CONDITIONING ON ONE-STEP SURVIVAL FOR BARRIER OPTION SIMULATIONS

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1 CONDITIONING ON ONE-STEP SURVIVAL FOR BARRIER OPTION SIMULATIONS PAUL GLASSERMAN Graduate School of Busness, Columba Unversty, New York, New York 10027, JEREMY STAUM 226 Rhodes Hall, Cornell Unversty, Ithaca, New York 14853, Prcng fnancal optons often requres Monte Carlo methods. One partcular case s that of barrer optons, whose payoff may be zero dependng on whether or not an underlyng asset crosses a barrer durng the lfe of the opton. Ths paper develops varance reducton technques that take advantage of the specal structure of barrer optons, and are approprate for general smulaton problems wth smlar structure. We use a change of measure at each step of the smulaton to reduce the varance arsng from the possblty of a barrer crossng at each montorng date. The paper detals the theoretcal underpnnngs of ths method, and evaluates alternatve mplementatons when exact dstrbutons condtonal on one-step survval are avalable and when not avalable. When these one-step condtonal dstrbutons are unavalable, we ntroduce algorthms that combne change of measure and estmaton of condtonal probabltes smultaneously. The methods proposed are more generally applcable to termnal reward problems on Markov processes wth absorbng states. 1. INTRODUCTION Barrer optons are dervatve securtes wth the defnng characterstc that the payoff may be zero, dependng on whether or not an underlyng varable crosses a specfed barrer durng the lfe of the opton. There are two broad types of barrer optons: knock-out optons, whch pay zero when there s a barrer crossng, and knock-n optons, whch pay zero unless there s a barrer crossng. A barrer opton s cheaper than the equvalent opton wthout a barrer, because t may expre worthless f knocked out (or not knocked n) n the same stuaton n whch the standard opton would have pad off. Perhaps because of ths property, barrer features are frequently ncorporated nto opton contracts on many dfferent types of underlyng assets. To prce an opton s to evaluate the ntegral of ts expected dscounted payoff under a rsk-neutral probablty measure. (See, e.g., Duffe 1996 or Hull 1993 for background on opton prcng.) In the case of barrer optons, ths payoff s dscontnuous over the space of all paths of the underlyng varables. In suffcently smple cases, there are analytcal formulas for the prce (e.g., Merton 1973, Kuntomo and Ikeda 1992, Rubnsten and Rener 1991, and Sdenus 1998). However, there wll not be useful formulas f the specfcaton of the barrer or stochastc processes used to model the underlyng varables s too complex or hgh dmensonal. Consequently, t s often necessary to prce va smulaton, whch s better suted to hgh-dmensonal and path-dependent problems than other numercal methods, and has the added beneft of provdng the estmate s standard error. A straghtforward smulaton proceeds by dvdng the lfetme of the opton nto several tme steps. Each path begns wth the state vector at a specfed ntal value, and uses an approxmate, dscretzed verson of the dynamcs to propagate ths random vector forward at each tme step. A smulaton for a knock-out opton has the specal feature that f at any tme the underlyng process should cross the barrer, the path may be mmedately abandoned, because t s already known that t results n a zero payoff. If a path survves by never crossng the barrer, then ts payoff s determned at the termnal value of the state vector. Ths standard Monte Carlo approach to prcng knock-out optons suffers from a pecular defect whch creates a possblty for mprovng the method: Some smulated paths survve, allowng for a postve payoff, whle other paths fal to survve and have zero payoff. The lower the probablty of survval, the more smulated payoffs are zero. Ths can make the average payoff among the survvng paths, and hence the varance among all paths, qute large relatve to the prce. All the varance due to the possblty of knock-out could be removed by mportance samplng f t were possble to use the condtonal dstrbuton of the state vector gven survval to maturty. However, barrer opton smulaton s challengng precsely because t s generally mpossble to sample condtonal on fnal survval. It becomes necessary to smulate an entre dscrete path wth observatons at each barrer-montorng pont to ascertan whether or not survval occurs. However, there are other possbltes for samplng measures, whch may result n more effcent estmators. Subject classfcatons: Smulaton, effcency: Varance reducton. Fnance, asset prcng: Computatonal methods. Area of revew: Smulaton X/01/ $ electronc ISSN 923 Operatons Research 2001 INFORMS Vol. 49, No. 6, November December 2001, pp

2 924 / Glasserman and Staum One technque presented here s to use the condtonal dstrbuton gven one-step survval, whch may be avalable, even though the condtonal dstrbuton gven fnal survval s not. By samplng condtonal on survval at each step, t s possble to ensure that all smulated paths survve and yeld nformaton about potentally postve payoffs. Then t s necessary to ncorporate a lkelhood rato to make up for the absence of paths whch get knocked out. The result s an unbased estmator of the opton prce wth reduced varance. The dea of usng a condtonal dstrbuton at each step n a smulaton was nvestgated by Glynn and Inglehart (1988, 11) n a dfferent context estmatng the steady-state mean of a real-valued Markov chan. Glasserman (1993) analyzes a contnuous-tme verson of ths dea. An example of usng one-step condtonal dstrbutons n barrer opton smulatons appears n Boyle et al. (1997). Independent of our work, Ross and Shanthkumar (1999) combne two examples from Boyle et al. (1997) to arrve at an estmator smlar to the smplest one consdered here; they do not consder the case of unknown condtonal probabltes for whch we propose several alternatve estmators. For other smulaton methods appled to prcng path-dependent optons, see, e.g., Duan and Smonato (1998), Glasserman et al. (1999a), Joy et al. (1996), Lemeux and L Ecuyer (1998), and Vázquez-Abad and Dufresne (1998). We propose and analyze a varety of estmators that can be dvded nto two broad categores defned by whether or not the estmator requres explct knowledge of the dstrbuton of the underlyng process condtonal on one-step survval. Ths condtonal dstrbuton enters the problem n two ways through samplng condtonal on survval and through evaluaton of a lkelhood rato. The second role turns out to be the more fundamental one; for settngs n whch the condtonal dstrbuton s unknown, we formulate methods whch mplement a change of measure by estmatng the requred lkelhood rato. The technques we study are most natural for dscretely montored barrer optons, whch are knocked out only f the barrer s crossed on specfed montorng dates. In ths case, there s an obvous way to fx the tme steps of the dscretzaton. However, we show that these methods also workn the case of contnous montorng, wth an arbtrarly chosen dscretzaton. Other extensons addressed n ths paper are knock-n optons and rebates, whch are pad f the opton s knocked out. For all of these stuatons where the central dea of condtonng on survval over one step s feasble, we propose algorthms and assocated estmators n 2. In 3, we analyze ther propertes, whle deferrng any mathematcal detals whch would detract from the flow of the argument untl Appendx A. Secton 4 reports numercal results for the performance of these algorthms for varous knock-out optons, whle 5 presents conclusons. 