MANAGEMENT SCIENCE Informs ® Vol

Efﬁcient Monte Carlo and Quasi–Monte Carlo Option Pricing Under the Variance Gamma Model

Athanassios N. Avramidis, Pierre L’Ecuyer Département d’Informatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada {[email protected], [email protected]}

e develop and study efficient Monte Carlo algorithms for pricing path-dependent options with the vari- Wance gamma model. The key ingredient is difference-of-gamma bridge sampling, based on the representation of a variance gamma process as the difference of two increasing gamma processes. For typical payoffs, we obtain a pair of estimators (named low and high) with expectations that (1) are monotone along any such bridge sampler, and (2) contain the continuous-time price. These estimators provide pathwise bounds on unbiased estimators that would be more expensive (infinitely expensive in some situations) to compute. By using these bounds with extrapolation techniques, we obtain significant efficiency improvements by work reduction. We then combine the gamma bridge sampling with randomized quasi–Monte Carlo to reduce the variance and thus further improve the efficiency. We illustrate the large efficiency improvements on numerical examples for Asian, lookback, and barrier options. Key words: option pricing; efficiency improvement; extrapolation; quasi–Monte Carlo; variance gamma History: Accepted by Wallace J. Hopp, stochastic models and simulation; received October 19, 2004. This paper was with the authors 7 months for 2 revisions.

1. Introduction on the “more detailed” (or continuous-time) sample Madan and Seneta (1990) and Madan et al. (1998) path and approximate the corresponding payoff by introduced the variance gamma (VG) model with extrapolation, with explicit bounds on the approx- application to modeling asset returns and option pric- imation error. We simulate the process paths over ing. The VG process is a Brownian motion with a given time interval by successive refinements via a random time change that follows a stationary bridge sampling truncated after a small number of gamma process. This model permits more flexibility steps and combine this with extrapolation and ran- in modeling skewness and kurtosis relative to Brow- domized quasi–Monte Carlo sampling. nian motion. We developed closed-form solutions for Quasi–Monte Carlo (QMC) methods have attracted European option pricing and provided empirical evi- a lot of interest in finance over the past 10 years dence that this model gives a better fit to market or so (Caflisch and Moskowitz 1995, Moskowitz and option prices than the classical Black-Scholes model. Caflisch 1996, Acworth et al. 1997, Åkesson and One approach to option pricing under VG is based on Lehoczy 2000, Owen 1998, Glasserman 2004, L’Ecuyer characteristic function representations and employs 2004a). Randomized QMC (RQMC) methods replace efficient techniques such as the fast Fourier trans- the vectors of independent uniform random num- form (Carr and Madan 1999); however, it does not bers that drive the simulation by sets or sequences appear to be sufficiently well developed for path- of points that cover the space more uniformly than dependent options. Monte Carlo (MC) methods offer random points, so the variance is reduced, and yet a simple way of estimating option values in that the points are randomized in a way that provides an context (Glasserman 2004). However, these methods unbiased estimator of the integral together with a con- can be very time consuming if a precise estimator is fidence interval. needed. In applications where RQMC really improves on Our aim in this paper is to develop and study ordinary MC, the integral has typically low effec- strategies that improve the efficiency of MC estima- tive dimension in some sense (L’Ecuyer and Lemieux tors in the context of pricing path-dependent options 2002). Roughly, this means that the variance of the under the VG model. Our approach exploits the key integrand depends (almost) only on a small number property that a VG process can be written as the dif- of uniforms, or is a sum of functions that depends ference of two (increasing) gamma processes. By sim- (almost) only on a small number of uniforms. Caflisch ulating a path of these two gamma processes over a and Moskowitz (1995) and Moskowitz and Caflisch coarse time discretization, we can compute bounds (1996) introduced a bridge sampling method for

