It is widely acknowledged that the standard. Using Brownian Bridge for Fast Simulation of Jump-Diffusion Processes and Barrier Options

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1 Usng Brownan Brdge for Fast Smulaton of Jump-Dffuson Processes and Barrer Optons STEVE A.K. METWALLY AND AMIR F. ATIYA STEVE A.K. METWALLY s a vce presdent at Lehman Brothers n New York Cty. metwally@rcn.com AMIR F. ATIYA s an assocate professor of computer engneerng at Caro Unversty n Egypt. amratya@lnk.net Barrer optons are one of the most popular dervatves n the fnancal markets. The authors present a fast and unbased Monte Carlo approach to prcng barrer optons when the underlyng securty follows a smple jump-dffuson process wth constant parameters and a contnuously montored barrer. Two algorthms are based on the Brownan brdge concept. The frst one s based on a samplng approach to evaluate an ntegral that results from applcaton of the Brownan brdge. The second approach approxmates that ntegral usng a Taylor seres expanson. Both methods sgnfcantly reduce bas and speed convergence compared to the standard Monte Carlo smulaton approach. For example, the frst method acheves zero bas. In addton, t s about 00 tmes faster than the conventonal Monte Carlo method that acheves acceptable bas. In developng the second algorthm, the authors derve a novel approach for obtanng a frst-passage tme densty ntegral usng a Taylor seres expanson. Ths approach s potentally useful n other applcatons, where the expectaton of some functon over the frst-passage tme dstrbuton needs to be derved. It s wdely acknowledged that the standard assumpton of lognormal stock prce dffuson wth constant volatlty (as n the Black-Scholes framework) s nadequate to characterze fnancal markets. Emprcal studes examnng market returns and dervatves valuatons all pont to excess kurtoss and skewness. To address these emprcal observatons, two domnant approaches have been researched: the stochastc volatlty approach, and the jumpdffuson approach. In ths artcle we consder the jump-dffuson approach. Jump-dffuson processes were orgnally proposed by Merton [976]. He derves an nfnte-sum formula for a plan vanlla opton, assumng normally dstrbuted jump szes (see also Trautmann and Benert [995] for an extenson). Snce then, many research artcles have studed jump-dffuson processes. The problem s that for many exotc optons analytcal expressons have been very dffcult to come by n a jump-dffuson framework. We consder barrer optons, one of the most popular exotc optons. Barrer optons can have a varety of possble features, but the general concept s that the payoff depends on whether the underlyng asset prce hts a specfc barrer level. There are generally two types of barrer optons: knock-out and knock-n. For the knock-out, the opton s vald only as long as the barrer s never touched durng the lfe of the opton. For the knock-n, the opton becomes vald whenever the barrer s touched durng the lfe of the opton. Because of a party relatonshp between knock-out and knock-n optons, t s generally suffcent to study one of the two types. In our case, we consder knock-out optons. We also consder a constant rebate payment to be pad to the optonholder at the frst tme the FALL 00 THE JOURNAL OF DERIVATIVES 43

2 barrer s ht, f there s any barrer crossng durng the lfe of the opton. Valuaton of barrer optons calls for soluton of the frst-passage tme problem, or boundary-crossng problem, whch has been a wdely studed topc n mathematcs for the last 50 years (see Karatzas and Shreve [99]). A closedform expresson for the case of Brownan moton has long been well-known (the so-called nverse Gaussan densty). Based on ths, an analytcal formula for barrer optons under the standard lognormal process has been derved (Hull [997], Wlmott [998]). Closed-form expressons for the case of jump-dffuson processes are, however, very few. Kou and Wang [000] derve an expresson for the frst-passage tme when jumpszes follow a double-exponental dstrbuton (see also Boyarchenko and Levendorsk [000], Mordeck [000]). Because of the dffculty n obtanng general analytcal expressons for barrer optons under the jumpdffuson framework, much of the work has focused on numercal or Monte Carlo valuaton methods. We consder only the latter. The typcal Monte Carlo approach would be to examne the equvalent rsk-neutral process, smulate jump-dffuson paths accordng to ths process, and compute the expected dscounted payoff by averagng