2. ESTIMATORS AND ALGORITHMS 2.1. The Mathematcal Problem We treat the followng problem n prcng a knock-out opton. There s a stochastc model of the evoluton of a state vector S t t 0 T of underlyng fnancal varables. We assume that S t has the Markov property. The state vector must be defned so that, f the opton has not been knocked out, the termnal value S T determnes the dscounted payoff f S T. For example, n a model wth stochastc nterest rates, t wll be necessary to nclude the accumulated dscount factor (whch s tself stochastc) as part of the underlyng process. For an Asan opton (whose payoff depends on the average level of the underlyng asset), the state vector must nclude the runnng sum, as well as the current prce of the underlyng asset. Also, we deal only wth European-style optons, whch can be exercsed only at maturty. To condense notaton, from now on we wll wrte the state vector at the montorng dates as S 1 S m (abbrevatng the more explct S t1 S tm ), and assume that the tme between consecutve montorng dates t +1 t s a constant t. Because of the knock-out feature, the state vector S takes a value n R d, where s an absorbng state. If S crosses the barrer at tme, the opton s knocked out, and for all j, S j =. Defne A to be the ndcator functon 1 S ; that s, A = 1 means that the opton s alve at tme t. The structure of ths problem s by no means lmted to barrer optons. It apples to any smulaton of expected termnal reward wth the feature that there s a known payoff when the state vector exts a specfed regon of the state space. We now gve three motvatng examples to whch we wll return later as well. Example 1. Black-Scholes Model One example s a one-dmensonal Black-Scholes model. The state vector s a sngle stockprce, governed by the dynamcs ds t = dt + dw S t t where W t s standard Brownan moton, and and are constants. Ths has exact dscretzaton S +1 = S exp (( 12 ) 2 t + ) t Z (1) where Z 1 Z m 1 are..d. standard normal. The barrer s a prce level H<S 0, so A = 1 f the stock prce has not crossed beneath the barrer H by step. A down-and-out call n ths model has dscounted payoff A m exp rt S m K +, where K s the strke prce (.e., the prce at whch the holder of the call opton may buy the stock), r the constant nterest rate, and T = t m the maturty. A down-and-out bnary call s dscounted payoff s A m exp rt 1 S m K. These examples can be prced n closed form usng the results of Merton (1973), but numercal methods become necessary when the barrer s tmevaryng or the parameters and are stochastc.

3 Example 2. Two-Dmensonal Geometrc Brownan Moton In ths example, the state vector contans two stockprces, obeyng dynamcs ds 1 t S 1 t ds 2 t S 2 t = 1 dt + 1 dw 1 t = 2 dt + 2 dw 2 t where W t = W 1 t W 2 t s two-dmensonal Brownan moton wth zero drft and covarance matrx 1 1 The dscretzed dynamcs are (( S 1 +1 = S 1 exp 1 1 ) t + 1 ) t Z 1 (( S 2 +1 = S 2 exp 2 1 ) t + 2 ) t Z 2 where Z 1 Z 2 = Y 1 = Y Y 2 and all Y are..d. standard normal. In ths settng, the barrer crossng may be determned by one asset and the fnal payoff by the other. For example, the dscounted payoff may be e rt S 2 m K + 1 mn m S 1 H Heynen and Kat (1994) have provded closed-form solutons for some such optons, but not, for nstance, where the barrer s a functon of tme more complcated than an exponental. Example 3. LIBOR Market Model Our fnal motvatng example s ndcatve of a more complex class of models n wdespread use for prcng nterest rate dervatve securtes. In ths settng, the underlyng state vector records nterest rates rather than asset prces. We descrbe a model of London Inter-BankOffered Rates (LIBOR), a key benchmarkfor prcng. For further background, see Musela and Rutkowsk (1997, 14.3). Gven a set of maturtes 0 <t 1 <t 2 < <t N +1, let F k t denote the forward nterest rate for the perod t k t k+1 as of tme t<t k. Ths s the nterest rate for the perod t k t k+1 that can be locked n at tme t. In a model wth proportonal volatltes, the dynamcs of the vector of forward rates take the form df k t = k F t t F k t (2) dt + F k t k t dw t k= 1 N Here, W t s a d-dmensonal standard Brownan moton (d N ), and k t s the kth row of an N d determnstc matrx t, expressng the nstantaneous dependence Glasserman and Staum / 925 of F k t on the d components of the Brownan moton. Each k, k = 1 N, s a determnstc functon of the vector of rates F t and the volatltes t. The specfcaton of k s fully determned by the requrement that the model be arbtrage-free, as explaned by Musela and Rutkowsk (1997, Equaton 14.55). The choce of t determnes the nstantaneous covarance t t. Usng an Euler dscretzaton of the logarthm of the forward rates, where the tme subscrpt as usual means t, the dscretzed dynamcs are ( ln F k k +1 = ln F + k F 1 ) 2 k 2 t + C k Z t (3) where Z s a vector of d ndependent standard normals and C k s the kth row of an N d matrx C whch satsfes C C =. The choce C = s always avalable, but we wll see that flexblty n the choce of C can be useful. Many types of nterest rate optons wth barrer features are commonly traded. We wll consder barrer swaptons n partcular. These are optons that are knocked out f some functon of the forward LIBOR rates crosses a barrer, wth a payoff that s essentally a call or a put on a swap whch s an agreement to exchange nterest payments at floatng (varable) LIBOR rates for payments at a fxed rate. The swap rate whch makes the ntal value of the swap zero s tself a complcated functon of the forward LIBOR rates. As n Musela and Rutkowsk (1997, Equaton 16.5), 1 B = M t M k=m+1 B (4) k s the swap rate at tme t m for a swap wth payment dates t m+1 to t M. Here, B s the prce (at tme t m ) of a bond payng $1 at tme t and s gven by B = 1 1 (5) k=m 1 + tfm k.e., by dscountng $1 at the forward LIBOR rates. A swapton on $1 of notonal prncpal has a tme t m payoff of ( ) M t B k max 0 K (6) k=m+1 where K s the strke rate, and the summaton s the tme t m value of a bond payng $1 at tmes t m+1 to t M. The key features of ths example for our purposes are that the underlyng model dynamcs are farly complex, a typcal payoff s a complcated functon of the state vector, and a barrer may be mposed on a nonlnear functon of the state vector (e.g., the swap rate). Before analyzng these examples, recall how a standard smulaton proceeds, as descrbed n the ntroducton. In any of the examples above, a standard Monte Carlo algorthm smulates S 1 S m, where S s a vector of