1930 Avramidis and L’Ecuyer: Efficient Monte Carlo and Quasi–Monte Carlo Option Pricing Management Science 52(12), pp. 1930–1944, © 2006 INFORMS 1931 generating the path of a Brownian motion at discrete until the gap between the two estimators becomes points over a finite time interval: First generate the zero; this procedure yields an unbiased estimator value at the end, then the value in the middle con- whose expected work requirement is often consid- ditional on that at the end, then the values at one erably smaller than full-dimensional path sampling. quarter and at three quarters conditional on the previ- Ribeiro and Webber (2003) also develop and study ous ones, and so on, halving each interval recursively. empirically a method for “correcting for simulation This method rewrites the integrand so as to concen- bias” for lookback and barrier options when estimat- trate the variance to the first few coordinates, and this ing continuous-time prices for models driven by Lévy makes QMC and RQMC much more effective. processes; this method heuristic does not yield error Path-dependent option pricing under the VG model bounds and risks increasing the bias. via QMC with bridge sampling was recently pro- We compare empirically the variance and efficiency posed and studied empirically by Ribeiro and Webber of MC and RQMC for three option pricing exam- (2004) and Avramidis et al. (2003). The algorithms ples on a single asset, namely an arithmetic average proposed there are similar in spirit and structure Asian option, a lookback option, and a barrier option. to the bridge sampling algorithm mentioned for the We experiment with different types of QMC point Brownian motion. Both are based on the premise that sets from those used by Ribeiro and Webber (2004) the bridge sampling methodology improves the effec- and randomize our QMC point sets to obtain unbi- tiveness of QMC by concentrating the variance on the ased estimators of both the mean and variance (which first few random numbers (i.e., by reducing the effec- Ribeiro and Webber do not have). Our bridge sam- tive dimension of the integrand). pling algorithms combined with RQMC improve sim- Our contributions are a theoretical and empirical ulation efficiency by large factors in these examples. analysis of a particular bridge sampling technique The remainder of the paper is organized as fol- named difference-of-gammas bridge sampling (DGBS) lows. Section 2, reviews the VG model and option (Avramidis et al. 2003). We exploit this in two ways: pricing under this model. In §3, we introduce sam- (i) to approximate continuous-time prices, or prices of pling algorithms for the gamma and VG processes. options whose payoffs are based on a large number In §4, we develop and study truncated DGBS, exam- of observation times, by simulating the process only ine special cases, and suggest a few estimators for over a limited number of observation times, and (ii) to each case. In §5, we compare the estimators (in terms improve the effectiveness of RQMC. of variance and efficiency) on a numerical example, The representation of a VG process that underlies with and without truncation and extrapolation, for DGBS yields a pair of estimators (named low and MC and RQMC. In §6, we briefly review Lévy pro- high) whose expectations are monotone along any cesses, generalize the approach to Lévy processes of sequence of paths with increasing resolution. Under finite variation, and state the practical difficulties that monotonicity assumptions on the payoff that hold for arise beyond the VG case. many frequently traded options, if bridge sampling is stopped at a coarser resolution than the one on which 2. Background and Notation the payoff is based, then the low and high estimators have negative and positive bias, respectively. By 2.1. The Variance Gamma Process and extrapolating estimators based on different levels of Option Pricing resolution, one can reduce the bias of these inexpen- We provide the relevant background for the VG model sive estimators and improve their efficiency relative to developed by Madan and Seneta (1990), Madan and estimators that compute the exact payoff. (As usual, Milne (1991), and Madan et al. (1998), with emphasis we define efficiency as one over the product of the on pricing path-dependent options via MC sampling mean square error by the expected computing time of methods. The notation X ∼ 2 means that X has the estimator.) the normal distribution with mean and variance 2. We focus on prototypical Asian, lookback, and bar- Gamma denotes the gamma distribution with rier options. For Asian options, we obtain the con- mean and variance 2. vergence rate of the expected gap between the low Let B = Bt = Bt t ≥ 0 be a Brownian and high estimators and develop an estimator, based motion with drift parameter and variance parame- on Richardson’s extrapolation, whose bias has a better . That is, B0 = 0, the process B has independent ter convergence rate. For the lookback case, we exhibit increments, and Bt + − Bt ∼ 2, for all the low and high estimators and provide empirical t ≥ 0 and >0. Let G = Gt = t t ≥ 0 be evidence that the extrapolation method can improve a gamma process independent of B, with drift >0 both convergence of the bias and efficiency. For and volatility >0. That is, G0 = 0, the process G the barrier case, we exhibit low and high estima- has independent increments, and Gt + − Gt ∼ tors and develop bridge sampling that continues Gamma2/ / for t ≥ 0 and >0. Avramidis and L’Ecuyer: Efficient Monte Carlo and Quasi–Monte Carlo Option Pricing 1932 Management Science 52(12), pp. 1930–1944, © 2006 INFORMS

A VG process with parameters is deﬁned 3. Bridge Sampling for Gamma and by Variance Gamma Processes X = Xt=Xt =BGt 1 t≥0 (1) 3.1. Sampling the Brownian Motion In simple terms, the VG process is obtained by sub- A Brownian motion B with parameters can be jecting the Brownian motion to a random time change sampled at arbitrary discrete times 0 = t0

ESt = S0 expr − qt (3) sampled at discrete times 0 = t0 0. We will as- and Gti = Gti−1 + )i, where sume (4) throughout the paper. ) ∼ Gammat − t 2/ / We consider the pricing of a general path-depen- i i i−1 dent option under the VG model. The payoff may and the )is are independent, for i = 1m. depend on the value of the process at a ﬁnite num- Gamma bridge sampling (GBS) relies on the obser- ber of observation times, say 0 = t

The GBS algorithm operates in essentially the same Parameters at Arbitrary Times 0 = t0 Brownian bridge sampling and gamma 0 y = T , and y y dense in 0T, but otherwise bridge sampling, these algorithms sample the VG pro- 1 2 3 cess at a time partition that becomes increasingly fine. arbitrary. This is the sequence of times at which the To simulate a path of the asset price dynamics, it suf- two gamma processes are generated, in order (i.e., first at y , then at y conditional on their values at y , then fices to generate a path of the VG process X and trans- 1 2 1 at y conditional on their values at y and y , and so form it to a path of the process S via the exponential 3 1 2 on). For each positive integer m, let 0 = t

Figure 2 Difference-of-Gammas Bridge Sampling of a VG Process X and with Parameters 1 at an Inﬁnite Sequence of Times y = 0, y = T , and y y in 0T(All Variates Are Inde- + + 0 1 2 3 U t = St exp2t−t +- t −- t pendent) m mi−1 mi−1 mi mi−1 + − = S0 exp2t + - tm i − - tm i−1 t1 0 ← 0; t1 1 ← T - +0 ← 0; - −0 ← 0 + for t 0, and all t ∈ 0T, we have }

Lmt ≤ Lm+1t ≤ St ≤ Um+1t ≤ Umt in this section. This implies in particular that with DGBS, one can stop (truncate) the sampling process Proof. At t = tm i, for 0 ≤ i ≤ m, we have Lmt = at any intermediate step and compute bounds on the Umt = St by definition. The increase of the process value of the VG process at all other times that are + − X = - − - in any subinterval of tm i−1tm i can- involved in the option payoff. From that, one can not exceed the increase of the process - + during that obtain bounds on the final payoff without having to + + interval, that is, cannot exceed - tm i − - tm i−1. simulate all the process values that determine the Similarly, its decrease cannot exceed the increase of - − payoff, and some intermediate value between these − during that interval, that is, cannot exceed - tm i − bounds (instead of the exact payoff) can be taken as - −t . Combining these observations with (8), we an estimator of the expected payoff. This truncated m i−1 obtain that Lmt ≤ St ≤ Umt. DGBS procedure can save work in exchange for some To show that U t ≤ U t, let t t be bias. In many practical settings, the saving can be m+1 m m j−1 m j the interval that contains the point ym+1. Outside significant, and the additional bias contributes little this interval t t , we have U t = U t. to the mean square error. Truncating early can there- m j−1 m j m+1 m We also have tm+1j−1 = tm j−1, tm+1j = ym+1, and fore improve the overall estimator’s efficiency by an t = t , so this interval splits into the two inter- important factor, especially when the computing bud- m+1j+1 m j vals t t and t t , when m in- get is small (so the squared bias can be dominated by m+1j−1 m+1j m+1j m+1j+1 creases to m + 1. For t ∈ t t , we have the variance). For certain types of options, the bias can m+1j−1 m+1j be completely eliminated. In other cases, the bias can U t = St exp2t − t be reduced by computing the truncated DGBS estima- m+1 m+1j−1 m+1j−1 + + tors at different truncation points and extrapolating. + - tm+1j − - tm+1j−1 This methodology is also very convenient for approx- = St exp2t − t imating the values of options based on continuous- m j−1 mj−1 + + time observation of the process. + - tm+1j − - tm j−1