over the many smulated paths. There s, however, a problem wth ths approach. Tme dscretzaton of the jump-dffuson paths ntroduces some bas n the prcng estmate. Ths bas should not be underestmated. In our smulatons we get bases as hgh as 0% (for a tme dscretzaton of one pont per tradng day). To reduce the bas, one should make the dscretzaton fne enough, but as a result the computaton becomes very slow. Our work takes account of these two ssues: the estmaton bas, and the computaton speed. Specfcally, we propose two Monte Carlo technques to value barrer optons for jump-dffuson processes. The frst method elmnates the bas completely, whle the second method reduces t sgnfcantly. In addton, both methods are computatonally orders of magntude more effcent than the conventonal Monte Carlo method. The new algorthms are based on the Brownan brdge concept (see Karatzas and Shreve [99]). Ths concept s used to obtan the boundary-crossng probablty densty, gven that we observe the two end-ponts of the process. The Brownan brdge concept has been used before n other related problems. Duffe and Lando [00] use the concept to obtan the condtonal dstrbuton of a frm s assets at tme t, gven no default before t. Beaglehole, Dybvg, and Zhou [997] and El Babsr and Noel [998] use the same concept to value optons whose payoffs depend on extreme values, usng pure dffuson processes for the underlyng assets (see also Andersen and Brotherton-Ratclffe [996]). Atya [000] consders jump-dffuson processes, and uses the Brownan brdge concept to compute the boundary-crossng probablty more effcently. We focus on a contnuously montored barrer, although a dscretely montored barrer s more common n practce. A number of authors take dscrete montorng nto account. For example, Broade, Glasserman, and Kou [997] derve a correcton to the barrer opton prce obtaned under a contnuously montored barrer n order to obtan an approxmaton of the prce under the dscretely montored barrer. Also, Nahum [999] prces look-back optons usng jump-dffuson processes and approxmates the dscrete montorng of the extreme value usng Brownan meander. For the two proposed approaches, we start by generatng the jump-nstants of the process, as well as the asset value mmedately before and mmedately after the jumpnstant. In between these generated ponts, we have a pure dffuson wth known end-ponts, hence a Brownan brdge. The two methods compute the expected payoff of the rebate porton of an nterjump nterval dfferently. The frst proposed method, the unform samplng approach, s based on samplng from a unform dstrbuton n a fashon smlar to the rejecton method. The second approach, s the seres expanson approach. As the name suggests, t evaluates the payoff ntegral correspondng to the nterjump nterval usng a seres expanson. I. MATHEMATICAL MODEL We assume the underlyng securty follows a smple jump-dffuson process that conssts of two parts, a contnuous and a dscontnuous part. The contnuous part s a geometrc Brownan moton wth constant nstantaneous drft µ and volatlty σ. The dscontnuous part represents the change n the securty value upon arrval of the rare event. Rare events nclude major dsasters or poltcal changes, or the release of unexpected frm or economc news. The process s: ds S t t µ dt σ dz V dq t t t () 44 USING BROWNIAN BRIDGE FOR FAST SIMULATION OF JUMP-DIFFUSION PROCESSES AND BARRIER OPTIONS FALL 00

3 where the frst two terms on the rght-hand sde represent the contnuous part, the thrd term represents the dscontnuous part, and q (q t ) t 0 s a homogeneous Posson process wth a non-negatve ntensty parameter λ (annual frequency of jumps). Let T, T,, denote the arrval tmes of the jumps. Moreover, let V T (S T /S T ), where V T, V T,, are..d. random varables representng the successve percentage changes n the securty value at the jump-events. Note that (S T /S T ) 0 snce the securty prce s assumed to be non-negatve at all tmes. Z (Z t ) t 0 s a standard Brownan moton Z T ~ N(0, T). We also assume that the processes Z, q, and V T are jontly ndependent. Let A be the logarthm of the rato of the securty value after and before the jump. We assume t s normally dstrbuted and state-ndependent (we could assume other dstrbutons as well): and A lns lns ln( V ) T T T f( A) ~ N( µ, σ ) Let J Σ m A be the sum of the log jump-szes n the nterval [0, T]. Under ths assumpton, J s also normally dstrbuted: J ~ N( µ λ T, [ µ σ ] λ T) Also let A A A A A P A k E ( VT ) exp A µ σ where P s the real-world