4 926 / Glasserman and Staum underlyng prces (or forward nterest rates) at the th dscretzaton tme. It proceeds n the usual fashon by generatng S +1 from S accordng to the law of the underlyng process, or perhaps an approxmate dscretzaton thereof. Then t uses ths path to evaluate the followng estmator. The estmated prce s the average of X m over multple paths. Standard Estmator. X m = Am f S m (7) 2.2. Known Transton Probabltes An alternatve to the standard estmator generates S +1 from S condtonal on A +1 = 1, f ths one-step condtonal dstrbuton s known. Of course, under ths scheme, A m = 1 always, and the average of f S m may be terrbly based. For ths reason, t s necessary to weght by a lkelhood rato. (See, e.g., Bratley et al for background on mportance samplng.) Defne 1 L = p S j (8) where j=0 p s = P S j+1 S j = s (9) We adopt the conventon that p S j = 0fA j = 0, because there s no chance of survvng to the next step f the barrer has already been crossed. (Ths formulaton s general enough to allow the probablty of one-step survval to depend on j because the tme ndex can be ncorporated n the state vector.) So L s the lkelhood of survvng steps va ths path, n a sense to be made precse n 3. Then the new estmator s: Exact Estmator wth Full Importance Samplng. X m = Lm f S m (10) Theoretcal propertes of ths and all subsequent estmators are presented n 3. In partcular, we show that ths estmator s unbased and has lower varance than the standard estmator. Example 1 contnued. In the example of prcng a downand-out call under the Black-Scholes model, t s easy to sample condtonal on one-step survval. An algorthm to sample S +1 uncondtonally mplements Equaton (1) by S +1 = S exp (( 12 ) 2 t + ) t 1 U (11) where U s unformly dstrbuted and s the standard normal cdf. It s easy to evaluate 1 numercally, as shown by Marsagla et al. (1994). Samplng condtonal on one-step survval uses the same equaton, except that U = 1 p S + Vp S (12) where V s unformly dstrbuted and p S =P S +1 H S (( ( ) S = ln + ( 12 ) /( t) H ) ) 2 t (13) The result s that U s unformly dstrbuted condtonal on beng at least as large as necessary to prevent knockout. That s, gven S and gven S +1 H, 1 U has the dstrbuton of ln S +1 ln S 2 /2 t / t. It s also possble (and perhaps faster) to sample from the tal of the normal dstrbuton usng acceptance-rejecton rather than 1 ; see Fshman (1996). The logarthmc evaluaton n (13) s easly avoded by storng the exponent n (11). Implementaton of (13) does ental the overhead of evaluatng a cumulatve normal probablty; fast approxmatons to are ncluded n many mathematcal software lbrares. Example 2 contnued. In the settng of Equaton (2), samplng condtonal on one-step survval works as follows: ( S 1 +1 = S 1 exp 1 t + 1 ) t 1 U 1 ( S 2 +1 = S 2 exp 2 t + 2 t 1 U 1 + ) U 2 where U 1 = 1 p S + V 1 p S U 2 = V 2 and V 1 V 2 are unformly dstrbuted and ndependent. The probablty of one-step survval p S s exactly as n Equaton (13), but wth S 1, 1, and 1 for S,, and, because the barrer condton nvolves only the frst asset prce. Example 3 contnued. Suppose that the barrer s a floor beneath the current LIBOR rate,.e., the opton s knocked out at step + 1fF s beneath the barrer. It s easest to mplement Equaton (3) by choosng the matrx C such that F depends on a sngle component of the drvng Brownan moton. That s, pck C +1 such that t has the form 0 0, e.g., through Cholesky factorzaton. (All rows of C wth ndex or less are zero, because these LIBOR rates refer to maturtes already n the past.) Much as n Example 2, frst smulate the current LIBOR rate condtonal on ts beng above the floor, and then smulate all forward LIBOR rates condtonal on ths value of the current LIBOR rate. For the current LIBOR rate, ln F =ln F +1 ( + +1 F ) t 2 + tc U 1

5 and the probablty of survvng one step s ( ( p F = ln F +1 /H + +1 F In general, for k>, ) t /( t ) ) ( ln F k k +1 = ln F + k F 1 ) 2 k 2 t + t d j=1 C k j 1 U j where j ndexes the components of the Brownan moton: U 1 U j = 1 p S + V 1 p S = V j j = 2 d and V 1 V d are unformly dstrbuted and ndependent. Note that n ths example, choosng a dfferent square root of the covarance matrx would have made t dffcult to fnd the probablty of survvng one step. An nherently dffcult case s when the barrer s a floor beneath more than one forward LIBOR rate smultaneously. The dstrbuton for the mnmum among those rates s nconvenent, regardless of the square root of the covarance matrx, and t s awkward to fnd the probablty of survval analytcally. If the barrer s a floor beneath the forward swap rate tself, ths s such a complcated functon of forward LIBOR rates that t s effectvely mpossble to determne the probablty of survval. Smlar to Equaton (4), the forward swap rate from tme t m to tme t M s m M = B m B M t (14) M k=m+1 B k (See Musela and Rutkowsk 1997, ) Ths example s featured n the next subsecton, whch develops methods for use when transton probabltes are unknown Unknown Transton Probabltes When the condtonal probablty of one-step survval p S s not known, the technque of the prevous subsecton s not applcable. However, as long as p S >0, t s always possble to sample condtonal on one-step survval by generatng uncondtonal successors to S, and keepng the frst one to survve. Ths stll leaves the problem of evaluatng the lkelhood rato when p S s unknown, a problem of potentally broader scope. To address t, we estmate p S n the smulaton tself. Ths s made possble by the observaton that the number of uncondtonal successors requred to generate a survvor (the watng tme) s geometrcally dstrbuted wth unknown parameter p S. Therefore, one Glasserman and Staum / 927 can try to estmate p S from data observed n the course of smulaton, and use the approach of 2.2 wth an estmated, nstead of an exact, lkelhood. A problem swftly emerges n the form of an rrtatng conundrum n elementary statstcs: Gven a sngle observaton, there s only one unbased estmator of the parameter of a geometrc dstrbuton, and t s perfectly useless for our purposes. If Y s the observed geometrc watng tme, then the only unbased estmator s the ndcator functon whch s 1 when Y = 1 and 0 otherwse (e.g., Cox and Hnkley 1974, p. 253). That s, f the frst uncondtonal successor to S survves, estmate that p S s 1, but f t does not survve, estmate that p S s 0. Of course, pluggng ths estmator nto the formula for L m results n A m, because the only values are 0 and 1, reducng the method to standard smulaton. However, the problem s not ntractable, because there s a good unbased estmator when more than one observaton s avalable. An algorthm ncorporatng t s ths: 1. Produce uncondtonal successors to S untl r of them survve; 2. Call the total watng tme Y, whch has a