4.1. Bounds on the VG Process ≤ Stm j−1 exp2t − tm j−1 + − Deﬁne 2 = + r − q, 2 = max20, 2 = max−20, + - +t − - +t and m+1j+1 m j−1 = Umt St = S0exp2t+Xt=S0exp2t+- +t−- −t

for t ≥ 0 (8) because tm+1j+1 = tm j . For t ∈ tm+1jtm+1j+1,we have We now develop bounds on the path of S from the + values of the two gamma processes at a finite set of Um+1t = Stm+1j exp2t − tm+1j + - tm+1j+1 observation times in the setting of the DGBS algo- − - +t rithm of Figure 2. m+1j + Define the two processes Lm and Um over 0T by = Stm+1j−1 exp2t − tm+1j−1 + - tm+1j − − − + − Lmt = Stmi−1exp2t−tmi−1−- tmi+- tmi−1 − - tm+1j − - tm+1j−1 + - tm+1j−1 − + + + = S0 exp2t − - tm i + - tm i−1 + - tm+1j+1 − - tm+1j Avramidis and L’Ecuyer: Efficient Monte Carlo and Quasi–Monte Carlo Option Pricing Management Science 52(12), pp. 1930–1944, © 2006 INFORMS 1935

+ ≤ Stm+1j−1 exp2t − tm+1j−1 + - tm+1j+1 If C is (conditionally) nonincreasing instead, the reverse inequality holds. In both cases, these bounds are narrowing − - +t m+1j−1 when m increases. = St exp2t − t + - +t m+1j−1 m+1j−1 m j Proof. This follows from Proposition 1 and + − - tm j−1 straightforward monotonicity arguments. Recall that d is the number of times at which the = U t m option contract requires the underlying asset to be observed. When d is large (possibly infinite), we sug- The inequality Lmt ≤ Lm+1t can be proved by a symmetrical argument. gest the following sampling strategy for estimating The following simplified (piecewise constant) ver- c = EC: Generate the process X by DGBS for some sion of the bounds of Proposition 1 will be convenient. integer k such that m = 2k 0, and all t ∈ 0T, we have = S0 exp2t + Xtm i + Xtm i−1/2 ∗ ∗ ∗ ∗ Lmt ≤ Lm+1t ≤ St ≤ Um+1t ≤ Umt for tm i−1 ≤ t ≤ tm i, or by the process defined by re- 4.2. Estimators Based on Truncated DGBS and placing X in (8) by its conditional expectation given

Extrapolation m, that is, Under monotonicity conditions on the option’s payoff ¯ as a function of the path of S, conditional on the val- SXmt ues observed so far, these bounds on S translate into = S0 exp2t + EXt bounds on the conditional payoff. Any value lying m between these bounds can be taken as a (generally = S0 exp2t + itXtm i + 1 − itXtm i−1 biased) estimator of the expected payoff. When the two bounds on the payoff coincide (which can hap- for tm i−1 ≤ t ≤ tm i, where it = t − tm i−1/tm i − G X pen for barrier options, for example), there is no bias. tm i−1. We shall denote by Cm and Cm the values ¯ ¯ To state this more formally, we define of Cm when S is replaced by SGm and SXm, respectively. Yet another approach is to let Cm be the payoff + − m = tm 1- tm 1 - tm 1 of the corresponding discrete-time option with obser- D + − vation times t t , which we denote by Cm . tm m- tm m - tm m (9) m 1 m m Suppose that d = and that the bias m = ECm which contains the values of the two gamma pro- − c has the asymptotic form cesses at the first m observation times. Denote by ∗ ∗ m = m−51 + Ogm (10) CLm, CUm, CLm, and CUm the discounted payoffs 0 (that correspond to C) when S is replaced in (5) ∗ ∗ for some constants 0 and 51 > 0 and a function g and (6) by Lm, Um, Lm, and Um, respectively. such that gm = om−51 . One frequently encounters Corollary 2. Suppose that conditional on m, C is a 5 = 1 (see the next subsections). Assume = 0 and monotone nondecreasing function of St for all values of t 1 0 replace the estimator Cm by not in tm 0tm m. Then, 51 2 Cm − Cm/2 ∗ ∗ C ≤ C ≤ C ≤ C ≤ C Cm = (11) Lm Lm Um Um 251 − 1 Avramidis and L’Ecuyer: Efficient Monte Carlo and Quasi–Monte Carlo Option Pricing 1936 Management Science 52(12), pp. 1930–1944, © 2006 INFORMS where m is assumed to be a power of 2 (for simplic- by a discrete-time, generally biased, Monte Carlo esti- ity). We have mator. We consider the DGBS algorithm of Figure 2.