measure. Applyng the Doléans-Dade stochastc exponental formula for semmartngales, we get a unque soluton to the stochastc dfferental equaton (see Metver [98], Jacod and Shryayev [987], and Duffe [00]): σ ST S0 exp r T σ ZT Q J Q λ Q E Q ( VT ) T () where Q s the equvalent martngale measure that makes the securty process a local martngale. Equaton () can also be wrtten as ds S where v ~ s the P-compensated Posson random measure, and the jump-sze space s the real lne R(Last and Brandt [995]). Pham [997] characterzes the equvalent martngale measures by ther Radon-Nykodm densty wth respect to P: dq dp where ε(.) s the exponental semmartngale of Doléans- Dade, θ s the market prce of the dffuson rsk, and p(a) s the market prce of the jump-rsk. θ and p(a) are related as follows: µ r θ σ λ VT [ p( A)] f( A) da whch s used to get a unque θ for a choce of p(a). Under the Q measure: f where ZT Q s a Q-Brownan moton J Q Q t t T T ε θ dzt ( p( A) ) v ( dt, da) λ µ dt σ dz V v ( dt, da) Q λ pa ( ) f( AdA ) pa ( ) f( A) ( A) pa ( ) f( AdA ) Z Z θ T T Q m Q A here m Q : the number of jumps n the nterval [0, T], where the Posson process q now has ntensty of λ Q. Q T T Q E ( V ) V f ( A ) da R 0 0 R R T R R t R t (3) FALL 00 THE JOURNAL OF DERIVATIVES 45

4 Ths gves us many equvalent martngale measures, whch leads us nto an ncomplete market. Each measure has a dfferent combnaton of the three parameters: the drft, the jump-ntensty, and the jump-sze dstrbuton. In such an ncomplete market, perfect hedgng s not possble. Pham [999] revews three dfferent hedgng crtera n an ncomplete market: superhedgng, mean-varance hedgng, and shortfall rsk mnmzaton. Because we focus on the computatonal technque, we consder only the smplest measure and do not prce the jump-rsk (as consdered by Merton [976]). Our proposed computatonal approach does apply wth mnor modfcatons to the general case where the jump-rsk s prced. Ths means that p(a) (the real-world and rskneutral dstrbutons for the jump component are the same). From (3) we get θ whch mples: λ ( µ r) σ Q Q Q T Substtutng ths n () we get: σ ST S0 exp r T σ ZT Q J λ kt Let x(t) lns t Then: λ, f ( A) f( A), E ( V ) k xt ( ) x( 0) ct σ Z J T Q σ ~ N([ r λ kt ] µ Aλ T, σ T [ µ A σa] λ T) (4) where c r [σ /] λk. The proposed algorthms consder manly the transformed process x(t). In ths artcle we apply our approach to down-andout barrer call optons, for whch we use the shorthand DOC. For these optons, a payoff of max(s T X, 0) s pad at maturty T f the barrer s never crossed durng the lfe of the opton. If a crossng occurs before maturty, a rebate s pad at the tme of the frst crossng, and there s no subsequent payoff. The opton value for a contnuously montored barrer s DOC exp( rt ) max(exp( x( T )) X, 0) PxT (( ) dx,nf xs ( ) > ln( H)) dx T 0 s T R exp( rt) ht ( ) dt 0 where exp( rt) s the dscountng term, X s the strke value, H s the barrer level, R s the constant rebate value, and h(t) s the densty of the frst-passage tme. The term P[x(T ) dx, nf x(s) > ln(h)] represents the jont dstrbuton of the random varable x(t ) and the event that 0 s T the asset prce stays above the barrer durng the lfe of the opton (Harrson [985]). Note that the frst-passage tme densty for jumpdffuson processes does not have a closed-form soluton, except n two cases: when the jumps are doubly exponental or exponentally dstrbuted (Kou and Wang [000], Mordeck [000]), or when the jumps can have only non-negatve values (Blake and Lndsey [973]). Our proposed algorthm represents effcent ways to evaluate the ntegrals n the valuaton formula. II. SUMMARY OF THE METHODS AND PRELIMINARIES ln X Descrpton of the Proposed Methods Frst, we generate the jump-nstants from the nterjump densty functon. Then, we generate the asset value mmedately before the frst jump usng a normal dstrbuton. Next, the value mmedately after the jump s generated, usng the jump-sze dstrbuton. Then, the value mmedately before the second jump s generated usng a normal dstrbuton. We contnue n ths manner untl we reach the expraton tme. We use a normal dstrbuton because n between any two jumps the process follows a pure dffuson, and hence, gven the prevous jump value, the endng value s normally dstrbuted. If any of the generated values are below the barrer, we know that at some tme the process has crossed the barrer. Yet there s also a chance that, prevous to that, n between any two jumps the path has crossed the barrer. To tackle ths stuaton, we consder the probablty densty of the barrer crossng-tme, gven the end-pont (5) 46 USING BROWNIAN BRIDGE FOR FAST SIMULATION OF JUMP-DIFFUSION PROCESSES AND BARRIER OPTIONS FALL 00