negatve bnomal dstrbuton; 3. Let S +1 be any of the r survvng successors; 4. Estmate p S j by r 1 Y 1 Replacng L m by an estmated lkelhood rato, a new estmator s then: Negatve Bnomal Estmator. m 1 r 1 f S m (15) =0 Y 1 Ths estmaton algorthm would clearly be less effcent than standard smulaton f there were no chance of knockout, because t would produce r>1 survvng successors at each step, then throw out r 1 of them. As knock-out gets more lkely, the standard method grows neffcent n that t does too much of ts workn generatng successors at early steps, because most paths termnate early. The standard method wastes some of the value of early successors n that early termnaton of a path means that the random varates already generated provde no nformaton about the fnal payoff or the probablty of survvng later steps. When the probablty of knock-out s hgh enough and the number of steps m s large enough, the negatve bnomal estmator wll outperform the standard estmator. A potental drawbackof ths negatve bnomal method s that there s no upper bound on the amount of tme t mght take to sample a survvng successor from the uncondtonal dstrbuton. In order to avod ths dffculty, we can sample a fxed number n of successors from the uncondtonal dstrbuton, and record the number N j whch survve, whch s bnomally dstrbuted. Then the estmate of p S j

6 928 / Glasserman and Staum s N j /n, but the lkelhood rato need not have the same form as before. It s now possble for a path not to survve, f at one step none of the n successors generated survve. In ths sense, the algorthm takes only partal advantage of mportance samplng. Ths requres a new estmator: Exact Estmator wth Partal Importance Samplng. X n m where L n = A m L n m f S m (16) = 1 j=0 p S j (17) 1 1 p S j n because 1 1 p S j n s the probablty of fndng at least one survvng successor n n attempts; 3 provdes a theoretcal explanaton of ths pont. However, the p S j are not known, so L must be estmated. One natural possblty s: Bnomal Estmator. m 1 A m f S m j=0 N j (18) n The product n (18) s an emprcal counterpart of the lkelhood rato; as shown n 3, t results n an unbased estmator. Ths bnomal estmator s a specal case of a more general class of estmators based on rght-censored geometrc random varables. The nsght s that for a large maxmum computatonal budget of n at step, fp S s not too small, t mght be neffcent (n terms of varance reducton for a fxed amount of work) to spend tme estmatng p S extremely precsely rather than to move on, savng the budget for smulatng more paths. For nstance, each step could nvolve r trals whch end after one successor survves or N = n/r successors fal to survve, whchever comes frst. Then the length of the tral s a rght-censored geometrc random varable, and the bnomal estmator s the specal case of N = 1 and r = n. Such a censored geometrc estmator s consstent but s not compettve because t s based at low values of r or N and does not mprove effcency for large r and N. For detals, see Staum (2001). The bnomal estmator (18) bears some resemblence to splttng or RESTART estmators consdered n, e.g., Vllén-Altamrano and Vllén-Altamrano (1994) and Glasserman et al. (1999b) for estmatng rare event probabltes. However, n splttng algorthms all survvng paths are smulated, whereas n (18), multple survvors are used to estmate the one-step survval probablty but smulaton contnues for just one survvng path Extenson: Contnuous Montorng Ths frameworkcan also apply to contnuously montored barrer optons. In ths case, the dates t 1 t m are merely for purposes of dscretzaton, and at step, the opton s knocked out not only f S t = S s on the wrong sde of the barrer, but f S t crossed the barrer at any tme t n the nterval t 1 t. In general, the knock-out condton can be expressed as an nequalty b S <H, where b s a functon of the underlyng asset prce. Then augment the state vector S to be S M, where M = mnt t 1 t b S t. If the jont dstrbuton of S and M s known, then the dscretzed smulaton can effectvely montor the barrer contnuously. Ths s also possble f b s a vector-valued functon, and the mnmum s taken coordnatewse. References for smulaton of barrer optons wth contnuous montorng are Andersen and Brotherton-Ratclffe (1996), Bald et al. (1999), and Beaglehole et al. (1997). Asmussen et al. (1995) consdered the related problem of smulatng the maxmum of Brownan moton. If the condtonal dstrbuton of M +1 gven S s not known, then the only choce s to use the methods of the prevous subsecton where transton probabltes are unknown, and the state vector s now defned to be the par S M. However, f ths condtonal dstrbuton s known, t would be desrable to mplement condtonng on one-step survval exactly, by smulatng M +1 gven S and the event A +1 = 1, then smulatng S +1 gven S and M +1. Unfortunately, ths s often not practcal. The dffculty arses n the latter step; often the dstrbuton of S +1 condtonal on S and M +1 s unknown. For nstance, even the case of one-dmensonal Brownan moton wth nonzero drft s complcated, because t matters at what tme the mnmum was acheved. Instead, take advantage once more of the trck of smulatng condtonal on survval by repeated uncondtonal samplng. To do ths, t s only necessary to know the margnal dstrbutons of S +1 and M +1, and the dstrbuton of M +1 condtonal on S and S +1. Repeat the followng process untl t succeeds n producng S Generate S +1 condtonal on S and b S +1 >H. 2. Generate M +1 condtonal on S and S If M +1 >H, accept ths value of S Compute p S from the margnal dstrbuton of M +1. In Example 1, for one-dmensonal Brownan moton wth drft, mplement step 1 exactly as specfed by Equatons (11) and (12) for samplng condtonal on survval n the dscrete case. Step 2 reduces to samplng the mnmum of a Brownan brdge, for whch the orgnal drft has become rrelevant. From Karatzas and Shreve (1991, ), we get P M +1 x S S +1 = exp ( ( ) 2 x 2 t ln ln S ( S+1 x )) (19)