Here, conditional on m, the discounted payoff C is EC = 251 c + m − c + m/2/251 − 1 m clearly a monotone nondecreasing function of St for 51 51 −51 −51 all t. Therefore, = 2 c + 2 0m + Ogm − c − 0m/2

51 − Ogm/2/2 − 1 CLm−1 ≤ CLm ≤ C ≤ CUm ≤ CUm−1 = c + Ogm for all m>1 and all sample paths, from Corollary 2. This implies that C and C are negatively and That is, C has reduced bias order compared with C . Lm Um m m positively biased estimators of c , respectively. A In case = 0, the bias order remains Ogm, and A 0 better estimator could be C defined in (11) for some the dominant (i.e., nonzero, slowest-decaying) bias m C between these bounds, if we know the correct term has a (generally) different constant. As an exam- m value of 5 . −52 1 ple, if m = 2m , where 2 = 0 and 52 >51 > 0, The upper bound CUm here can be written as −52 then it is easy to see that Cm has bias c2m , where + c = 251 − 252 /251 − 1; the bias remains in order 1 T 2 2 C = e−rT U tdt − K −52 Um m m , but the constant in front has opposite sign and T 0 larger absolute value than . Further, a simple calcu- 2 where lation shows that the absolute bias may also increase if the dominant term in (10) alternates sign as a func- 1 T Umtdt tion of m. T 0 m This method is an application of a general class of tm i 1 + − techniques called extrapolation to the limit or Richard- = S0 exp2t + - tm i − - tm i−1 dt T t son’s extrapolation (Joyce 1971, Conte and de Boor i=1 m i−1 m 1972), also proposed by Boyle et al. (1997) in a sim- 1 tm i = S0 exp- +t − - −t exp2tdt ulation-based security-pricing context. Our numerical T m i m i−1 i=1 tm i−1 experiments in §5 show that the bias reduction can  S0 m be significant, typically without a significant increase  + −  exp- tm i − - tm i−1 of the variance. This bias reduction permits one to  2T  i=1 obtain an estimator with a given (target) mean square   · exp2tm i − exp2tm i−1 2 = 0 error using a smaller value of m (i.e., with a smaller =  m amount of work). S0 Although many traded options meet the mono-  exp- +t − - −t  T m i m i−1 tonicity assumption of Corollary 2, there exist actively  i=1  traded types of options that fail the assumption, · tm i − tm i−1 2 = 0 for example, if the payoff is the realized volatil- Similarly, ity (sample-path standard deviation), or a monotone T + function of it, or the first (in time) large drop or −rT 1 CLm = e Lmtdt − K increase of the underlying; in all cases, the condi- T 0 tional payoff has “interactions” between the not-yet- where sampled path values, destroying monotonicity. For  S0 m such options, truncated DGBS would yield biased  exp−- −t +- +t estimates, and the error bounds of Corollary 2 would  2T mi mi−1  i=1 not apply.   ·exp2t −exp2t 2 =0 In the remainder of this section, we specialize these 1 T mi mi−1 Lmtdt = bounds to specific option types, namely Asian options T  m 0 S0 in §4.3, lookback options in §4.4, and barrier options  exp−- −t +- +t  T mi mi−1 in §4.5. For an Asian option with continuous-time  i=1  ·t −t 2 =0 observation, we also prove that (10) holds with 51 = 1 mi mi−1 (where 0 could be zero) for an equal-length partition Besides CLm and CUm, natural choices of Cm are (i.e., tm i = iT /m) for both CLm and CUm. D G X CAm, Cm , Cm , and Cm defined in §4.2, and the sym- D 4.3. Asian Options metrical version of Cm defined by Suppose we want to estimate the continuous-time S −rT 1 Asian call price c = cA = ECA where Cm = e S0 + ST /2 m T + + −rT 1 C = CA = e Stdt − K + St1 +···+Stm−1 − K T 0 Avramidis and L’Ecuyer: Efficient Monte Carlo and Quasi–Monte Carlo Option Pricing Management Science 52(12), pp. 1930–1944, © 2006 INFORMS 1937

G X This implies that whenever C ≤ C ≤ C for all m, For the Asian option, Cm and Cm can be computed Lm m Um via the following expressions (we leave out the details ˜ of the derivations): ECm − cA=/m + o1/m (13) 1 T + as m →, where 0 ≤ ˜ ≤ < . G −rT ¯ 0 Cm = e SGmtdt − K T 0 Proof. To prove the inequality in (12), define )- + m i T + + − − −rT 1 = - iT /m − - i − 1T /m, )-m i = - iT /m − = e S0 exp2t + Xtm i − T 0 - i − 1T /m, and observe that + 0 ≤ CUm − CLm + Xtm i−1/2dt− K 1 m m −rT S0 ≤ e Um i − Lm i −rT m = e expXtm i − Xtm i−1/2 i=1 2T i=1 1 m 2 +T + ≤ e−rT Si − 1T /m exp + )- + m m m i · exp2t − exp2t − K i=1 m i m i−1 2 −T − exp − − )- − and m m i 1 T + X −rT ¯ We take expectations, observing that Si − 1T /m, Cm = e SXmtdt − K T 0 )- + , and )- − are independent, with ESiT /m = m i m i m S0 expr − qiT /m from (3); Eexp)- + = S0 tm iXtm i−1 − tm i−1Xtm i m i = e−rT exp 1 − / −T/m (the finiteness of this moment T t − t p p i=1 m i m i−1 follows from (4), which implies / < 1); and p p expD t − expD t + Eexp−)- − = 1 + / −T/m. We obtain · m i m i m i m i−1 − K m i p p Dm i ECUm − CLm where D = Xt − Xt /t − t and m i m i m i−1 m i m i−1 1 m−1 we assumed 2 = 0; obvious modifications apply when ≤ e−rT S0 Q1/m − Q1/m expr − qiT /m m + − 2 = 0. i=0 Proposition 2 establishes that for CLm ≤ Cm ≤ CUm and equal-length time discretizations of the form and summing the geometric series proves the inequal- −1 ity in (12). The convergence part of (12) follows by tm i = iT /m, the bias converges to zero as Om when m → and as om−1 when extrapolating with L’Hôpital’s rule. In view of (12) and the fact that EC ≤ c ≤ 51 = 1. Lm A ECUm from Corollary 2, we have that ECm − Proposition 2. Assume (4). Consider equal-length cA=O1/m. It is either o1/m, in which case discretizations of the form tm i = iT /m, where m = (13) holds trivially with ˜ = 0, or of the form (13) for 1 2 We have >˜ 0. For the bias model (10), a numerical experiment mEC − C Um Lm with one example suggests that for estimators C ,   Lm  e−rT − e−qT  C , CD, CG, and CX, we have 5 = 1 and gm =  S0Q1/m − Q1/m  Um m m m 1  + − 1 − e−r−qT /m  m−2. For CS and C , we observed 5 ≈ 2.   m Am 1 if r = q Consider now an Asian option based on discrete ≤ → <0 (12)   observation times 0 = t