5 values (whch are already generated and are hence known). The dervaton of such a densty s based on the Brownan brdge concept descrbed below. In the frst proposed method, we randomly generate a value for the dscounted payoff correspondng to the partcular nterjump nterval usng that densty (that s, the condtonal barrer crossng-tme densty). Because there s also a non-zero probablty that no crossng has occurred n that nterval, we use the concept of rejecton samplng, where rejecton of a generated pont means no crossng occurred n the nterval. Ths procedure s performed sequentally startng from the nterval t 0 untl the frst jump, then the nterval from the frst jump untl the second jump, and so on untl expraton tme. If stll no crossng has happened at that pont, then the opton has not been knocked out. We then generate the termnal asset value and compute the correspondng dscounted payoff. In the second approach, rather than generatng the dscounted payoff correspondng to any nterjump nterval randomly, we evaluate the expectaton of the dscounted payoff correspondng to that nterval (agan gven the two end-ponts). That expectaton, however, s an ntegral that s hard to evaluate n closed form. We approxmate the ntegral usng a seres expanson. Ths method also proceeds n a sequental manner untl reachng the opton expraton tme. Brownan Brdge Concept Let the jump-nstants be T,, T K. As mentoned, these are the frst varables to be generated. Let x(t ) be the process value mmedately before the -th jump and x(t ) the process value mmedately after the -th jump. These values are generated sequentally, that s, frst x(t ), then x(t ), then x(t ), then x(t ), and so on. The process follows a pure Brownan moton n between any two jumps. Snce we know the ntal and termnal values, however, the Brownan moton s actually a ted-down Brownan moton or a Brownan brdge (Karatzas and Shreve [99], Revuz and Yor [994]). Let B s be a Brownan brdge n the nterval [T, T ] wth B T x(t ), B T x(t ), and τ (T T ) where dx t cdt σdz tq. From Karatzas and Shreve [99], we can obtan the probablty that the mnmum of B s s always above the barrer n the nterval τ: P P( nf B > ln H B x( T ), B x( T )) T s T s T T [ ] [ ] ln H x( T H x T ) ln ( ) exp τσ 0 where ln H s the barrer level. We are also nterested n obtanng the densty functon of the frst-passage tme t, gven the two end-pont values. Frst, defne C(t) as the event the process crosses the barrer for the frst tme n the nterval [t, t dt]. The condtonal frst-passage densty s defned as: g () t pc ( () t dt x( T ), x( T )) From Feller [968]: g() t PC ( () t dt x( T), x( T )) PCt ( ( ) dt, xt ( ) dx xt ( )) PxT (( ) dx ( xt)) PCt ( ( ) dt xt ( )) * PxT ( ( ) dx Ct ( ), xt ( )) PxT (( ) dx ( xt)) PCt ( ( ) dt xt ( )) * PxT ( ( ) dx xt ( ) ln H) PxT (( ) dx ( xt )) From Rogers and Wllams [994], we have the frstpassage tme (nverse Gaussan) densty: xt ( ) lnh PCt ( ( ) dt xt ( )) ( t T ) πσ [( xt ) ln H ct ( T )] exp ( t T ) σ (8) The other two components of the formula are the normal densty functons (Feller [968]): PxT (( ) dx ( xt)) π( T T) σ [( xt ) xt ( ) ct ( T )] exp ( T T) σ (9) f 3 xt ( ) > lnh otherwse (6) (7) FALL 00 THE JOURNAL OF DERIVATIVES 47

6 PxT (( ) dx () xt ln H) π( T t) σ [( xt ) ln HcT ( t)] exp ( T t) σ (0) The prce of the down-and-out barrer opton can then be computed as the expectaton over {T, x(t ), x(t ):,, K } of Equaton (5), whch represents the dscounted expectaton (wth respect to the other random components) of the payoff: Substtutng (8), (9), and (0) n (7), after some algebrac groupng, we get: xt ( ) lnh g() t ( t T ) ( T t) yπσ where 3 (( xt ) ln HcT ( t)) (( xt ) ln H ct ( T )) exp ( T t) σ ( t T ) σ () where U DOC R P exp( rs) g ( s) ds j T j T A exp( rt ) R P I I j j I ( A )exp( rt)max(exp( x( T)) X, 0) P I K j j (5) [( xt xt c ) ( ) τ] y exp πτσ στ () A I f I 0 0 f I 0 Ths condtonal frst-passage densty s used n the rebate payoff porton correspondng to the nterval [T, T ]. III. THE PROPOSED SIMULATION ALGORITHMS For ease of notaton, let T 0 0 and T K T. Also let I represent the ndex of the frst jump, f any, that crosses the barrer durng the lfe of the opton: j j I mn( : x( T ) > ln H ; j,...,, x( T ) > lnh j,...,, x( T ) ln H) (3) If no such I exsts, then I 0. Let M(s) M denote the ndex of the nterjump