7 To generate M +1, nvert ths cdf, and evaluate at a unformly dstrbuted random varable U : ( ( 1 M +1 =exp ln S 2 S +1 ( ( )) 2 S ln 2 tln U )) S 2 (20) +1 Ths s smlar to a result of Asmussen et al. (1995, 4.5). Also, the margnal dstrbuton of M +1 (condtonal on S, but not on S +1 ) s nverse Gaussan, as n Corollary B.3.4 of Musela and Rutkowsk (1997). Evaluatng t at H, p S = P M +1 >H S ( ) ( ln S t H 2 ln S = exp ) H t 2 ( ) ln S t H (21) t The procedure functons smlarly for Examples 2 and 3, because a sngle dmenson of the underlyng Brownan moton determnes the barrer crossng. In these cases, use the above method to smulate ths sngle component, then sample the rest of the state vector condtonal on t Extenson: Rebates Knock-out optons are sometmes wrtten so that the buyer receves a rebate f the opton s knocked out. Dependng on the contract specfcaton, ths rebate can be payable ether at maturty or at the tme of knock-out. The technques developed n ths paper are well suted to handlng rebates payable at knock-out, but treatment of the topc s lmted to ths subsecton n order to lghten the burden of notaton elsewhere. The only further assumpton necessary s that the present value of the rebate payable f knock-out occurs at tme t be a functon of the state vector at tme t 1. For rebates pad at knock-out, ths assumpton s not very restrctve. The present value of the rebate s the product of the nomnal value (the amount pad) and a dscount factor. The nomnal value of the rebate s generally constant, and at step 1 both the dscount factor up to tme t 1 and the nterest rate r 1 for the nterval t 1 t are known. The dscrete dynamcs of the dscount factor D are D = D 1 exp r 1 t, sod s known at t 1. Wrte the present value of the rebate earned at tme t as g S 1. Then the standard estmator, the realzed value of the opton on a smulated path, s X m = A m f S m + A 1 A g S 1 (22) =1 The expresson A 1 A s an ndcator functon whch s one when the opton s knocked out at step. Ths formula Glasserman and Staum / 929 also holds f g S 1 s the expected present value of the rebate at step. The exact estmator wth full mportance samplng defned n Equaton (10) s now X m = L m f S m + L 1 1 p S 1 g S 1 (23) =1 Rebates payable at maturty do not n general ft ths frameworkbecause the dscountng nvolves nterest rates condtonal on knock-out. (If dscountng s not stochastc or s ndependent of the rest of the process, ths s not an objecton.) To handle a knock-out opton wth rebate payable at maturty, decompose t nto the sum of an ordnary knock-out opton and a bnary knock-n opton whch knocks n and pays the rebate at maturty precsely when the other knocks out. The next subsecton treats knock-n optons Extenson: Knock-In Optons Dealng wth knock-n optons s not so smple, but s possble f there s a known expresson f S for the present value of a barrerless opton, receved at tme t when the state vector s S, whose payoff wll be f S m at tme t m. Ths s the case for suffcently smple knock-n optons, and n partcular for valung rebates payable at maturty as dscussed n the prevous subsecton. In that case, f S = rd B t m, where r s the nomnal rebate amount and B t m the prce at tme t of $1 pad at tme t m. We contnue to use A to mean the ndcator functon whch s one f the barrer has not been crossed by tme t, whch n ths stuaton means that the opton has not yet been knocked n. As before, p S s the probablty of not crossng the barrer over ths step. Then, a dfferent standard-type estmator takng advantage of the prncple that a knocked-n barrer opton s effectvely transformed nto a barrerless opton s X m = A 1 A f S (24) =1 To take advantage of full mportance samplng, t s necessary to complcate the samplng scheme. At each step there must be two successors to S. Smulatng condtonal on no knock-n produces S +1, whle smulatng condtonal on knock-n produces S+1. The S +1 are not part of the path, but are used to estmate the value of the opton should t be knocked n at step + 1, snce n ths case t s not realstc to expect that ths value should be known at tme. The estmator s: X m = =1 L 1 1 p S 1 f S (25) 3. PROPERTIES OF THE ESTIMATORS Ths secton wll explore statstcal propertes, such as bas, varance, and consstency of the estmators whch the prevous secton dscussed. We defer proofs to an appendx.

8 930 / Glasserman and Staum Precse statements of some of the propertes of the estmators wll requre a dscusson of probablty measures as they relate to varous smulaton algorthms. Let P be the measure under whch the underlyng state vector process S 1 S m has ts usual jont dstrbuton on a unverse called. In a standard Monte Carlo smulaton, the smulated prce vectors obey the law of P. Let be the subset of on whch the prevously defned ndcator functon A equals 1,.e., where the opton s alve at tme, and let F be the sgma-algebra generated by S 1 S. Then P s defned on m relatve to F m through the condtonal dstrbutons P S +1 Q S = P S +1 Q S A +1 = 1 (26) where S 0 = s 0 s fxed. Smulatng under P means samplng the next state vector condtonal on survval at the next step. Of course, ths mples that A m = 1 wth probablty 1 under P;.e., all paths smulated under P survve untl the end. Our frst result shows that t s possble to compensate wth a lkelhood rato. Let Ê denote expectaton wth respect to P. Lemma 1. L, defned n (8), s the lkelhood rato relatng the measures P and the restrcton of P to as follows: Ê L Y = E A Y (27) for any F -measurable functon Y such that the expectaton E A Y exsts and s fnte. Ths s useful because we can defne the expected payoff at step n the smulaton, whch s an F -measurable functon: X = E Xm F X = Ê X m F These are not observable durng the smulaton, except when = m, whch corresponds to the fnal estmator. (Recall that X m and X m are estmators defned n (7) and (10), respectvely.) Throughout ths secton, we also requre the nonrestrctve techncal condton E X 2 m < (28) n order to allow the use of Lemma 1 and ensure that varances exst. Lemma 2. The Condton (28) mples Ê X m 2 < and m E A X <. Under ths assumpton, a drect consequence of Lemma 1s Theorem 1. The estmator X m defned n Equaton (10) s unbased. That s, Ê X m = Ê L m f S m = E A m f S m = E X m whch equals the prce of the opton. Note that whle X m s defned as an exact estmator wth full mportance samplng, we are not n the usual settng of mportance samplng, because n nondegenerate examples, t s not the probablty measure P but only ts restrcton to m whch s absolutely contnuous wth respect to P. (A smlar lackof absolute contnuty arses n other applcatons of mportance samplng; see, e.g., Asmussen 1987, 14.7, and Glynn and Iglehart 1989.) A smple extenson of Theorem 1 s: Theorem 2. The estmator for knock-out optons wth rebates n (23) s unbased. Ê L m f S m + =1 L 1 1 p S 1 g S 1 = E A m f S m + A 1 A g S 1 Ths result reles on lnearty of expectaton and Ê L 1 p S Y = E A A +1 Y, whch s much lke Lemma 1. The followng lemma s useful for analyzng the reducton n varance whch X m provdes, whch may be expressed as a sum of expected one-step condtonal varances. Lemma 3. The varance of X m can be expressed as Var X m = =1 E Var X F 1 (29) =1 We are nterested n algorthms whch mprove upon the one-step condtonal varance n the standard estmator, Var X F 1, whch can be expressed as follows. Lemma 4. For the standard estmator, the one-step condtonal varance s Var X F 1 = p S 1 Var X F 1 A = 1 and ts expectaton s + p S 1 1 p S 1 E X F 1 A = 1 2 (30) E Var X F 1 = E X 2 E X2 1 (31) Fndng the expected one-step condtonal varance leads to the followng theorem, whch summarzes the varance reducton results. In the theorem and throughout, Var denotes varance wth respect to P. Theorem 3. The estmator X m has reduced varance Var X m = E L m X 2 m X2 0 E X2 m X2 0 = Var X m (32) The nequalty s strct f E A m <1 and E f S m > 0,.e., f there s any chance of knock-out and postve payoff.

