OPTIMAL EXERCISE POLICIES AND SIMULATION-BASED VALUATION FOR AMERICAN-ASIAN OPTIONS

Transcription

1 OPTIMAL EXERCISE POLICIES AND SIMULATION-BASED VALUATION FOR AMERICAN-ASIAN OPTIONS RONGWEN WU Deparmen of Mahemacs, Unversy of Maryland, College Park, Maryland 20742, MICHAEL C. FU The Rober H. Smh School of Busness, Unversy of Maryland, College Park, Maryland , Amercan-Asan opons are average-prce opons ha allow early exercse. In hs paper, we derve srucural properes for he opmal exercse polcy, whch are hen used o develop an effcen numercal algorhm for prcng such opons. In parcular, we show ha he opmal polcy s a hreshold polcy: The opon should be exercsed as soon as he average asse prce reaches a characerzed hreshold, whch can be wren as a funcon of he asse prce a ha me. By explong hs and oher srucural properes, we are able o parameerze he exercse boundary, and derve graden esmaors for he opon payoff wh respec o he parameers of he model. These esmaors are hen ncorporaed no a smulaon-based algorhm o prce Amercan-Asan opons. Compuaonal expermens carred ou ndcae ha he algorhm s very compeve wh oher recenly proposed numercal algorhms. Receved Aprl 2000; revson receved December 2001; acceped December Subjec classfcaons: Fnance, secures: opon prcng. Smulaon: perurbaon analyss and sochasc approxmaon. Dynamc programmng, models: srucure of opmal polces. Area of revew: Fnancal Servces. 1. INTRODUCTION Asan opons are dervave secures wh payoffs ha depend on he average of an underlyng asse prce over some specfed perod. Because of her relavely small exposure o rsk, hey have become one of he mos popular exoc opons raded over he couner. The purposes of hs paper are o rgorously esablsh a characerzaon of he opmal exercse polcy for Amercan-Asan opons 1 and o develop a Mone Carlo smulaon-based mehod ha explos he esablshed srucural properes o effcenly prce such opons. Secons of Karazas and Shreve 1998) provde a farly comprehensve survey on he properes of he early exercse boundares for ordnary vanlla Amercan opons; however, exoc Amercan-Asan opons are no consdered here. These opons dffer from ordnary Amercan opons n many aspecs. Frs, snce her payoff s a funcon of he average asse prce, he payoff upon exercse depends on he prce pah of he asse, raher han only he asse prce a he exercse dae. Second, a any exercsable dae, he asse prce remans nfluenal n deermnng he early exercse decson. Ths nerplay beween he curren asse prce and he average sock prce makes he analyss of Amercan-Asan opons more complcaed. Pror work by Gran e al. 1997) provdes plausble heursc argumens for he form of he opmal exercse boundary. Our work provdes rgorous mahemacal proofs esablshng he srucure of he opmal exercse polcy for Amercan-Asan opons. Assumng he asse prce evolves accordng o a Markovan model n a que general seng, we rgorously show ha he opmal exercse polcy for a fxed srke Amercan-Asan call opon s a hreshold polcy: The opon should be exercsed as soon as he average asse prce reaches a characerzed hreshold, whch can be wren as a funcon of he asse prce a ha me. Furhermore, we prove ha he hreshold level s unbounded, and under a mld condon, nondecreasng n he asse prce a ha me, and for a large class of models, he hreshold level s also convex. A closely-relaed purpose of hs paper s o prce Amercan-Asan opons. Asan opons have proven o be much more dffcul o value han regular asse opons. Because of her pah dependences, sandard echnques end o be mpraccal or naccurae. There are a few approxmaon mehods for European-syle.e., whou early exercse feaures) Asan opons appearng n he leraure e.g., Turnbull and Wakeman 1991, Vors 1992, Levy 1992, Levy and Turnbull 1992, Geman and Yor 1993). Mone Carlo smulaon seems o be a popular approach, especally for praconers, o prce European-syle Asan opons e.g., Fu e al. 1999). As for Amercan-Asan opons, here are even fewer alernaves. Hull and Whe 1993) propose a modfcaon of he bnomal mehod, bu provde no proof of convergence. Neave 1994) provdes a frequency dsrbuon approach based on a bnomal ree, bu hs mehod sll requres ON 4 compuaon me, where N s he number of me seps n he lace. Hansen and Jorgenson 2000) provde analyss of he floang srke case ha leads o a closed-form soluon for geomerc averagng and an approxmaon for arhmec averagng, bu her mehodology canno be appled o he more common) fxed srke case. There are recenly developed paral dfferenal equaon PDE) approaches Barraquand Operaons Research 2003 INFORMS Vol. 51, No. 1, January February 2003, pp X/03/ $ elecronc ISSN