perod n whch the tme s falls. Ths means that s [T M, T M ]. Then: PC ( ( s) ds xt ( ), xt ( ),,..., K ) M gm() s Pj f M < I or I 0 j M M gm() s Pj Pjδ( s TI ) f M I j j 0 f M > I where δ s the Drac s delta functon. (4) and I f I 0 U K f I 0 The ntegral n Equaton (5) cannot be evaluated n closed form. We have derved a seres expanson method to evaluate t. By retanng up to the second-order term of the expanson, we can get a very accurate approxmaton. The detaled formulas are descrbed n the appendx. We use ths approxmaton n the second proposed approach. Unform Samplng In the unform samplng method we evaluate the ntegral n Equaton (5) by sequentally generatng a value from a dstrbuton n the approprate range, and then evaluatng the ntegrand at the generated value of s. Specfcally, the dea of the algorthm s to consder the nterjump perods sequentally. Consder, for example, the perod [T, T ]. We generate a varable from a dstrbuton unform n an nterval startng from T, but extendng beyond T by a factor of /( P ) (steps 4b and 4c n the algorthm below). Ths adjusts for the fact that the total probablty of crossng the barrer any tme wthn perod s ( P ). 48 USING BROWNIAN BRIDGE FOR FAST SIMULATION OF JUMP-DIFFUSION PROCESSES AND BARRIER OPTIONS FALL 00

7 If the generated pont s falls n the nterjump nterval [T, T ], a barrer crossng has occurred at that generated pont. We then sample the ntegrand functon g (s) at that generated pont. Or, f the generated pont s falls outsde the nterval [T, T ] (whch happens wth probablty P ), that pont s rejected. Ths means no barrer crossng has occurred n the nterval, and we proceed to the next nterval and repeat the whole process agan. The steps of the algorthm are as follows:. For n to N perform Monte Carlo runs as follows (steps -5):. Generate jump-nstants T by generatng the nterjump tmes (T T ) accordng to the gven densty (e.g., exponental). 3. For to K(K s the number of jumps that occur durng the lfe of the opton): a. Generate x(t ) from a Gaussan dstrbuton of mean x(t ) c(t T ) and standard devaton σ T T. The ntal state s x(0) x(t 0 ). b. Generate the sze of jump, A, accordng to the gven jump-sze dstrbuton. c. Compute the post-jump value: x(t ) x(t ) A. 4. For ntervals to K : a. Compute the ntraperod probablty of no barrer crossng P accordng to Equaton (6). b. Let b (T T )/( P ) c. Generate s from a dstrbuton unform n the nterval [T, T b]. d. If s [T, T ], then the frst-passage tme to the barrer occurred n ths nterval [T, T ]. In ths case, we evaluate g (s) by substtutng the generated s nto Equaton (). Then: DscPayoff(n) Rbg (s)exp( rs). Ext loop, and perform another Monte Carlo cycle (steps -5). e. If s [T, T ], then the frst-passage tme has not yet occurred. f. If x(t ) ln H, then the jump crossed the barrer (ndex I as defned n Equaton (3) equals ). The payoff becomes: DscPayoff(n) R exp( rt I ). Ext loop, and perform another Monte Carlo cycle (steps -5). g. If x(t ) > ln H, then examne the next nterval, that s, ncrement, and perform another teraton of step No crossng occurred durng the lfe of the opton. The payoff s gven by: DscPayoff(n) R exp( rt ) max (exp (x(t )) X, 0). Perform another Monte Carlo cycle (steps -5). 6. If n N,.e., we have completed all cycles of the Monte Carlo smulaton, obtan the estmate for the opton prce: N DOC DscPayoff () n N n Much of the complexty n the frst algorthm comes from smulatng the exact pont wthn the nterval at whch the barrer s crossed. We also test a much smpler verson of the unform samplng approach. To save on computatonal effort, rather than evaluatng the lengthy formula of g (s) n step 4d, we smply assume that f nterval [T, T ] s accepted as a crossng nterval, the barrer crossng occurs at the mdpont: (T T )/. Ths s only an approxmaton, but f r s small, a small msestmaton of the exact barrer crossng locaton wll have lttle effect on the dscounted value of the rebate.* Seres Expanson In the second approach we focus on approxmatng the ntegral appearng n Equaton (5), rather than evaluatng t usng a Monte Carlo samplng approach. The ntegral cannot be evaluated n closed form, unless r 0, n whch case t equals P. Snce r s usually small, a Taylor seres expanson n r, retanng a few terms, would be farly accurate (see the appendx for detals of the seres expanson). We agan proceed n a sequental way as n the unform samplng approach. Here are the detals of the algorthm:. For n to N, perform Monte Carlo runs as follows (steps -5):. Intalze DscPayoff(n) 0. Generate T by generatng the nterjump tmes (T T ) accordng to the gven densty (e.g., exponental). 3. For to K(K s the number of jumps that occur durng the lfe of the opton): a. Generate x(t ) from a Gaussan dstrbuton of mean x(t ) c(t T ) and standard devaton σ T. The ntal state s x(0) x(t 0 ). T b. Generate A accordng to the gven jump-sze dstrbuton. FALL 00 THE JOURNAL OF DERIVATIVES 49