9 The varance reducton reles on the comparson of E L m Xm 2 and E X2 m. The greater the probablty of knockout, the smaller L m tends to be, so the greater the reducton n varance. To analyze estmaton schemes wth a maxmum computatonal budget of n forward smulatons at each step, defne the measure P n, whch governs the process where the path survves a step when at least one of n potental successors survves. Both P and P are specal cases, when n = 1 and n =, respectvely. The nterpretaton of P n s that t takes partal advantage of mportance samplng, where P takes full advantage of t and P takes none. P n s defned on relatve to F m by P n S +1 = S = 1 p S n (33) P n S +1 Q S = 1 1 p S n P S +1 Q S A +1 = 1 for Q +1 (34) where S 0 = s 0 s fxed, because 1 1 p S n s the P-probablty that at least one of n successors of S survves. Theorem 4. The estmator Xm n of Equaton (16) s unbased,.e., E n Xm n = E X m. Ths follows from E n A Y L n = E A Y, n the sprt of Lemma 1. Theorem 5. The estmator Xm n acheves an ntermedate reducton n varance: Var n Xm n = E Ln m X2 m X2 0, hence Var X m Var n X n m Var X m (35) where the nequaltes are strct for 1 <n< f E A m <1 and E f S m > 0,.e., f there s any chance of knock-out and postve payoff. Also nterestng s the behavor of estmators when p S are not known. It s hghly desrable that these estmators be consstent as the number of sample paths goes to nfnty, n order that barrer optons may be prced to arbtrary accuracy by ncreasng the number of paths. Because the smulated prce s an average of the values of an estmator realzed on each path, t would be best f ths estmator were unbased. Ths s true of both the negatve bnomal estmator (15) under P and the bnomal estmator (18) under P n. Both nvolve products of factors whch are ndvdually condtonally unbased for what they seekto estmate, namely p S, and whose errors are uncorrelated. Theorem 6. The negatve bnomal estmator s unbased. Ê f S m m 1 =0 m 1 E f S n m =0 r 1 Y 1 N n = Ê X m Theorem 7. The bnomal estmator s unbased. = E n X n m Glasserman and Staum / 931 The proofs are based on condtonal ndependence of Y and S +1 n conjuncton wth the unbasedness of the ndvdual estmates of p S ; see the appendx. It s dffcult to produce varance comparsons for estmators where the one-step survval probabltes are unknown. Instead, we rely on numercal comparsons. 4. NUMERICAL RESULTS Secton 3 contaned theorems formalzng the dea that condtonng on one-step survval produces a reducton n varance, compared to standard Monte Carlo smulaton. Ths suggests that at least the exact estmator X m defned n (10) should be superor to the standard estmator X m of (7). On the other hand, X m requres a greater average number of transtons smulated per path than does X m. Ths s because an effcent mplementaton of the standard smulaton scheme wll cease work on a path as soon as knockout occurs, whereas an algorthm whch condtons on survvng each step necessarly smulates every path for m steps. Algorthms whch generate a maxmum of n potental successors at each step have an ntermedate number of expected transtons smulated. It s necessary to test the performance of the varous estmators numercally. As the followng results show, the effectveness of the new methods depends on the specfc problem to a great extent. We examne the performance of the proposed estmators relatve to the standard technque for specfc benchmark optons. For each of the three examples used n 2, we analyze one opton wth a moderate knock-out probablty and one wth a hgh knock-out probablty. We do not consder optons wth low knock-out probablty because our methods do not produce substantal varance reducton for them, as noted after the statement of Theorem 3. Example 1a. The underlyng process s a sngle stock prce whch, under the rsk-neutral measure, obeys geometrc Brownan moton wth annual drft = 5% and volatlty = 60%. The stock s ntal prce S 0 = 100, and there are barrers at H l = 95 and H u = 105, so that the opton s knocked out f t crosses ether of these. The opton has K = S 0 and ts maturty T = 0 25, wth three montorng dates. Example 1b. The opton s specfcaton s the same, except that the lower barrer H l = 100. Example 2a. The state vector contans a stockprce and an ndex level whch obey geometrc Brownan moton. They both have annual drft = 5%, and whle the ndex has volatlty 1 = 40%, the stockhas volatlty 2 = 60%. Ther correlaton s = 0 5. The ntal value of the ndex s S 1 0 = 1,000, and the stock s ntal prce s S 2 0 = 100. The barrers on the ndex are at H l = 950 and H u = 1,050, and the strke for the stock prce s K = 100. The opton stll has maturty T = 0 25 and three montorng dates. Example 2b. Ths example s lke the prevous, but the maturty of the opton s T = 3 years, wth quarterly

10 932 / Glasserman and Staum montorng. The ndex has volatlty 1 = 15% and the barrers are H l = 900 and H u = 1,050. The stockhas volatlty 2 = 25%, and correlaton = 0 5 wth the ndex. Example 3a. Ths example uses the LIBOR market model based on bonds wth maturtes 0.5 years apart, and ths equals the smulaton tme step t. All forward rates are ntally 5%. The drvng Brownan moton has dmenson d = 1, and each k s a constant 0.3. The contract s a 2 nto 2 payer swapton wth strke K = 5%. That s, n two years the owner has the opton to enter nto a swap to pay 5% nterest and receve the floatng rate for two years. The owner wll exercse the opton f the swap rate for years 2 to 4 (.e., Steps 4 to 8) at maturty s above K see Equaton (4). At T = 2 years, the payoff s the maxmum of zero, and the present value of recevng nterest at rate % for two years see Equaton (6). There are barrers of H l = 5% and H u = 7% on the current LIBOR rate. Example 3b. The settng s smlar, but the swapton s now 4 nto 2. The maturty s T = 4, and agan we consder a payoff based on a two-year swap rate whch s now The barrers are H l = 5% and H u = 6%, but they only take effect n the second two years of the opton s lfe; that s, there s only montorng at steps 4 through 8. We compare the performance of the estmators based on the product of the average number of transtons smulated per path (N) and the varance per path (V). Ths fgure of mert approprately penalzes the new estmators for smulatng every path to maturty whle the standard method abandons a path as soon as a barrer