2 and Pude 1996, Zvan e al. 1997), for whch specal care needs o be aken n order o ge an accurae opon value. Also, he compuaonal requremens for he bnomal and PDE approaches become mpraccal for models ncorporang sochasc volaly and sochasc neres raes. Mone Carlo smulaon was frs nroduced o fnance n Boyle 1977). Snce ha me, smulaon has been successfully appled o a wde range of asse prcng problems Boyle e al. 1997). However, unl recenly he echnque has no been appled o he valuaon of Amercan-syle opons. The major dffculy les n he need o esmae an opmal exercse polcy, whch s usually obaned va a backward nducon algorhm, whereas smulaon s a forward-based process. In he pas decade, a number of Mone Carlo smulaon-based approaches have been proposed o address he problem of prcng Amercan-syle opons. For an overvew of he approaches, see Broade and Glasserman 1997b) or Fu e al. 2001). Of he work surveyed here, only Gran e al. 1997) address specfcally he prcng of Amercan-Asan opons. Ther procedure mmcs he backward nducon soluon mehod of sochasc dynamc programmng. A every exercsable dae, he opmal hreshold parameers are esmaed by esng all possble values from a preseleced fne parameer grd. The algorhms proposed by Broade and Glasserman 1997a, 1997c) are based on smulaed pahs and lead o based hgh esmaors and based low esmaors ha converge o he rue value n he approprae lm. Unforunaely, snce here s no proper ranson probably densy funcon for Amercan-Asan opons, he sochasc mesh mehod Broade and Glasserman 1997c) does no appear applcable o Amercan-Asan opons. Alhough one can exend he smulaed ree mehod Broade and Glasserman 1997a) o Amercan-Asan opons, a large number of smulaed rees need o be generaed n order o ge an accurae opon value, whch s mpraccal from he perspecve of compuaon coss. Over he las decade, here has been a lo of research on smulaon-based approxmae dynamc programmng see Bersekas and Tsskls 1995). Recenly, Longsaff and Schwarz 2001) and Tsskls and Van Roy 2001) appled hs approach o he prcng of Amercan-syle opons by approxmang he holdng value funcon a each me sep usng a lnear combnaon of bass funcons fed o he smulaed daa va leas square regresson. In parcular, Longsaff and Schwarz 2001) demonsrae he effcency of her leas square approach hrough several numercal examples, and Tsskls and Van Roy 2001) rgorously esablsh he general convergence properes of he mehod. An alernave o approxmang he value funcon s o approxmae he exercse boundary,.e., he boundary a whch he holdng value equals he exercse value. However, n order for hs approach o be effecve, some knowledge on he srucure of he opmal polcy s crucal, and hus he heorecal resuls on he form of he polcy are exploed o hs end. Our smulaon-based approach o value Amercan-Asan opons parameerzes he exercse Wu and Fu / 53 boundary and maxmzes he expeced dscouned payoff wh respec o he early exercse hreshold parameers. Smlar deas are also used n Fu and Hu 1995) o prce ordnary Amercan call opons. Once a parameerzaon s assumed, he mos dffcul and challengng par of our approach s o fnd a good graden esmaor for Amercan- Asan opons, a ask ha s more complcaed han ha for ordnary Amercan opons Fu e al. 2000). We derve he gradens wh respec o assocaed parameers va perurbaon analyss PA)Ho and Cao 1991, Glasserman 1991, Fu and Hu 1997), a sample pah mehod for graden esmaon. Then we ncorporae he PA esmaors no a sochasc approxmaon algorhm o esmae he opmal hreshold parameers, and consequenly oban an esmae for he opon prce. Usng examples from Gran e al. 1997), we compare our algorhm wh her algorhm and wh he algorhm from Longsaff and Schwarz 2001), and fnd ha our approach s que compeve, f no superor, for he esbed of problems consdered. In sum, our work conrbues o he research sream on prcng Amercan-Asan opons n sgnfcan ways: We provde rgorous proofs esablshng varous srucural properes of he opmal exercse polcy n a Markovan seng more general han geomerc Brownan moon). We derve graden esmaors for he opon payoff wh respec o model parameers. By explong he srucural properes, we apply he graden esmaes o a parameerzed exercse boundary n order o provde a compuaonally effcen smulaonbased prcng mehod. Durng fnal preparaon of he nal verson of our paper, we were made aware of relaed work by Ben Ameur e al. 2002), who also develop a numercal mehod for prcng Amercan-Asan opons based on dynamc programmng combned wh fne-elemen pecewsepolynomal approxmaon of he value funcon. Independenly from us, hey also esablsh some smlar heorecal properes for he opmal exercse sraegy for Amercan-syle Asan opons n he Black-Scholes seng. Many of our resuls hold n a more general seng han hers, because our proof echnques dffer from hers, n ha he soppng mes are carred ou hroughou our proofs, whereas her proofs proceed by backward nducon on me seps of he dynamc programmng opmaly equaon. The res of he paper s organzed as follows. Secon 2 nroduces he problem seng. Secon 3 descrbes he varous srucural properes for he opmal exercse polcy. All proofs of he resuls can be found n he Appendx. The perurbaon analyss esmaors are presened n 4, wh he dealed dervaons n he Appendx. In 5, we parameerze he exercse boundary, smplfy he esmaors derved n he prevous secon, and provde he smulaonbased valuaon algorhm, whch s esed on some numercal examples n 6. Secon 7 conans concludng remarks.