4.4. Lookback Options As a prototypical barrier option, we consider an We now examine a prototypical lookback option, up-and-in call, whose continuous-time payoff is the floating-strike call. The approach extends easily to other related payoff types; for these and further −rT + CB = e ST − K I sup St > b (18) information, see Hull (2000). The continuous-time 0≤t≤T price of this option is c = c = EC with F F where b>S0 is the barrier, K is the strike price, and payoff I denotes the indicator function. Conditional on m −rT CF = e ST − inf St (14) for m>0, this payoff is nondecreasing in St for 0≤t≤T each t, so the first part of Corollary 2 applies. The low Conditional on ST , this payoff is a nonincreasing and high estimators are function of St for each t, which implies from the second part of Corollary 2 that C ≤ C ≤ C . −rT + Um F Lm CLm = e ST − K I sup Lmt > b These bounds are narrowing when m increases. The 0≤t≤T low and high estimators are −rT + = e ST − K I max Stm i>b 1≤i≤m −rT CUm = e ST − inf Umt 0≤t≤T and −rT = e ST − min Stm i (15) −rT + 0≤i≤m CUm = e ST − K I sup Umt > b 0≤t≤T and −rT + = e ST − K I max Um i >b 1≤i≤m −rT CLm = e ST − inf Lmt 0≤t≤T The gap between the low and high estimators van- ishes if ST ≤ K and also whenever the indicator −rT = e ST − min Lm i (16) 0≤i≤m function takes the same value in both cases, that is, whenever respectively. A numerical experiment with one example indi- max Um i ≤ b or max Stm i>b 1≤i≤m 1≤i≤m cates that for an equal-length discretization where One approach to estimating the continuous-time price tm i = iT /m, when Cm is taken as either CUm or CLm, −1 is to increase m in DGBS until this gap is closed. m has approximately the form m = 0m + Om−5/3. Thus, the extrapolation from either of these The smallest m at which the gap is closed is a ran- high and low estimators reduces the bias (empirically) dom variable, say M, and the resulting estimator from Om−1 to Om−5/3 approximately. CLM = CUM = CM is unbiased. This unbiased estima- The discretely observed version of this option has tor may require considerably less expected computa- discounted payoff tional time than a (generally biased) estimator based on a large but fixed value of m. To avoid the possi- −rT bility of excessively large values of m, one may select CFd = e ST − min Sti (17) ∗ 0≤i≤d some upper bound m and increase m only up to minMm∗. For the cases where m reaches m∗, one We have CUd ≤ CFd ≤ CLd, and therefore CUm ≤ can use an extrapolation-based estimator as in the CFd ≤ CLm whenever tm 1tm m ⊆ t1td. previous examples. These cases would happen when the path gets very close to the barrier, without cross- 4.5. Barrier Options ing it at an observation time tm i. Barrier options appear to pose greater computational Our approach extends easily to other related pay- challenges relative to Asians and lookbacks. Ribeiro off types. For example, for a down-and-in call option, and Webber (2004), pricing with the VG model and we have bb” using BGBS sampling, achieved substantial efficiency in the indicator function of (18) by “inf0≤t≤T St < b,” improvement for Asian and lookback options but with the corresponding replacements in the low and very little efficiency improvement for barrier options. high estimators. The cases up-and-out call, down- A barrier option can be classified as either knock-in and-out call, and the put versions can be handled in or knock-out. A knock-in option has zero payoff unless a similar way. the underlying asset price reaches a barrier, whereas a The computationally most challenging setting for knock-out option has zero payoff whenever the under- an up-and-in call option occurs when the barrier is lying asset price reaches a barrier. For further infor- far from the initial asset price (i.e., b S0), mak- mation, see Hull (2000). ing the barrier crossing an event of small probability. Avramidis and L’Ecuyer: Efficient Monte Carlo and Quasi–Monte Carlo Option Pricing Management Science 52(12), pp. 1930–1944, © 2006 INFORMS 1939