s crossed. For the estmators of 2.3 (.e., wth unknown transton probabltes), N counts all canddate transtons generated at every step, not just the number of survvors. The computatonal effort per transton s the same n these methods as n the standard estmator, and the total computatonal effort n both cases s essentally proportonal to the number of such transtons. Thus, comparng the performance of the estmators of 2.3 wth the standard estmator based on N*V s essentally equvalent to comparng them based on the product of average computer tme per path and varance per path. For the exact estmator of 2.2, a comparson based on N*V does not reflect the dfference n tme requred to generate a condtonal transton and an uncondtonal transton. Ordnarly, generatng a condtonal transton wll take longer, but not much longer. An exact comparson turns out to be extremely senstve to the precse mplementaton of the method (e.g., how one generates normal random varables and how one computes normal probabltes). In contrast, the product N*V should be nearly ndependent of the mplementaton. The overhead nvolved n generatng condtonal transtons s llustrated by the comparson of (11) and (13). By smulatng ln S, and exponentatng only to get S m (to compute the termnal payoff), we can accelerate the basc smulaton (11) and avod the evaluaton of the logarthm n (13) after a one-tme computaton of ln H. The overhead n Table 1. Example Bnary Standard 1a 1 5% 1 9% 2a 4 5% 6 1% 3a 0 4% 3 9% (11) prmarly conssts of generatng a condtonal normal rather than an uncondtonal normal, and of one evaluaton of. For ths example, we fnd that generatng a condtonal transton takes approxmately 50% longer than an uncondtonal survvor. We vew ths as close to a worst case precsely because the basc model s so smple. For more complcated models, the tme per transton n a standard smulaton s greater; the addtonal effort to generate a condtonal transton should be smlar to that for (13) n absolute terms, and thus smaller as a percent of the tme to generate an uncondtonal transton. Tables 1 4 report numercal results for the examples lsted above. In each case, we report the product N*V by normalzng the correspondng product for the standard method to be 100%. In Table 1, we present the performance of the P Estmator of Equaton (10) relatve to the standard estmator of Equaton (7) for the typcal optons. For each benchmark opton, we report results for two versons: the standard verson and a bnary verson, n whch the payoff s 1 when the payoff of the standard verson s postve, and 0 otherwse. The bnary payoffs have less varance condtonal on fnal survval. The methods proposed n ths paper reduce only the varance assocated wth knock-out, and have no effect on varance condtonal on fnal survval. However, t should be possble to combne them wth other methods whch do reduce that part of the varance. Next, we test values of the estmator parameter r (the number of trals per step) n order to gve gudelnes for choosng the parameter n practce. Table 2 gves the results for Examples 1a, 2a, and 3a, whle Table 3 contans the results for the hgher knock-out probablty optons of Examples 1b, 2b, and 3b. We focus on bnary optons, for whch the methods are most effectve. Increasng r mproves the accuracy of the lkelhood rato estmaton, but ths accuracy comes at the prce of ncreased Table 2. Method r Example 1a Example 2a Example 3a Bnomal 2 52% 72% 83% 3 46% 70% 82% 4 43% 72% 81% 6 43% 80% 79% 8 46% 88% 77% Negatve 2 173% 205% 162% Bnomal 3 104% 152% 118% 4 85% 142% 102% 6 82% 152% 89% 8 90% 173% 84%

11 Table 3. Method r Example 1b Example 2b Example 3b Bnomal 2 37% 35% 27% 3 24% 32% 15% 4 19% 29% 12% 6 15% 24% 11% 8 14% 22% 10% Negatve 2 11% 0.29% Bnomal 3 3.7% 0.14% 4 2.2% 0.08% n. a % 0.06% 8 1.3% 0.06% computatonal effort. The margnal beneft of large r s decreasng n r; that s, the more effort has been expended on estmatng the lkelhood rato accurately, the less value there s to expendng further effort on ths task. Ths s reflected n several examples n whch ncreasng r too much eventually ncreases the computatonal expense. Table 3 hghlghts one of the shortcomngs of the negatve bnomal estmator. It s mpractcal to use t for Example 3b, n whch the barrer only takes effect after two years. If the forward rate s too far outsde the barrer, there s a very low probablty that an uncondtonal successor wll survve the step n whch the barrer frst takes effect. Ths makes the expected watng tme unreasonably long. Ths dffculty of pantng oneself nto a corner llustrates a potental shortcomng of our methodology. What one would really wsh to do to reduce varance s to condton on fnal survval. As ths s seldom possble, condtonng on one-step survval can be an effectve substtute, but s not precsely the same. In general, the barrer at tme has no effect on smulaton of steps j< 1, so for certan problems, the state vector S can have a very large probablty of beng n a regon n whch there s mnuscule chance of survvng step. Ths stuaton weakens the effectveness of all of the methods, but especally the negatve bnomal. Table 4 examnes the characterstcs of a barrer opton whch make condtonng on one-step survval an effectve technque. We modfy Example 1 so that t s a down-andout call, for whch there s only a lower barrer. Example 1c. The stock s ntal prce s S 0 = 100, and t has drft = 0 and volatlty = 30%. The opton s maturty s T = There s a sngle barrer at H = 94 30, and monthly montorng (m = 3). The strke s K = H, so that the bnary opton pays 1 unless t s knocked out. Table 4. T H m Bnary Standard 30.0% 0 25 $ % 58% 73.7% 0 25 $ % 48% 30.0% 1 5 $ % 48% 30.0% 0 25 $ % 42% 30.0% 0 25 $ % 155% Glasserman and Staum / 933 Ths means that the prce of the bnary opton equals the probablty of fnal survval, because there s no dscountng. The prce of the opton s 50 cents. Then we construct four new scenaros n each of whch one parameter dffers: The volatlty = 73 7%. The maturty T = 1 5 years, but there are stll only three montorng dates. The barrer H = The montorng s daly, so m = 63. All of these changes produce the same new, lower prce of 33.9 cents. All else beng equal, a lower probablty of fnal survval s assocated wth a greater beneft to these methods, because there s more varance due to survval to be elmnated. However, dfferent factors affect