3 54 / Wu and Fu 2. PROBLEM SETTING We begn by nroducng he followng noaon o be used hroughou: S = asse prce a me r = annualzed rskless neres rae compounded connuously) = volaly of he underlyng asse K = srke prce of he opon conrac T = expraon dae of he opon conrac For ease of exposon, r, and K wll be assumed consan. If r or s sochasc, can also be easly ncorporaed no our conex. Whou loss of generaly, we desgnae he presen me as me 0. We consder a dscree arhmec Amercan-Asan opon, where he averagng sars a me 0 and s he equally-spaced nerval beween he averagng daes. Le N be he number of prce average daes f held o expraon T so T = 0 + N 1), and < N be he number of early exercsable daes for he Amercan-Asan opon see Fgure 1), denoed as = 1 2. For noaonal convenence we also denoe T = +1. Defne as he se of all average daes and E as he se of all exercsable daes,.e., = N 2 T and E = = Noe ha E. IfE =, hen he opon can be exercsed a any prce averagng dae. For any, defne n as he number of averagng daes up o and ncludng me. Le S denoe he asse prce process, whch we assume hroughou o be Markovan. We defne he soppng me o be a random varable ha akes values n E such ha each even of he form = E s an elemen of he algebra F, he flraon generaed by he relevan prce processes up o me n he economy. Wre x = x K + maxx K0. Then he mmedae exercse value of he Amercan-Asan call opon a me E s gven by S, where S s he average prce up o and ncludng me,.e., S = S 0 + S S n We assume ha he fnancal marke s arbrage free, so ha by he fundamenal heorem n asse prcng e.g., Harrson and Plska 1981), here exss an equvalen rskneural prcng measure Q such ha e r S s a marngale under measure Q. Snce he sock prce process S s Markovan, he fuure sock prce pah S > only Fgure 1. Averagng and exercse daes for he dscree Amercan-Asan opon. depends on he curren sock prce S. Arbrage-free valuaon heory mples ha he value of he Amercan-Asan call opon a me E s gven by akng he supremum over all soppng mes of he expeced dscouned payoff of he opon under rsk-neural prcng measure Q: sup E Q e r S S S 1) Throughou, all expecaons wll be aken under he Q measure, so for ease of noaon, he superscrp Q wll be omed. 3. STRUCTURE OF THE OPTIMAL EXERCISE POLICY In hs secon, we characerze he srucure of he opmal exercse polcy for he Amercan-Asan call opon. Analogous deas can be used for a dscusson of he Amercan- Asan pu opon. Specfcally, we frs show ha he opmal exercse sraegy s a hreshold polcy,.e., he opon should be exercsed as soon as he average asse prce reaches a characerzed hreshold, whch can be wren as a funcon of he asse prce a ha me. Then we ry o explore furher he properes of he hreshold,.e., he shape of he exercse boundary. Noe ha a each exercsable me E, he opon holder mus choose wheher o exercse mmedaely or o connue he lfe of he opon and revs he exercse decson a he nex exercsable dae. The payoff upon mmedae exercse a me wh average asse prce S s gven by S. We nroduce he noaon cxy o denoe he connuaon value of he opon,.e., he value of he opon condonal on he opon no beng exercsed a or pror o me, wh curren average asse prce S = x and curren asse prce S = y. Then we have cxy = sup E e r S S = xs = y > where he supremum s aken over all soppng mes >, nsead of. Snce he value of an Amercan opon a any me s he maxmum of he payoff upon mmedae exercse and he connuaon value, one may also wre he opon value a me, gven by 1), as maxc S S S. Snce he opon holder exercses as soon as he mmedae exercse value s greaer han or equal o he value of connuaon, he exercse regon a exercse pon E, denoed as R, can be characerzed by R = xycxy x The hreshold polcy s based on he followng observaon: For he same curren asse prce, a hgher curren runnng average wll have a hgher connuaon value, bu he dfference n connuaon value s no greaer han ha n he curren runnng average. Lemma 1. For any xy > 0 E, we have 0 cx + y cxy

4 Remark 1. I s easy o nfer from Lemma 1 ha for any fxed y and Ec y as a funcon of s frs varable s nondecreasng and unformly connuous. Theorem 1. The opmal exercse polcy s a hreshold polcy,.e., here s a funcon F a me E such ha s opmal o exercse he opon whenever S F S K, where F y = nfxcxy x By esablshng he exsence of a hreshold polcy, exercse decsons are compleely deermned by he funcon F s s 0 + a each early exercsable dae =. The remander of he resuls n hs secon characerze he shape of hs funcon, whch wll be useful n formulang an effecve parameerzaon for developng numercal prcng algorhms. The frs resul esablshes ha he boundary goes off o posve nfny wh ncreasng values of he curren sock prce. Theorem 2. For any E TF s unbounded,.e., f y, hen F y. The nex resul esablshes he monooncy of he boundary. In order o show hs, we need o make a mld assumpon on he model of he underlyng asse. Assumpon 1. If y 1 >y 2, hen cxy 1 cxy 2 for all E. Inuvely, Assumpon 1 mples ha for he same curren runnng average a any fxed me, a hgher curren asse prce canno lead o a lower call opon value. One can see ha mos of he models n pracce sasfy he assumpon. Defnon 1. An asse prce model s mulplcave f can be represened by he form S = S X for any >, where X > 0 s a random varable ndependen of all S u u and only a funcon of quanes defned on. Inuvely, f he prce of he sock a me doubles, hen he sock prce a me would double. For example, geomerc Brownan moon falls no hs caegory: S = S e r 2 /2 + Z 2) where Z s a N0 1 random varable. The general jump dffuson model Meron 1976) s also mulplcave: S = S e r 2 /2 + Z 0 + q =1 Z 2 /2 3) where Z N0 1 d, q Posson wh beng he jump arrval rae, and he jump szes are..d. lognormally dsrbued LN 2, wh = 2 /2 and = > 0. Wu and Fu / 55 Defnon 2. An asse prce model s addve f can be represened by he form S = S + Y for any >, where Y s a random varable ndependen of all S u u and only a funcon of quanes defned on. Noe ha Y s allowed o be negave so our model could also handle negave sock prces). I s easy o check ha boh mulplcave and addve models whch nclude a very general se of sochasc processes wh saonary ndependen ncremens called Lévy processes wll sasfy Assumpon 1. For example, consder he mulplcave model. For any soppng me >, we have E e r S K + S = xs = y 1 n = E e r x + y 1 X + + X + X + +2 ) ) X + X /n K n E e r x + y 2 X + + X + X + +2 ) ) X + X /n K = Ee r S K + S = xs = y 2 where he nequaly follows from y 1 >y 2 and he fac ha a + b + f a>b. Takng he supremum wh respec o all soppng mes >yelds he resul of Assumpon 1. Noe ha he same proof goes hrough for he addve model. Acually, he class of asse prce models ha sasfy Assumpon 1 exend far beyond jus he mulplcave and addve models llusraed here. Bergman e al. 1996) demonsrae ha as long as a ceran no-crossng propery holds, he prce of a call opon s nondecreasng n he underlyng asse prce. For example, hey show ha all he one-dmensonal dffuson processes, ds = S d + S S dw where he nsananeous volaly can be a funcon of he concurren asse prce, sasfy he no-crossng propery. Noe ha he processes are generally no mulplcave. The Consan Elascy of Varance CEV) Cox and Ross 1976) and he local volaly model Dupre 1994, Derman and Kan 1994), wo of he mos wdespread models among praconers, fall no hs caegory. Under Assumpon 1, he monooncy resul saes ha he exercse boundary s ncreasng n he nonsrc sense) as a funcon of he curren asse prce. Theorem 3. If Assumpon 1 holds, F s nondecreasng.