8 c. Compute the post-jump value: x(t ) x(t ) A. 4. For to K : a. Compute P accordng to Equaton (6). b. Compute the ntegral accordng to Equatons (A-) or (A-) n the appendx. c. DscPayoff(n) DscPayoff(n) RJΠ P j j. d. If x(t ) ln H, then ext loop and perform another Monte Carlo cycle (steps -5). Note that n such a case, P j 0 for j, so the dscounted payoff for subsequent ranges wll be zero. e. If x(t ) ln H, then the jump crossed the barrer (ndex I as defned n Equaton (3) equals ). The payoff becomes: DscPayoff(n) DscPayoff(n) R exp ( rt I )Π P j j. Ext loop, and perform another Monte Carlo cycle (steps -5). f. If x(t ) > ln H, then examne the next nterval, that s, ncrement, and perform another teraton of step No crossng occurred durng the lfe of the opton. Ths means that all values of x(t );,, K and x(t );,, K are above the barrer level. The payoff s gven by: DscPayoff(n) DscPayoff(n) exp ( rt) K Π P j j max(exp(x(t)) X, 0). Perform another Monte Carlo cycle (steps -5). 6. If n N,.e., we have completed all cycles of the Monte Carlo smulaton, obtan the estmate for the opton prce: N DOC DscPayoff () n N n T J exp( rs) g ( s) ds T A Note on Estmaton Bas The unform samplng method provdes an unbased estmate of the opton prce. The reason s that DscPayoff(n) s drawn from the dstrbuton P( DscPayoff dxx( T), x( T ),,..., K ) where DscPayoff represents the true dscounted payoff (note that we generate s, and s along wth x(t) unquely determnes the dscounted payoff). Because T, x(t ), and x(t ) are generated accordng to ther jont-dstrbuton, we have E(Dscpayoff(n)) DscPayoff and hence t s an unbased estmate of the true opton prce. On the other hand, the mdpont approxmaton of the unform samplng approach ntroduces some bas. Also, the Taylor seres approach produces some bas because of seres truncaton. As we wll see n the smulaton experments, for these two approaches the bas s very small. IV. SIMULATION RESULTS We test the proposed algorthms on a number of examples. To obtan an dea of the comparable advantage of the proposed method, we also mplement the standard Monte Carlo procedure on these same examples. In the standard Monte Carlo approach, we dscretze tme fnely (let be the dscretzaton nterval). Startng from zero tme, we sequentally generate values of the rsk-neutralzed asset prce by smulatng the stochastc dfferental equaton forward, and n the process we generate the Wener ncrements from a normal dstrbuton. We smulate the standard method usng a varety of possble tme dscretzaton szes. We run MATLAB programs on a Pentum III 733 MHz computer: mllon Monte Carlo teratons for each method, except for the unform samplng method we run 0 mllon teratons. The reason for the dfference s that we know the unform samplng method has zero bas. We therefore use ts smulaton result to obtan the real value of the opton prce. Hence we use more runs to obtan a more accurate estmate. We measure the bas and the varance per teraton and the CPU tme per mllon teratons for each method. We also compute a measure that combnes speed and accuracy, by multplyng mean square (MS) error by the CPU tme per teraton. Example In Example we use parameter values as follows: S 0 50, X 55, H 45, R, r 0.05, σ 0.3, λ 8, σ A 0.05, µ A 0, and T year. Results are shown n Exhbt. 50 USING BROWNIAN BRIDGE FOR FAST SIMULATION OF JUMP-DIFFUSION PROCESSES AND BARRIER OPTIONS FALL 00

9 E XHIBIT Example Comparsons Std Dev CPU Tme MS Error* Bas Bas (per (per mllon CPU Method (absolute) (%) teraton) teratons) (per teraton) Standard Monte Carlo ,93.36 Standard Monte Carlo , Standard Monte Carlo , Unform Samplng , Unform Samplng Mdpont Approxmaton Seres Expanson , Standard devaton s per teraton, whle the CPU tme s per mllon teratons. The true value of the barrer opton s *Mean square error. E XHIBIT Example Comparsons Std Dev CPU Tme MS Error* Bas Bas (per (per mllon CPU Method (absolute) (%) teraton) teratons) (per teraton) Standard Monte Carlo , Standard Monte Carlo , Standard Monte Carlo , Unform Samplng Unform Samplng Mdpont Approxmaton Seres Expanson , Standard devaton s per teraton, whle the CPU tme s per mllon teratons. The true value of the barrer opton s *Mean square error. E XHIBIT 3 Example 3 Comparsons Std Dev CPU Tme MS Error* Bas Bas (per (per mllon CPU Method (absolute) (%) teraton) teratons) (per teraton) Standard Monte Carlo , Standard Monte Carlo , Standard Monte Carlo ,4.955 Unform Samplng Unform Samplng Mdpont Approxmaton Seres Expanson ,737.6 Standard devaton s per teraton, whle the CPU tme s per mllon teratons. The true value of the barrer opton s *Mean square error. FALL 00 THE JOURNAL OF DERIVATIVES 5