the results n slghtly dfferent ways. For nstance, the scenaro wth the tghter barrer produces superor results for the standard call. Tghtenng the barrer produces the greatest mprovement n varance reducton because t nether ncreases the varance condtonal on survval nor causes wld sample paths wth hghly varable lkelhoods. Most sgnfcantly, we see that wth ths knock-out probablty and a large number of steps, the exact method can underperform the standard method. Wth the total knockout probablty held constant, ncreasng the number of steps weakens the performance of the exact method for two reasons. For one, the rato of the expected number of steps smulated by the standard and exact methods s 1 when m = 1 and decreases to some lmt as m ncreases. Also, from Theorem 3, the varance of the exact estmator s Ê L 2 m f S m 2 X0 2, whch ncreases wth the varance of L m. When m = 1, the lkelhood rato L m s constant and has zero varance, and ts varance grows wth m. Ths s another llustraton that successve one-step survval s not the same as fnal survval; the more steps there are, the less condtonng on one-step survval resembles condtonng on fnal survval. Table 5 llustrates ths pont usng Example 1d wth a bnary payoff. The rows labeled N and N*V contan, as usual, the rato of the estmate for the exact P estmator dvded by that for the standard estmator. The opton prce approaches a lmt whch s the prce wth contnuous montorng. Lkewse, the rato of N approaches a lmt whch s T dvded by the expected tme untl knock-out or maturty wth contnuous montorng. For m = 1, the exact estmator has zero varance for ths bnary opton. Despte the ncreasng probablty of knock-out as m ncreases, the effcency of the exact method decreases. Example 1d. Ths s the same as Example 1c, except that there are two barrers, H l = and H u = The frequency of montorng m vares. 5. CONCLUSIONS Condtonng on one-step survval at each step of a barrer opton smulaton s a natural extenson of mportance samplng whch takes advantage of the partcular structure of

12 934 / Glasserman and Staum Table 5. m Prce N 100% 176% 248% 321% 402% 441% 461% 487% N*V 0% 0 1% 0 5% 1 3% 3 2% 5 5% 7 0% 9 2% barrer optons. We have proposed an estmator (10), whch mplements ths technque when the state vector s dstrbuton condtonal on one-step survval s explctly known. The estmator s unbased and has less varance than a standard estmator, but also requres more computatonal effort on average. For typcal barrer optons, t produces a substantal mprovement n effcency, as measured by the product of workand varance. When the probablty of knock-out s hgh, ths estmator can be far more effcent than a standard Monte Carlo estmate. It s possble to mplement the same concept even when the one-step condtonal dstrbuton s unknown. Usng a properly estmated lkelhood rato results n consstent estmators. Standard Monte Carlo smulaton s seen to be a specal case of ths type of algorthm, where each step s contrbuton to the lkelhood rato s estmated by a sngle Bernoull tral. The computatonal expense of usng more smulatons per step to estmate the lkelhood rato s justfed by a suffcent reducton n varance only when the probablty of knock-out s hgh. Consequently, the structure of the estmated lkelhood rato can have a sgnfcant effect on the estmator s performance, makng t potentally dffcult to choose a sutable estmator before havng analyzed the problem already. Nonetheless, t s possble to gve some gudelnes about when condtonng on one-step survval wll be most effectve. In general, the probablty of knock-out s hgh when the underlyng asset s close to the barrer, relatve to ts volatlty and the maturty of the opton. A fxed barrer s effectvely closer when volatlty s hgh or maturty s long. However, large volatltes and maturtes are assocated wth hgher payoff varance, whch can reduce the effectveness of the method. An mportant pont s that condtonng on survval reduces only the varance assocated wth knock-out, not the varance of the payoff, whch remans condtonal on fnal survval. When the former type of varance s small compared wth the latter, ths varance reducton method wll not prove very effectve f appled alone. If there s sgnfcant varance condtonal on fnal survval, condtonng on one-step survval may be used n conjucton wth other varance reducton methods such as control varates or antthetc varates. Another stuaton n whch condtonng on survval s effectve s when knock-out s dsproportonately lkely to occur late n the smulaton. In ths case, standard Monte Carlo smulaton wll waste a lot of tme n computng paths that get knocked out, makng t attractve to smulate under a scheme where paths have a hgher probablty of survvng each step. Ths stuaton can arse f a combnaton of the drft of the process or tme dependence of the barrer or volatlty make the barrer ntally dstant from the underlyng asset, but very lkely to be close after the elapse of some tme. For typcal barrer optons, condtonng on one-step survval s an effectve varance reducton technque when the one-step condtonal dstrbuton s known. If t s unknown, the lower the probablty of fnal survval, the more computatonal effort should be spent at each step n estmatng the contrbuton to the lkelhood rato. However, algorthms whch expend too much effort n ths drecton may underperform standard Monte Carlo smulaton. APPENDIX A. ANALYSIS OF THE ESTIMATORS Although Lemma 1 s n some respects standard (see, e.g., Glynn and Iglehart 1989 for related results), we detal the proof because the fact that P and P do not have common support requres some care. Proof of Lemma 1. Let be a path up to tme : S 1 = s 1 S = s, and let be the set of all such paths. The space s the Cartesan product j=1 j, where each j, the outcome space for S j,sacopyof d, for some d. The subsets of may be naturally dentfed wth the elements of F by mappng each pont s 1 s to the set S 1 S m S 1 = s 1 S = s. Then we may stretch notaton by treatng P and P as measures on, and usng j to refer to the projecton of j onto j. Wth ths machnery, the proof of the lemma s as follows: E A Y = A YdP = A YdP S S 1 =s 1 dp S 1 S 0 =s 0 1 = YdP S S 1 =s 1 dp S 1 S 0 =s = YL 1 j=0 p S j dp S S 1 =s 1 dp S 1 S 0 =s 0 dp S = YL S 1 =s 1 dp S 1 S 0 =s 0 1 p S 1 p S 0 = YL dp S S 1 =s 1 A =1 1 dp S 1 S 0 =s 0 A 1 =1 = YL d P S S 1 =s 1 d P S 1 S 0 =s 0 1