5 56 / Wu and Fu Lasly, we are able o esablsh convexy of he exercse boundary under a furher resrcon from Assumpon 1, sasfed by, for example, he geomerc Brownan moon model 2). Theorem 4. If he asse prce model s mulplcave or addve, hen F s convex. 4. PERTURBATION ANALYSIS ESTIMATORS 4.1. Movaon Armed wh knowledge on he srucure of he opmal polcy, we now ry o prce he Amercan-Asan opons by parameerzng he early exercse boundary and hen formulang he opmal soppng problem as he followng opmzaon problem: max EL 4) where R p s he p-dmensonal vecor of neres, e.g., he parameers of he early exercse boundary o be esmaed, L s he sample) dscouned payoff of an Amercan-Asan opon, a compac se n R p, and an elemen n he probably space of neres, e.g., a sample pah n smulaon. We wll apply sochasc approxmaon e.g., Kushner and Yn 1997) o he above opmzaon problem. Bascally, we aemp o fnd he soluon o 4) by mmckng seepes-decen algorhms from he deermnsc doman of nonlnear programmng usng he followng erave search scheme: n+1 = n + a n ĝ n 5) where n = n 1 n p represens he nh erae, ĝ n represens an esmae of he graden of EL wh respec o he parameer vecor a n a n s a posve sequence of numbers convergng o 0, and denoes a projecon on. In order o mplemen he algorhm, he key feaure s he avalably of a graden esmae, whch could eher be a drec esmae or a fne dfference esmae. However, a drec esmae generally wll provde a superor convergence rae. Nex, we derve such a drec graden esmaor va perurbaon analyss PA) Ho and Cao 1991, Glasserman 1991, Fu and Hu 1997) Dervaon In 3, we esablshed ha he opmal polcy follows a hreshold polcy,.e., a any me E, he opon holder exercses he opon whenever S F S. We wll le F denoe an approxmae form of F. For ease of noaon, for he res of he paper we wre F F and n n for E. Therefore, he value of he Amercan-Asan call opon can be wren as 4), wh he sample performance L gven by { 1 } L = 1 S j <F j S j S F S S Ke r =1 { + 1 } S j <F j S j S T K + e rt 6) where 1 denoes he ndcaor funcon. Expresson 6) smply represens he dscouned opon payoff as a sum of payoffs a each exercsable dae, where exercse can occur a mos once over he lfe of he conrac. In order o derve he PA esmaors, we assume ha F s convex for any E. As demonsraed n he prevous secon, convexy holds for a large class of sock prce models, ncludng geomerc Brownan moon. In fac, we can also use he deas presened here o derve he PA esmaors for he case where F s concave. Noe ha S = n 1 S n + S n Frs, we need he followng resul n order o derve he PA esmaor. Lemma 2. If F s convex, assumng ha S = z fxed and { y n } 1z + y F n y 7) hen we can always fnd L and U such ha n 1z + y n F y L z y U z where L and U may ake on he values 0 + and +, respecvely, wh he subscrp noaon x and x + denong he correspondng respecve lef-hand and rgh-hand lms. Remark 2. In Lemma 2, we assume 7) holds. Acually, f s empy, we may choose L z = U z = consan. I wll no affec our dervaon of he PA esmaor, snce he negral on any se wh measure zero s zero. Now we wll derve he PA esmaor. Snce S s Markovan, we assume he asse prce dynamcs follow he form S = hz S 0 r for some random varable Z ndependen of he parameers and nal sock prce S 0. In parcular, S + = hz + S r for any, wh ndependen Z + f, he approprae probably densy funcon. We also assume ha h s monoone n he frs varable. Noe ha our dervaon could adm dfferen forms of h and f for dfferen ; however, for ease of exposon, we assume he same form for all. Also, for noaonal convenence, we wll henceforh om explc dependence on r and n he dsplay of funcon h. For he geomerc Brownan moon model 2), h s gven by hz S = Se r 2 /2+ Z 8) where Z s a sandard N0 1 random varable wh densy funcon fx= 1 2 e x2 /2.

6 Our goal s o fnd an esmae for EL, where can be any parameer of he model, alhough our neres n he nex secon focuses on parameers of he early exercse boundary. The dealed dervaon of he PA esmaor for = 2 s gven n he Appendx. The exenson of he PA esmaor o he general case s gven by he followng: { 1 } 1 S j <F j S j =1 { h 1 L S S + fh 1 L S S 1 E L S j <F j S j S S = L S E S Ke r S S = L S h 1 U S S fh 1 U S S 1 E L S j <F j S j S S = U S + 1 =1 E S Ke r S S = U S { 1 } S j <F j S j S F S S Ke r { } + 1 S j <F j S j S T K + e rt 9) Alhough expresson 9) for he esmaor appears que complcaed, he mplemenaon s farly sraghforward once he L and U funcons defned n Lemma 2 are known, as we see n he nex secon. The erms nvolvng / and f are all readly avalable on he orgnal smulaed sample pah, whereas he condonal expecaon quanes requre evaluang he expeced payoff on pahs generaed by specal sarng condons. 5. PARAMETERIZATION OF EARLY EXERCISE BOUNDARY In order o develop a numercal prcng algorhm, we consder he PA esmaor specfcally appled o parameers of he early exercse boundary. There are many ways o parameerze he exercse boundary. Here we consder a lnear approxmaon of he exercse boundary, as n Gran e al. 1997). For oher forms of he exercse boundary, smlar deas could be followed o smplfy he PA esmaors } Fgure 2. Wu and Fu / 57 Early exercse boundary for he Asan opon. derved n 4. For he call opon consdered, he exercse regon s aken as follows see Fgure 2): S s and S S + v 10).e., we approxmae he exercse boundary a by a pecewse lnear funcon: { s F y = f y s v 11) y + v f y>s v where s v are parameers o be esmaed such ha hey maxmze he expeced payoff of he opon. Now we wll proceed o fnd L and U from he exercse condon S F S. Frs, we rewre he exercse condon ha compares S wh FS defned by 11) as follows: ) If S s v, hen S n s n 1 S,.e., n s n 1 S S s v. ) If S >s v, hen S S n v,.e., s n 1 v < S S n v. n 1 Therefore, he opon s exercsed a me f and only f n s n 1 S S s v or s v <S S n v n 1 Noe ha: ) If S >s + v, hen we have n n 1 s n 1 S <s v < S n v. So he above exercse condon can be smplfed as n s n 1 S S n 1 S n v. In hs case, L n 1 and U can be aken as L S = n s n 1 S and U S = S n v n 1 ) Conversely, f S s + v, hen n n 1 s n 1 S s v S n v. In hs case, he opon can n 1 be exercsed f and only f S = s v. Ths s a rval case, where L and U can be smply chosen as L = U = s v. Noe ha n he second case, hose erms n he PA esmaor 9) ha are drecly relaed o L and U cancel each oher. Furhermore, as ndcaed n he proof

