 n 5 n+1 *.If5 ≤ 0 and . ≥ 0, then for all n ≥ 0 I c9. 5 *=.If5 = 0 n with Q = qn. Incidentally, we have also shown that n P + and .<0, then for all n ≥ 0 I c9. 5 * =  e.x Hh −* dx = n n k=1 n k n c n .c Qn k = 1.
Hhn−*e .

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2 In other words, we have a recursion, for n ≥ 0 In = Letting y = x −  / yields −e.c /. Hh 5c − * + 5/.I with n n−1 2 −t  /2 n−1 n √  fZ+ n  t = e e   I = 28 e.x$−5x + * dx i=1 i −1 c t/− t/ − y − n−1 1 √ × √ e−y2/2 dy 2 28 .* . . − n− 1! 28 = exp + % −5c + * +  5 5 252 5 2 e /2 = √  n−1ne−tHh −t/ +  Solving it yields, for n ≥−1, 28 n−1 e.c n 5 i 5 n+1 I =− Hh 5c − * + I because 1/n − 1! a a − yn−1e−y2/2 dy = Hh −a. The deriva- n . . n−i . −1 − n−1 i=0 tion of fZ− n  t is similar. i=1 i e.c n 5 n−i Case 2. PZ + n  ≥ x and PZ − n  ≥ x. From (B9), it is =− Hh 5c − * i=1 i i=1 i . . i clear that i=0 √ 2 5 n+1 28 .* .2 . n ne /2  t + exp + % −5c + * + P Z +  ≥ x = √ e−t Hh − +  dt 2 i n−1  . 5 5 25 5 i=1  28 x where the sum over an empty set is defined to be zero. ne2/2 1 Case 2. 5<0 and .<0. In this case, we must also have, for = √ I x9− − − n−1  .x  28 n ≥ 0 and any constant .<0 e Hhn5x−* → 0asx →, thanks n to (B5). Using integration by parts, we again have the same recur- by (B6). We can compute PZ − i=1 i ≥ x similarly.
sion, for n ≥ 0 I =−e.c /. Hh 5c −*+5/.I , but with a dif- n n n−1 Theorem B.1. With 8 1= PNT = n = e− T  Tn/n! and I in ferent initial condition n n Proposition B.2, we have √  I = 28 e.x$−5x + * dx −1 2 c e1 T/2  n √ √ PZT  ≥ a = √ 8 P  T k 28 .* .2 .  28T n n k 1 =− exp + % 5c − * −  n=1 k=1 2 5 5 25 5 1 √ × I a − T 9 − − √ − T Solving it yields (B8), for n ≥−1.
k−1 1  T 1

2 e2 T/2  n √ B.3. Sum of Double Exponential and the Normal k + √ 8n Qn k T2 Random Variables  28T n=1 k=1 Proposition B.3. Suppose {  } is a sequence of i.i.d. expo- 1 √ 1 2 × I a − T 9  √ − T nential random variables with rate >0, and Z is a normal random vari- k−1 2  T 2 able with distribution N0 2. Then for every n ≥ 1, we have: (1) The a − T density functions are given by + 8 % − √  0  T e2/2 t n −t fZ+ n  t =  √ e Hhn−1 − +  (B9) Proof. By the decomposition (B2), i=1 i  28   n 2 √ e /2 t PZT  ≥ a = 8 P T +  TZ+ Y ≥ a n t n j fZ− n  t =  √ e Hhn−1 +   (B10) i=1 i  28  n=0 j=1 √ (2) The tail probabilities are given by = 80PT +  TZ≥ a n n  n k  2 1 √ P Z +  ≥ x = √ e /2I x9− − − (B11) + 8 P P T +  TZ+ + ≥ a i n−1  n n k j i=1  28 n=1 k=1 j=1 n n  n k  2/2 1 √ P Z − i ≥ x = √ e In−1 x9 −  (B12) −  + 8n Qn kP T +  TZ− j ≥ a  i=1  28 n=1 k=1 j=1 n n Proof. Case 1. The densities of Z + i and Z − i We i=1 i=1 The result now follows via (B11) and (B12) for  > 1 and have 1  2 > 0.
f n t = fn t − xf xdx Z+ i i Z i=1 − i=1 t ext − xn−1 1 = e−tn √ e−x2/22 dx References − n− 1!  28 Abramowitz, M., I. A. Stegun. 1972. Handbook of Mathematical t t − xn−1 1 = e−tne2/2 √ e−x−22/22 dx Function, 10th Printing. U.S. National Bureau of Standards, − n− 1!  28 Washington, D.C.

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Accepted by Paul Glasserman; received November 29, 2001. This paper was with the authors 2 months and 3 weeks for 2 revisions.

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