In such instances, truncating the DGBS algorithm at over another, in our examples, we estimate the con- the random stopping time M typically yields a signif- stants involved in (20). icant computational saving, because EM tends to be The constants c0 and c1 depend on the sampling small relative to a reasonable preselected value of m. algorithm and on the method used for generating the Empirical evidence is provided in §5.4. normal, gamma, and beta random variables. In our The saving can also be important for the discretely experiments, all random variables were generated by observed version of this option, with discounted inversion (Law and Kelton 2000, Hörmann et al. 2004) payoff to make them compatible with RQMC. To approximate the normal and gamma inverse distribution −rT + CBd = e ST − K I max Sti>b functions, we used the rational Chebyshev approx- 1≤i≤d imation of Blair et al. (1976), which gives 16 deci- We have CLm ≤ CBd ≤ CUm whenever tm 1 mal digits of accuracy, and the algorithm of Moshier tm m ⊆ t1td, so we can use the same truncated (2000), respectively. Because we worked exclusively DGBS procedure, stopped at m = minMd. with dyadic partitions, all required beta random variables have density that is symmetric with respect to 5. Numerical Results 1/2, and we developed a special method to approxi- We illustrate the efficiency improvement provided by mate the inverse distribution function for such sym- truncated DGBS with MC and RQMC. We work with metric beta random variables (?). This special method dyadic partitions (§3.3) in all examples. is much faster than all available methods for inverting the beta distribution with general parameters. The 5.1. Efficiency ratio-of-gammas method for generating beta random The efficiency of our MC estimators depends on the variables should not be used in this context because truncation parameter m and on the number of inde- it becomes unstable when the density of the beta dis- pendent replications n. Suppose that the truncated tribution concentrates near 0 and 1. estimator Cm has expected computing cost (e.g., CPU With these inversion methods, the normal random time) cm, bias m = ECm − EC, and variance variables are faster to generate than the symmetric 2 VarCm = m. Let Cm n be the average of n i.i.d. betas (roughly by a factor of 20), and the gamma ran- copies of Cm. The efficiency of Cm n is dom variables are the slowest to generate (roughly by a factor of 10 compared with the symmetric betas, 1 EffC = and this factor increases when the shape parameter m n 2 2 ncm m + m/n of the gamma distribution becomes smaller). There- 1 fore, GBS is significantly faster than GSS (typically = (19) cmn2m + 2m by a factor of 10 or more) for sampling the gamma processes when using these variate-generation meth- This expression goes to zero if m is fixed and n → ods. In what follows, efficiency comparisons are for , reflecting the fact that increasing n eventually the DGBS algorithm only. Our simulations were per- becomes a waste of computing resources if m is formed with the Java simulation library SSJ (L’Ecuyer fixed, because the bias eventually dominates the mean 2004b). square error. This means that as the available comput- For comparison, we ran MC experiments where all ing budget increases, both m and n should increase random variates were generated by the fastest non- simultaneously, in a way that depends on the func- inversion methods available in SSJ (an acceptance/ tions c, , and 2. complement ratio method for the normal, a rejection Suppose that cm = c + c m, m = m−5 , and 0 1 0 method with log-logistic transform for the beta, and 2m = 2 for some positive constants c , c , , 5, 0 0 1 0 an acceptance/rejection method for the gamma). The and . This is often a good approximation to reality. 0 MC simulation ran faster with these methods than Then with inversion when m is small but became slower 1 EffC = (20) when m is large. Roughly, in our examples, they cut m n 2 −25 2 c0 + c1mn0m + 0 the CPU time in half for m = 16 and take about the same time for m = 512. The explanation is that For a fixed computing budget b = c + c mn, this ex- 0 1 the performance of these beta and gamma generators pression is maximized by taking depends highly on the distribution’s parameters and, 252b 1/25+1 as m increases, we get into the parameter range where m = m∗ = 0 (21) c 2 the methods are less efficient. 1 0 For RQMC, we tried various types of methods That is, m must increase proportionally to b1/25+1.To implemented in SSJ. The results reported here are quantify the efficiency improvement of one estimator only for the F2wLFSR point sets taken from Panneton Avramidis and L’Ecuyer: Efficient Monte Carlo and Quasi–Monte Carlo Option Pricing 1940 Management Science 52(12), pp. 1930–1944, © 2006 INFORMS

(2004), randomized by a linear matrix scrambling plus Figure 3 Estimated Absolute Bias m of Various Estimators a random digital shift (Owen 2003). One of the alter- without Extrapolation, as a Function of m, for the Asian natives are the well-known Sobol’ point sets with the Option Example same randomization, but with the available imple- L S 1 mentation of these point sets we were limited to m ≤ 10 U G 128, due to an upper bound on their dimension. The D X A variance reduction with the Sobol’ point sets was actu- 100 ally better than for the selected point sets for some –1 small values of m. To estimate the variance and efﬁ- 10 ) –2 ciency with RQMC, we performed N = 100 indepen- m

( 10 dent randomizations of the 2m-dimensional n-point β F2wLFSR point set, for several values of n equal to 10 –3 powers of 2. In this context, if Cm n N denotes the general average, we get 10 –4

10 –5 EffCm n N 2 4 8 16 32 64 128 1 m = (22) 2 Note. In the legend, “L” refers to C , “U” refers to C , and so on. The Nnc + c m m−25 + 2m n/N Lm Um 0 1 0 dashed line indicates the half-width of the confidence interval on the bias. Bias estimates that fall below that line are too noisy to be reliable. in which c0 and c1 are easily estimated, whereas 2mn depends on both m and n in a complicated way. We could not easily fit an expression to this half-width of the confidence interval on c ). The 2mn, so in our efficiency computations, we sim- A rate estimates were 5 ≈ 2 for C and CS and 5 ≈ 1 ply replaced it by its empirical value for each m and 1 Am m 1 n. Expression (22) always reaches its maximum for for the other estimators (the slopes in Figure 3 seem N = 1, but we need to take N>1 to be able to esti- to converge to −2or−1 when m →). These estimate the variance. mates may not apply generally. Figure 4 summarizes the bias of all extrapolated estimators based on the 5.2. An Asian Option associated estimated rate and of the extrapolated esti- S We borrow VG model parameters from an unpub- mators CAm and Cm with 51 = 1. The empirical bias lished draft of Hirsa and Madan (2004), where the of all estimators extrapolated with the rate 1 is also VG model parameters were calibrated against options well modeled by (10) and converges as Om−2.Of D S on the S&P 500 index using data for June 30, 1999. these, the best performer is Cm , followed by Cm , In our example, T = 040504, = 01927, = 02505, G X Cm , and Cm , which are almost indistinguishable; last =−02859, r = 00548, and q = 0. These parameter come CLm, CUm, and CAm, which are almost indis- values can be considered representative for the case tinguishable. The best estimators in terms of bias are of intermediate-maturity options for a nondividend- S CAm and Cm using the rate 2; the bias is below our paying asset. Our first example is an Asian option confidence interval half-width for m ≥ 16. The esti- with parameters S0 = 100 and K = 100. Our aim is S G X mators Cm , Cm , and Cm without extrapolation are to estimate the continuous-time price cA. To estimate the bias as a function of m for the estimators Cm, we computed very precise RQMC esti- Figure 4 Estimated Absolute Bias m of Various Estimators with k mates of ECm for m = 2 , k = 111. For that, we Extrapolation (with 1 = 1, Unless Otherwise Indicated) as a used 100 copies of a 2m-dimensional F2wLFSR point Function of m, for the Asian Option Example set with n = 218 points, randomized independently by 101 an affine linear scramble. Figures 3 and 4 show the L S U G estimated absolute bias as a function of m (in loga- 100 D X A with γ =2 rithmic scale), without and with Richardson’s extrap- A 1 –1 S with γ =2 olation, respectively. To estimate this bias, we used 10 1 ) m an estimated value of cA obtained by extrapolat- ( –2