7 58 / Wu and Fu of Lemma 2, s necessary ha L 0 +. Therefore, he PA esmaor 9) wh respec o he hreshold parameer can be s or v ) can be smplfed o 1 =1 { 1 S j <F j S j S >s + v } n 1 { 1L S >0 h 1 L S S fh 1 L S S 1 E L S j <F j S j S S = L S s Ke r h 1 U S S fh 1 U S S 1 E L S j <F j S j S S = U S + S v ) } n 1 K e r where L S = n s n 1 S and U S = S n v. The las wo erms n 9) are zero, because n 1 he underlyng asse prce process s ndependen of he hreshold parameers. In parcular, for he geomerc Brownan Moon model 8), he nverse of h s gven by h 1 ys= lny/s r 2 /2/ so we have h 1 L S S n = s n s n 1 S h 1 U S S = 0 s h 1 L S S = 0 v h 1 U S S n = v n v n 1 S The PA esmaor wh respec o s s gven by { 1 1 S j <F j S j S >s + v } n 1 { 1n s >n 1 S n e lnn s n 1 S /S r 2 /2/ 2 /2 2n s n 1 S 1 E L S j <F j S j S S = n s n 1 S s Ke r and ha wh respec o v becomes { 1 1 S j <F j S j S >s + v } n 1 { n e lnn 1 S n v /n 1S r 2 /2/ 2 /2 2n v n 1 S 1 E L S j <F j S j S S = S n ) v n 1 S v ) } n 1 K e r + } Thus, he PA esmaors for he dervave w.r.. parameers a he h early exercse dae have hree ypes of erms: An ndcaor funcon whch s based on wheher or no he average sock prce exceeds a ceran level a every exercse dae up o he h, a densy rae quany whch nvolves he sock prce and he average sock prce one averagng dae ) pror o he h early exercse dae and a number of model parameers, and he dfference of wo dscouned expeced payoffs a he h early exercse dae one wh he average sock prce jus below a ceran level e.g., s n he esmaor w.r.. s ) and he oher wh he sock prce jus above he same level. The laer payoff s smply he dscouned) mmedae exercse value, and s hus avalable on he smulaed sample pah, bu he former payoff s a connuaon value and requres some addonal smulaon o esmae. 6. NUMERICAL RESULTS We now repor numercal resuls on prcng Amercan- Asan opons by ncorporang he perurbaon analyss esmaors no a sochasc approxmaon algorhm accordng o 5). We consder examples from Gran e al. 1997), usng he followng sengs: nal sock prce S 0 = 100, srke prce K = , expraon dae T = 120 days, neres rae r = 009, and volaly = Averagng sars a day 0 = 91, and he averagng nerval s one day,.e., = 1. The earles 365 me 1 for exercse s he end of day 105. In oher words, he average ncludes a leas 15 observaons of he asse prce. We consder hree values for he number of early exercse opporunes: = 1 3, and 5. For he sep-sze sequence, we choose he harmonc seres,.e., a n = a/n wh a = 50, and decrease he sep sze only f he graden drecon has changed from he prevous eraon,.e., he nner produc of he curren and prevous graden drecons s negave. The sarng values are s = K and v = 0, for = 1 2, wh consran condon s K v unconsraned) and he projecon operaor defned by smply akng s = K for any volaed consrans. We ake observaon lenghs of 50 for = 3 and 5 and 40 for = 1, where he observaon lengh s he number of pahs of he asse prce generaed for he graden esmaon n each

8 Table 1. = 02r = 009S 0 = 100. CPU seconds are for compung he hreshold parameers only. Wu and Fu / 59 = 105, 120 = 105, 110, 115, 120 = Prce Sd. Error CPU Prce Sd. Error CPU Prce Sd. Error CPU K = 90 PASA DP DIFF K = 95 PASA DP DIFF K = 100 PASA DP DIFF K = 105 PASA DP DIFF K = 110 PASA DP DIFF eraon. For each prce pah, we use 10 replcaons o esmae he condonal expecaon porons n he PA esmaors. Frs, we compare our resuls wh he smulaon algorhm of Gran e al. 1997), akng her recommended parameer sengs n mplemenng her algorhm. In boh our and her procedures, opon valuaon s formulaed as a maxmzaon problem wh respec o he assocaed hreshold parameers. Therefore, comparson of he algorhms s carred ou by esmang he expeced dscouned payoff a he parameer sengs obaned by he correspondng algorhm, where a hgher esmae of he opon prce mples superor performance. To make he comparsons more precse, we run 2,000,000 smulaons afer he parameer sengs are obaned for each algorhm, n order o accuraely esmae he expeced opon payoff. The resuls are provded n Tables 1 and 2, where all Table 2. = 03r = 009S 0 = 100. CPU seconds are for compung he hreshold parameers only. = = = Prce Sd. Error CPU Prce Sd. Error CPU Prce Sd. Error CPU K = 90 PASA DP DIFF K = 95 PASA DP DIFF K = 100 PASA DP DIFF K = 105 PASA DP DIFF K = 110 PASA DP DIFF