10 Example In Example, we use parameter values as follows: S 0 00, X 0, H 95, R, r 0.05, σ 0.5, λ, σ A 0., µ A 0, and T year. Results are shown n Exhbt. Example 3 In Example 3, we use parameter values as follows: S 0 00, X 0, H 85, R, r 0.05, σ 0.5, λ, σ A 0., µ A 0, and T year Results are shown n Exhbt 3. The standard devaton numbers are per teraton, so for M teratons we would dvde the standard devaton by M. One can see that the standard Monte Carlo method produces sgnfcant bas even as we shorten the tme-step sze at the expense of the CPU tme. Also note that the unform samplng method, n addton to producng zero bas, s sgnfcantly faster than the other two methods. For acceptable bas for the standard Monte Carlo method, t seems that one should choose For such a choce, the unform samplng method s typcally more than 00 tmes faster. The mdpont verson of the unform samplng approach s even a lttle faster, although at the expense of a lttle more bas. The seres expanson approach s also consstently better than the standard Monte Carlo method n terms of both bas and CPU tme. The bas n the seres expanson approach s due to approxmaton of the ntegral usng only up to the second term n r n the Taylor seres. A surprsng observaton here s that the standard devaton per teraton does not dffer much across methods. The unform samplng method produces only one pont per Monte Carlo path, whle the seres expanson method goes through great pans to compute expectatons. Its mprovement n standard devaton s not as great as we would have expected. V. CONCLUSION Obtanng effcent and accurate valuaton of exotc optons has been a major goal for academcans and fnancal nsttutons. Because of the lack of analytcal solutons, mprovng the computatonal methodology has been of sgnfcant mportance. As a step n ths drecton, we have presented a fast and low-bas approach to prcng downand-out barrer optons when the underlyng securty follows a jump-dffuson process and the barrer s contnuously montored. The use of the Brownan brdge sgnfcantly reduces bas and speeds convergence compared to the standard Monte Carlo approach. Another approach for approxmatng the frst-passage tme densty ntegral uses Taylor seres expanson. We show that the seres expanson approach s stll a sgnfcantly better approach than the standard Monte Carlo approach. In follow-up work we plan to examne the ssue of dscrete montorng of the barrer crossng, and ncorporate stochastc volatlty of the jump-dffuson process. APPENDIX Integral of the Frst-Passage Densty The ntegral T exp( rs) g ( s) ds T appearng n Equaton (5) cannot be evaluated n closed form, unless r 0, n whch case t wll equal P. Takng off from the fact that r s usually small (for example r 0.05), we propose an approxmaton of the ntegral by a Taylor seres expanson n r. Specfcally, we obtan the Laplace transform of the ntegral. Then we take the Taylor seres expanson of the Laplace transform n terms of r. We retan up to the second-order terms of r. For the case of r 0.05, for example, the thrd-order term s , whch s neglgble. We then take the nverse Laplace transform of the truncated seres. For smplcty, let us shft the tme axs by subtractng T. The ntegral becomes J() τ exp( rt ) exp( rt)ˆ g () t dt where τ (T T ), and g^ (t) s the tme-shfted verson of g (t). From prevous dervaton we can use Equaton () to get: exp( rt ) τ J() τ fth () ( τ tdt ) y 0 [ ] exp( rt) x( T ) lnh 3 (( xt H ct ft t ) ln ) () exp πσ tσ 0 (( xt ) ln Hct) ht () t exp πσ tσ τ 5 USING BROWNIAN BRIDGE FOR FAST SIMULATION OF JUMP-DIFFUSION PROCESSES AND BARRIER OPTIONS FALL 00