β 10 ing the values for k = 10 and 11. This estimated exact value is 368538 ± 0000048 with 95% confidence. 10 –3 In this example, (10) holds well for all estimators, 10 –4 and we estimated 51 and 0 by fitting the linear regression model log m = log − 5 log m to the 0 1 10 –5 data points for which the estimated bias differed sig- 2 4 8 16 32 64 128 nificantly from zero (i.e., was more than twice the m Avramidis and L’Ecuyer: Efficient Monte Carlo and Quasi–Monte Carlo Option Pricing Management Science 52(12), pp. 1930–1944, © 2006 INFORMS 1941

Figure 5 Estimated MC Efﬁciency for the Asian Option Example, Figure 7 Estimated Absolute Bias m of Various Estimators without without Extrapolation Extrapolation, as a Function of m, for the Lookback Option Example

L S 101 L 80 U G D X U A 100 A Eff. 40 10 –1 ) m (

0 β 4 8 16 32 64 128 256 512 10 –2 m doing practically as well in terms of bias as the extrap- 10 –3 olated CLm, CUm, and CAm with 51 = 1. Figures 5 and 6 show the estimated efficiency of 10 –4 standard MC as a function of m, without and with the 4 8 16 32 64 128 256 extrapolation, respectively. Without the extrapolation, m C , C , and CD do very poorly for moderate m Lm Um m D (they lie close to the horizontal axis in the figure), C16 have efficiencies 30.5 and 0.10, respectively. The S most efficient RQMC estimator, CD, is approximately, whereas Cm with m = 16 is the best combination for 16 this example. With the extrapolation, CS with 5 = 2 3,000 times more efficient than the naive estimator m 1 D D C . That is, it requires a total CPU time approx- is the best performer, followed by Cm and then by 2048 S imately 3,000 times smaller to estimate the option Cm with 51 = 1. The efficiency of all extrapolated S value with a given precision. estimators decreases for m>64. For CAm and Cm , extrapolating with 51 = 2 versus 51 = 1 increases effi- 5.3. A Lookback Option ciency, and the efficiency peaks at smaller m, because Our second example is a lookback option, with the of the faster decay of bias. same parameters as for the Asian option. The bias 14 16 18 With RQMC, for n = 2 ,2, and 2 , the effi- and efficiency are estimated in the same way as in ciency as a function of m behaves very similarly to the Asian option case, for the three estimators C , that under MC, except that it is significantly larger. Lm CUm, and CAm considered in §4.4. The estimated The best estimator in our MC experiments, exclud- exact value is 939805 ± 000015 with 95% confidence. D ing those extrapolated with 51 = 2, is C8 and has Figure 7 shows the bias without extrapolation; the (empirical) efficiency 130. For comparison, the naive points for CLm and CUm fall well along a straight line, estimators CD and CD, which correspond to esti- 2048 8 and the rate estimates are 51 ≈ 1. The points for CAm mating the continuous-time option by a discrete-time show mild concave nonlinearity, suggesting extrapo- one with 2,048 and 8 observation times, respectively, lation may be less effective. Figure 8 shows the bias have efficiencies 2.2 and 0.03. For RQMC with n = 212, when we heuristically use 5 = 1 in the extrapolation D D 1 C16 has (empirical) efficiency 6,973, whereas C2048 and for all estimators, and some concave nonlinearity is

Figure 6 Estimated MC Efﬁciency for the Asian Option Example, with Figure 8 Estimated Absolute Bias m of Various Estimators with Extrapolation Extrapolation, as a Function of m, for the Lookback Option Example 150 L S 100 U G L D X U A A with γ = 2 1 10 –1 100 γ A S with 1 = 2

) –2

m 10 ( Eff. β 50 L 10 –3 U

–4 0 10 4 8 16 32 64 128 256 512 4 8 16 32 64 128 256 m m Avramidis and L’Ecuyer: Efﬁcient Monte Carlo and Quasi–Monte Carlo Option Pricing 1942 Management Science 52(12), pp. 1930–1944, © 2006 INFORMS

Figure 9 Estimated RQMC Efﬁciency for the Lookback Option Example, Figure 10 Estimated RQMC Efﬁciency for the Lookback Option without Extrapolation Example, with Extrapolation

400 600 L L U U 300 A 400 A Eff. 200 Eff. 200 100

0 0 4 8 16 32 64 128 256 512 1,024 4 8 16 32 64 128 256 512 1,024 m m visible. A linear regression to the points in this figure improvement ratio gives slopes of about −5/3, that is, the bias converges c + c m∗ approximately as Om−5/3. Extrapolation reduces the qm∗ ≈ 0 1 ∗ c0 + c1EminMm bias order of CLm, CUm, but results in a slight increase in the bias of CAm. for some constants c0 and c1. This is simply the ratio Figures 9 and 10 display the estimated efficiency for of computing costs, because the estimator takes the RQMC, without and with extrapolation. The MC effi- same value in both cases. Table 1 displays values of ciencies are smaller but have a very similar behavior. EminMm∗ and qm∗ for m∗ equal to powers of 2. U These values are approximately the same for both MC The best estimator in our MC experiments, C64 , has (empirical) efficiency 46.2. For comparison, the naive and RQMC because their computing costs are about estimator CU , which corresponds to estimating the the same. The table also gives the estimated values 2048 6 continuous-time option by a discrete-time one with based on n = 10 replicates of CLm∗ with RQMC, with- d = 2048, has efficiency 2.03. The best RQMC estima- out extrapolation (est. value) and with extrapolation ∗ tor is CU, with (empirical) efficiency 585. It is approx- (est. value extra.). Despite the fact that EminMm 256 ∗ imately 300 times more efficient than the naive MC converges rather slowly with m , the estimated value stabilizes rapidly (i.e., the bias quickly becomes negli- estimator CU . 2048 gible), especially when we use extrapolation. Further 5.4. A Barrier Option evidence of the efficiency of DGBS with MC, for a We report results for a barrier option with b = 120 wider set of examples than presented here, is pro- and all other parameters as in the Asian option. In vided in Avramidis (2004). The efficiency of the RQMC estimator truncated at this case, the DGBS randomly truncated at M,as minMm∗ is approximately 12 times that of the cor- explained in §4.5, gives exactly the same payoff as a responding truncated MC estimator. So the efficiency “full-dimensional” estimator that would sample the improvement factor of this RQMC estimator com- entire path at all m∗ observation times. Therefore, pared with the standard MC estimator truncated at the efficiency improvement factor can be character- m∗ is approximately 12qm∗. This improvement fac- ized by the ratio of expected work between these two tor is approximately 12,800 for m∗ = 214 = 16386, for estimators. instance. The efficiency is estimated in the same way as in the Asian option case, for the two estimators CLm∗ ≡ ∗ 6. Generalization to Finite-Variation CBm and CUm∗ considered in §4.5. The estimated exact value is 21575 ± 00010 with 95% confidence. Lévy Processes If we compare an estimator truncated at m∗ with The DGBS method for option pricing under the VG one truncated at minMm∗, we obtain the efficiency model can in principle be extended to option-pricing