9 60 / Wu and Fu sandard errors ndcaed n he able column SdErr are no more han one cen. The resuls for he sochasc approxmaon mehod based on he perurbaon analyss esmaors are ndcaed by PASA and hose based on he smulaon-based) dynamc programmng of Gran e al. 1997) by DP, wh DIFF he dfference n he opon prces. CPU mes ndcaed are for approxmang he hreshold parameers only, snce he fnal prce esmaon requres he same compuaonal burden for boh algorhms. All cases are run on he same plaform: a Sun Ulra 60 Unx worksaon. We fnd ha he sochasc approxmaon algorhm based on he perurbaon analyss esmaors converges very quckly. Wh jus en eraons for each case o compue he assocaed parameers, we oban rapd convergence. In mos cases, he daa n he row of DIFF are posve, whch means ha he opon values based on our mehod are hgher han hose based on Gran e al. 1997),.e., our approach ouperforms hers. Furhermore, we use less CPU me o compue he assocaed hreshold levels. In he fve early exercse opporuny case, PASA ypcally needs abou 0.15 seconds, whle DP needs abou 0.32 seconds. Fgures 3 and 4 provde a ypcal graphcal comparson of he wo approaches for he case = 02, K = 100. In hese examples, PASA fnds beer early exercse boundares wh less compuaonal cos, ndcang ha our smulaon-based approach s very promsng. We also compare our approach wh he leas squares LS) smulaon algorhm of Longsaff and Schwarz 2001) cf., Tsskls and Van Roy 2001) usng he same esbed. We do 20,000 smulaons for each approach. For he LS algorhm, all polynomal erms on he curren asse prce and he runnng average, up o hrd order, are used as bass funcons. Thus, we use a oal of en bass funcons n he regressons. The resuls are repored n Tables 3 and 4, where LS ndcaes he LS algorhm. Agan, boh approaches provde lower bounds for he opon value subjec o sascal error). Mos of he opon values obaned from LS are smaller han hose from Fgure 3. Comparson of opon value for wo mehods. Fgure 4. Comparson of CPU me for wo mehods. PASA, alhough he dfferences are mosly whn one sandard error. Snce CPU mes for PASA are smaller han hose for LS, our approach s a leas compeve wh, f no superor o, he approxmae dynamc programmng approach for hs small esbed. 7. CONCLUSION Our work has llusraed he praccal benefs of nerplay beween heorecal analyss and compuaonal mplemenaon. Frs, we rgorously esablshed varous srucural properes of opmal exercse polces for Amercan-Asan opons. These properes provde a bass for characerzng he form of he exercse boundary a each poenal exercse pon, so ha by parameerzng he exercse boundary, he opon valuaon can be cas as a parameerzed opmzaon problem. By dervng sochasc graden esmaors, we provde an effcen smulaon-based algorhm for prcng hese ypes of opons. Ths algorhm can be used n sengs for whch Mone Carlo smulaon becomes he preferred numercal echnque, e.g., problems nvolvng mulple sochasc processes such as neres raes and volales. Furhermore, he resuls hold for a broad class of underlyng Markovan asse prce models, ha of Lévy processes generaed by ndependen ncremens) and general one-dmensonal dffuson processes. Fuure drecons nclude more general opmal soppng problems ha frequenly occur n fnancal engneerng. APPENDIX Proof of Lemma 1. For any soppng me >,wehave E e r S K + S =xs =y ) =E e r n x+s + + +S + K S n =y

10 Table 3. = 02r = 009S 0 = 100. CPU seconds nclude me for compung he opon value. Wu and Fu / 61 = = = Prce Sd. Error CPU Prce Sd. Error CPU Prce Sd. Error CPU K = 90 PASA LS DIFF K = 95 PASA LS DIFF K = 100 PASA LS DIFF K = 105 PASA LS DIFF K = 110 PASA LS DIFF ) E e r n x++s + + +S + K S n =y =Ee r S K + S =x+s =ycx+y where we have used he fac ha S s a Markov process. Takng he supremum over all soppng mes >yelds cxy cx + y Conversely, for any soppng me >, we also have E e r S K + S = x + S = y = E e r n x + + S + + +S + K) n S = x + S = y Table 4. = 03r = 009S 0 = 100. CPU seconds nclude me for compung he opon value. = = = Prce Sd. Error CPU Prce Sd. Error CPU Prce Sd. Error CPU K = 90 PASA LS DIFF K = 95 PASA LS DIFF K = 100 PASA LS DIFF K = 105 PASA LS DIFF K = 110 PASA LS DIFF