11 From Gradshteyn and Ryzhk [000], we have the Laplace transforms: and I at I t 3 Usng these, together wth the shft and convoluton theorems, we get the Laplace transform of J(τ): Js () exp( rt ) a exp π exp( t a s) a 4 > 0 a exp 4t c s exp xt ( ) xt ( ) [ xt ( ) lnh] σ σ c σ y [ xt ( ) lnh ] c exp s r σ σ c s r σ We take the Taylor seres expanson of J(s) n terms of r, and keep up to the second-order term of r. Then we take the nverse Laplace transform of the resultng seres. After some algebra and smplfcaton, the fnal expresson for J turns out to be: For the case that x(t ) > ln H: rxt ( ) lnh exp( rt rt) ( A CB) 8σ For the case that x(t ) ln H: [ ] [ ] xt ( ) ln H xt ( ) lnh J() τ exp( rt )exp τσ [ ] [ ] rxt ( ) lnh exp( rt rt) J() τ exp( rt ) 8σ exp( a s) π s (A-) ( ) A CB (A-) where A C r xt ( H xt H [ xt xt ] ) ln ( ) ln σ τ ( ) ( ) exp τσ πτ where Φ(z) s the cumulatve normal dstrbuton: z y Φ( z) exp dy π The accuracy of the approxmaton has been verfed usng a number of numercal examples. ENDNOTES The authors are grateful for helpful comments provded by Mark Broade, Darrell Duffe, Paul Glasserman, and Xaolu Wang. They also thank Jun Lu and Jun Pan for frutful dscussons and the referees for ther careful, crtcal, and constructve comments. * The mdpont approxmaton approach was suggested by Paul Glasserman. REFERENCES [ ] [ ] xt ( xt ) ( ) xt ( ) xt ( ) lnh exp Φ τσ τσ [ ] [ ] [ ] r B 8 rτ xt ( ) xt ( ) xt ( ) xt ( ) lnh σ rτ A [ xt ( ) xt ( ) lnh σ ] [ xt ( ) ( ) ] xt ( ) ( ) C πτ exp xt xt Φ τσ τσ Andersen, L., and R. Brotherton-Ratclffe. Exact Exotcs. Rsk, Vol. 9 (October 996), pp Atya, A. A Fast Monte Carlo Algorthm for the Level- Crossng Problem For Jump-Dffuson Processes. Workng paper, Calforna Insttute of Technology, 000. Beaglehole, D.R., P.H. Dybvg, and G. Zhou. Gong to Extremes: Correctng Smulaton Bas n Exotc Opton Valuaton. Fnancal Analysts Journal, January/February 997, pp FALL 00 THE JOURNAL OF DERIVATIVES 53

12 Blake, I., and W. Lndsey. Level-Crossng Problems for Random Processes. IEEE Transactons Informaton Theory, Vol. IT-9 (973), pp Boyarchenko, S., and S. Levendorsk. Barrer Optons and Touch-and-out Optons under Regular Levy Processes of Exponental Type. Research Report 45, MaPhySto, Unversty of Aarhus, 000. Broade, M., P. Glasserman, and S. Kou. A Contnuty Correcton For Dscrete Barrer Optons. Mathematcal Fnance, Vol. 7, No. 4 (October 997), pp Duffe, D. Dynamc Asset Prcng Theory, 3rd ed. Prnceton: Prnceton Unversty Press, 00. Duffe, D., and D. Lando. Term Structure of Credt Spreads wth Incomplete Accountng Informaton. Econometrca, Vol. 69, No. 3 (May 00), pp El Babsr, M., and G. Noel. Smulatng Path-Dependent Optons: A New Approach. The Journal of Dervatves, Wnter 998, pp Feller, W. An Introducton to Probablty Theory and ts Applcatons, Vol.. New York: Wley, 968. Gradshteyn, I.S., and I.M. Ryzhk. Table of Integrals, Seres, and Products. San Dego: Academc Press, 000. Harrson, J.M. Brownan Moton and Stochastc Flow Systems. New York: Wley, 985. Hull, J. Optons, Futures, and Other Dervatves, 3rd ed. Englewood Clffs, NJ: Prentce-Hall, 997. Metver, M. Semmartngales. Berln: Walter de Gruyter, 98. Mordeck, E. Optmal Stoppng and Perpetual Optons for Levy Processes. Workng paper, Unversty of the Republc of Uruguay, 000. Nahum, E. On The Prcng of Lookback Optons. Ph.D. thess, Unversty of Calforna, Berkeley, Sprng 999. Pham, H. Hedgng and Optmzaton Problems n Contnuous-Tme Fnancal Models. Lecture presented at ISFMA Symposum on Mathematcal Fnance, Fudan Unversty, August Optmal Stoppng, Free Boundary and Amercan Optons n a Jump Dffuson Model. Appled Mathematcs and Optmzaton, 35 (997), pp Revuz, D., and M. Yor. Contnuous Martngales and Brownan Moton. New York: Sprnger-Verlag, 994. Rogers, L.C.G., and D. Wllams. Dffusons, Markov Processes and Martngales, Vol.. New York: Wley, 994. Trautmann, S., and M. Benert. Stock Prce Jumps and Ther Impact on Opton Valuaton. Workng paper, Johannes Gutenberg-Unverstat Manz, Germany, 995. Wlmott, P. Dervatves The Theory and Practce of Fnancal Engneerng. New York: Wley, 998. To order reprnts of ths artcle please contact Ajan Malk at amalk@journals.com or Jacod, J., and A. Shryayev. Lmt Theorems for Stochastc Processes. New York: Sprnger-Verlag, 987. Karatzas, I., and S. Shreve. Brownan Moton and Stochastc Calculus. New York: Sprnger-Verlag, 99. Kou, S.G., and H. Wang. Frst Passage Tmes of a Jump Dffuson Process. Workng paper, Columba Unversty, December 000. Last, G., and A. Brandt. Marked Pont Processes on the Real Lne The Dynamc Approach. New York: Sprnger-Verlag, 995. Merton, R.C. Opton Prcng when the Underlyng Stock Returns are Dscontnuous. Journal of Fnancal Economcs, 3 (976), pp USING BROWNIAN BRIDGE FOR FAST SIMULATION OF JUMP-DIFFUSION PROCESSES AND BARRIER OPTIONS FALL 00