Table 1 Expected Truncation Value and Efﬁciency Improvement Ratio Under MC, and Estimated Value with RQMC with n = 220, as a Function of m∗, for the Barrier Option Example

m∗ 4 16 64 256 1,024 4,096 16,386 65,536

EminMm∗ 2204 2554 2894 3221 3598 3893 4159 4527 qm∗ 113 2054188 71 277 1,065 4,094 Est. value 19877 20980 21402 21528 21561 21569 21570 21571 Est. value extra. 20543 21428 21553 21572 21571 21571 21571 21571 Avramidis and L’Ecuyer: Efficient Monte Carlo and Quasi–Monte Carlo Option Pricing Management Science 52(12), pp. 1930–1944, © 2006 INFORMS 1943 models driven by any Lévy process whose paths have explicitly for other pure-jump finite-variation Lévy finite variation. (The variation of a function f over processes. According to Asmussen (1999, pp. 88–89 n the interval ab is Vf= sup i=1 fxi − fxi−1, and references therein), the increment density of a where a = x0 Gaussian process (Barndorff-Nielsen sion beyond the VG case and point to the difficul- 1998) is not a subordinator, and a Cauchy subordi- ties that arise. Exponentials of Lévy processes are nator is unlikely to be useful because the increment increasingly used as models of asset prices in the has an infinite mean. So at this point there is no clear derivative-pricing literature, for example, the CGMY practical generalization of our approach beyond the model (Carr et al. 2002) that contains VG as a spe- case where the subordinators are gamma processes. cial case. We make a minimal introduction to Lévy In particular, we do not have an algorithm to sample processes, following Asmussen (1999). from the conditional density of the increment for the A Lévy process X = Xt t ≥ 0 is defined as a CGMY process. continuous-time real-valued process with stationary independent increments and X0 = 0. This class of 7. Conclusions and Future Research processes is in one-to-one correspondence with the An insight that emerges from this paper is that in class of infinitely divisible distributions via the distri- certain applications the idea of integrand structur- bution of, say, X1. A Lévy process can be written ing to concentrate the variance to a few coordinates as the independent sum Xt = ct + Bt+ Jt of a can also be used to reduce the work by eliminat- deterministic drift ct, a Brownian component Bt, ing sampling of uninteresting coordinates altogether, and a pure-jump Lévy process J t t ≥ 0 character- given some assurance that the resulting estimator’s ized by its Lévy measure C, which can be any non- bias is acceptable. This is the theme behind the esti- negative measure on satisfying C 0 = 0 and mation approach proposed in §4, where it is pos- sible to bound the bias a priori (pre-sampling) for 2 minx 1Cdx< (23) Asian options, or a posteriori (after-sampling) for − lookback and barrier options. An extreme case is coor- A rough description of the process J is that jumps of dinate elimination, arising with barrier options, where size x occur with intensity (rate per unit time) Cdx. there is an a posteriori guarantee—once the stopping The paths of J t t ≥ 0 over any finite time interval condition is satisfied—that the full-dimensional and are functions of finite variation if and only if truncated estimators are equal. Special properties of the variance gamma process have been exploited to minx 1Cdx< (24) enable this structuring. − Our theoretical and numerical results suggest the A subordinator is a nondecreasing Lévy process. A attractiveness of difference-of-gammas bridge sam- pure-jump Lévy process can be written as the dif- pling combined with Richardson extrapolation when ference of two independent subordinators defined by pricing path-dependent options under the variance the restriction of C to 0 and the negative of the gamma model. In our numerical illustrations, we restriction of C to − 0, respectively, if and only if observed that combining this approach with RQMC its Lévy measure satisfies (24). improved the efficiency by yet another large factor. Generalizing the setting we have studied so far, The integrand structuring proposed here, via DGBS, we can replace the VG process by any pure-jump not only permits one to truncate with bounds on the Lévy process X of finite variation, that is, with Lévy bias, it also boosts the effectiveness of RQMC. Our measure satisfying (24). Because of the representation illustrations focused on a particular set of VG model X = X+ − X− where X+ and X− are subordinators, parameters to study the bias, variance, work, and effi- the pathwise bounding of X is straightforward if one ciency in some depth; we expect these measures to samples the subordinators via the bridge method. For be fairly representative of what one may encounter in the general case of bridge sampling, assuming knowl- typical applications. + − + edge of the increment density pt and pt of X t and It would be desirable to know the asymptotic (con- − X t, respectively, for &1

Perhaps the most natural control variates are the pow- Carr, P., H. Geman, D. Madan, M. Yor. 2002. The ﬁne structure of ers of the asset price or the powers of the two gamma asset returns: An empirical investigation. J. Bus. 75 305–332. processes sampled, for example, at an equal-length Conte, S. D., C. de Boor. 1972. Elementary Numerical Analysis: An partition of time; the means are directly derived from Algorithmic Approach, 2nd ed. McGraw-Hill, New York. Glasserman, P. 2004. Monte Carlo Methods in Financial Engineering. the moment-generating function of the gamma distri- Springer-Verlag, New York. bution (see the proof of (12)) and are only ﬁnite for Hirsa, A., D. B. Madan. 2004. Pricing American options under vari- powers r

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