11 62 / Wu and Fu = E e r S K+ n ) + S = xs = y E e r S K + + n ) S n = xs = y E e r S K + S = xs = y + where he frs nequaly follows from a + b + a + + b for any b>0, and he second nequaly resuls from n he fac ha < 1 and so e r n <. Agan, akng n n he supremum over all soppng mes >yelds cx + y cxy+. Proof of Theorem 1. A expraon dae T, he opon wll be exercsed as long as S T K, so he hreshold polcy follows for he ermnal case. Now we consder he case <T. For any fxed S = y a me E T, we can always fnd a value x such ha cxy x To see hs, frs we noe ha for any me E wh >, we have E e r S + + S S n S = y n 1 n + 1 Ee r S + + e 2r S +2 = 1 n + 1 n n y + + e r S S = y 1 n + 1 N n y 12) where he frs nequaly follows from n n + 1 and second nequaly follows from he marngale propery of e r S. Suppose s he nex exercsable dae afer me. For any soppng me >,wehave E e r S + +S S n S =y T =E e r S + +S S 1 = = n S =y T e S + +S S n S =y = E +2 n +1 N n y 13) where he las nequaly resuls from 12). We denoe he bound as C +2 N n n +1 y, whch s ndependen of soppng mes >. Takng x max n +1 Kn n + 1C, for any soppng me >,wehave E e r S K + S = xs = y = E e r n x K n ) + S ) + + +S + S n = y E e r n x n x n ) + n + 1 K + C n x n + 1 K ) + K + S ) + + +S S = y n ) + + x n + 1 = x K = x where he frs nequaly follows from a + b + a + + b for any b>0, he second nequaly follows from 13) and n n + 1, and he las lne follows from he choce of x. Takng he supremum over all soppng mes leads o cxy x, so ha he se over whch F y s defned s nonempy: xcxy x By defnon of F y, here exss a decreasng sequence x k ha approaches F S such ha x k F y and cx k y x k for every k. By he connuy of c y for he frs varable from Lemma 1) and, we know ha cf y y F y 14).e., he nfmum s aanable and well defned. Therefore, suffces o show ha cf y + y F y ) for any >0, whch means ha s opmal o exercse he opon a me f he asse prce s S = y and average asse prce s S = F y +. Usng 14), we can wre cf y + y F y + cf y + y F y + cf y y F y = cf y + y cf y y F y + F y F y + F y where he las lne follows from Lemma 1. Snce F y K, F y + K+ F y K+ = esablshng 15) and concludng he proof. Remark on Theorem 1. If we defne = nf S F S T denoes he mnmum operaor), hen from Theorem 1 we know ha s an opmal soppng me. So he resuls from Theorem 1 mply he exsence of an opmal soppng me. Proof of Theorem 2. Suppose, on he conrary ha F s bounded,.e., here exss a consan M, such ha F y M

12 for all y. Then by 14), cf y y F y = F y K+ should also be bounded, so suffces o show he conradcon ha cf y y as y. Suppose s he nex exercsable dae afer me and consder =,a fxed soppng me. Then we have cf yy E e r S K + S =F ys =y n F =E e r y+s + ++S K + n S =y n F ) + e r E S =y K n y+s + ++S n =e r n F y + 1 ) + e r ++e r y K n 16) where he second nequaly follows from Jensen s nequaly and he las equaon resuls from he marngale propery of e r S. I s easy o see ha as y, he rgh-hand sde of 16) goes o nfny, so cf y y. Proof of Theorem 3. Suppose on he conrary ha he oppose s rue. Then here would exs a par of prces y 1 and y 2 wh y 1 <y 2, such ha F y 1>F y 2 I follows from Theorem 1 ha he opon wll no be exercsed a me f he asse prce s S = y 1 and he average asse prce s S = F y 2,.e., cf y 2 y 1 >F y 2 17) On he oher hand, by he defnon of F, we know ha F y 2 cf y 2 y 2 18) Snce y 2 >y 1, by Assumpon 1, we have cf y 2 y 2 cf y 2 y 1 19) Combnng 18) and 19) leads o a conradon of 17). Proof of Theorem 4. We provde he dealed proof only for he mulplcave case. The proof for he addve case s essenally dencal, wh he addve relaonshp subsued n he approprae places. For y 2 <y 1, we wll show ha F y y 2 F y F y 2 20) for any 0 1. Wre y = y y 2. B y Theorem 1 suffces o show ha cf y F y 2 y F y F y 2 21) Wu and Fu / 63 because 21) s he exercse condon for an opon a me wh asse prce S = y and average asse prce S = F y F y 2, whch by Theorem 1 s equvalen o he condon S = F y 1+1 F y 2 F y. Now le > be he opmal soppng me for he sae wh average asse prce S = F y 1+1 F y 2 and asse prce S = y a me. Then we have cf y 1+1 F y 2y =E e r n F y 1+1 F y 2 +S + + +S / n K ) + S =y =E e r n F y 1+1 F y 2 +yx + + +X + X / n K ) + =E e r n F y 1+y 1 X + +X + X X + X /n K) +1 n F y 2+y 2 X + +X + X X + X /n K) + E e r n F y ) 1+S + + +S + K S =y E e r S =y 2 n F n y 2+S + + +S n supee r S K + S =F y 1S =y 1 > K) + +1 supee r S K + S =F y 2S =y 2 > =cf y 1y 1 +1 cf y 2y 2 F y 1+1 F y 2 by defnon of F =F y 1 K + +1 F y 2 K + =F y 1+1 F y 2 K =F y 1+1 F y 2 snce F K where he frs nequaly follows from a + b + a + + b +. Proof of Lemma 2. Noe ha F s nondecreasng, and v = n 1 z + y n n s a sragh lne as a funcon of y wh slope 1 > 0. We n consder four possble cases: ) If he enre sragh lne s above he curve v = F y, we can have L z = 0 + and U z =+

13 64 / Wu and Fu Fgure 5. Deermnng L and U : Case ). ) If he sragh lne nersecs wh v = F y a only one pon y 1, bu s no angen o he curve, hen eher L z = 0 + and U z = y 1 or L z = y 1 and U z =+ see Fgure 5). If he lne s angen o he curve, we have L z = U z = y 1 ) If he equaon n 1 z + y = F n n y 22) has wo soluons y 1 and y 2 wh y 1 <y 2, by he convexy of F see Fgure 6), we have L z = y 1 and U z = y 2 v) If Equaon 22) has more han wo soluons, le y 1 and y 2 denoe he smalles and larges soluons, respecvely, where y 2 could be +. Then, by he convexy of F, s easy o show ha 22) s sasfed y y 1 y 2. Thus, we may choose L z = y 1 and U z = y 2 Dervaon of PA Esmaor for he = 2 Case. The sample performance for he = 2 case s gven by L = 1 S 1 F 1 S 1 S 1 Ke r 1 Fgure S 1 <F 1 S 1 S 2 F 2 S 2 S 2 Ke r S 1 <F 1 S 1 S 2 <F 2 S 2 S T K + e rt 23) Deermnng L and U : Case ). Recall ha F = 1 2, are used o characerze he hreshold levels and depend on some parameers of neres. Furhermore, hey are nondecreasng funcons. Takng he expecaon of he frs erm of L gven by 23), we have E1 S 1 F 1 S 1 S 1 K + e r 1 h 1 U 1 S 1 S 1 = E h 1 L 1 S 1 S 1 n1 1 S 1 + hz S 1 n 1 ) K e r 1 f zdz 24) Noe ha here L 1 and U 1, whch are defned n Lemma 2, are dependen on he parameers of neres and S 1 F 1 S 1 L 1 S 1 S 1 U 1 S 1 Inuvely, hs mples ha he opon wll be exercsed a me 1 f and only f he asse prce a me 1 does no pull downward or upward he average asse prce oo much. Specfc dervaon of L 1 and U 1 can be seen more clearly from he example gven n 5. Dfferenang 24) and assumng an nerchange of dfferenaon and expecaon, we have E1 S 1 F 1 S 1 S 1 Ke r 1 h 1 U 1 S 1 S 1 = E E fh 1 U 1 S 1 S 1 S 1 Ke r 1 S 1 S 1 = U 1 S 1 h 1 L 1 S 1 S 1 E E fh 1 L 1 S 1 S 1 S 1 Ke r 1 S 1 S 1 = L 1 S 1 + E 1 S 1 F 1 S 1 S 1 Ke r 1 For he second erm of L gven by 23), we have E1 S 1 <F 1 S 1 S 2 F 2 S 2 S 2 Ke r 2 = E + h 1 U 1 S 1 S 1 h 1 L 1 S 1 S 1 h 1 U 2 S 2 S 2 n1 E 1 S 1 h 1 L 2 S 2 S 2 +hz 1 S 1 + +hz 2 S 2 / ) ) ) n2 K e r 2 fz 2 dz 2 S 2 S 2 ) fz 1 dz 1

14 where L 2 and U 2 are defned as n Lemma 2. Therefore, we have E1 S 1 <F 1 S 1 S 2 F 2 S 2 S 2 Ke r 2 h 1 L 1 S 1 S 1 = E fh 1 L 1 S 1 S 1 E1 S 2 F 2 S 2 S 2 Ke r 2 S 1 S 1 = L 1 S 1 h 1 U 1 S 1 S 1 E fh 1 U 1 S 1 S 1 E1 S 2 F 2 S 2 S 2 Ke r 2 S 1 S 1 = U 1 S 1 E 1 S 1 <F 1 S 1 h 1 L 2 S 2 S 2 fh 1 L 2 S 2 S 2 S 2 Ke r 2 S 2 S 2 = L 2 S 2 + E 1 S 1 <F 1 S 1 h 1 U 2 S 2 S 2 fh 1 U 2 S 2 S 2 S 2 Ke r 2 S 2 S 2 = U 2 S 2 + E 1 S 1 <F 1 S 1 S 2 F 2 S 2 S 2 Ke r 2 Smlarly, for he hrd erm of L gven by 23), we have E1 S 1 <F 1 S 1 S 2 <F 2 S 2 S T K + e rt h 1 L 1 S 1 S 1 =E fh 1 L 1 S 1 S 1 E1 S 2 <F 2 S 2 S T K + e rt S 1 S 1 =L 1 S 1 h 1 U 1 S 1 S 1 E fh 1 U 1 S 1 S 1 E1 S 2 <F 2 S 2 S T K + e r 2 S 1 S 1 =U 1 S 1 Wu and Fu / 65 E 1 S 1 <F 1 S 1 h 1 U 2 S 2 S 2 fh 1 U 2 S 2 S 2 E S T K + e rt S 2 S 2 =U 2 S 2 +E 1 S 1 <F 1 S 1 h 1 L 2 S 2 S 2 fh 1 L 2 S 2 S 2 E S T K + e rt S 2 S 2 =L 2 S 2 +E 1 S 1 <F 1 S 1 S 2 <F 2 S 2 S 2 K + e rt Combnng all hese resuls, we oban he PA esmaor for = 2 case: h 1 L 1 S 1 S 1 fh 1 L 1 S 1 S 1 EL S 1 S 1 = L 1 S 1 E S 1 Ke r 1 S 1 S 1 = L 1 S 1 h 1 U 1 S 1 S 1 fh 1 U 1 S 1 S 1 EL S 1 S 1 = U 1 S 1 + E S 1 Ke r 1 S 1 S 1 = U 1 S S 1 <F 1 S 1 h 1 L 2 S 2 S 2 fh 1 L 2 S 2 S 2 EL S 1 <F 1 S 1 S 2 S 2 = L 2 S 2 E S 2 Ke r 2 S 2 S 2 = L 2 S S 1 <F 1 S 1 h 1 U 2 S 2 S 2 fh 1 U 2 S 2 S 2 EL S 1 <F 1 S 1 S 2 S 2 = U 2 S 2 + E S 2 Ke r 2 S 2 S 2 = U 2 S S 1 F 1 S 1 S 1 Ke r S 1 <F 1 S 1 S 2 F 2 S 2 S 2 Ke r S 1 <F 1 S 1 S 2 <F 2 S 2 S T K + e rt ENDNOTE 1. Here, as n much of he relaed leraure we ce, we ake Amercan-Asan opons o mean Asan opons wh

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