EFFICIENCY IMPROVEMENTS FOR PRICING AMERICAN OPTIONS WITH A STOCHASTIC MESH: PARALLEL IMPLEMENTATION 1

Transcription

1 EFFICIENCY IMPROVEMENTS FOR PRICING AMERICAN OPTIONS WITH A STOCHASTIC MESH: PARAE IMPEMENTATION Absrac Thanos Avramds 2, Yury Znchenko 3, Thomas F. Coleman 4, Arun Verma 5 We dscuss a parallel mplemenaon of Mone Carlo smulaon algorhms for esmang he prce of Amercan-syle opons. We focus on he sochasc mesh mehod orgnally proposed n [3]. The mehod s sascal effcency was mproved by a basreducon echnque developed n [] and [2]. We repor resuls on he effcency of he parallel mplemenaon of hese wo algorhms on an SGI Orgn 2000 compuer wh up o 32 processors. Our concluson s ha he algorhm gans almos lnear performance mprovemen wh respec o he number of processors engaged n compuaons for moderae o large mesh szes.. Inroducon In he fnancal markes, sophscaed, complex producs are connuously offered and raded. Wh he ncreasng complexy of hese producs, Mone Carlo smulaon s seadly becomng an mporan ool used n valung and hedgng complex producs. In hs paper, we focus on Amercan opons, where we assume ha he opon can be exercsed dscreely, as opposed o connuously--ha s, he opon holder can exercse he opon a a fxed se of me pons (also called exercse opporunes, or sages up o expraon. Unl recenly, he prevalng opnon was ha Amercan opons could no be handled by Mone Carlo smulaon; e.g., Hull (997, p Recen developmens, however, have sared o pave he way for esmang Amercan opon prces va smulaon [,2,3,4]. An mporan mehod developed recenly for prcng Amercan opons va smulaon s he sochasc mesh mehod proposed n Broade and Glasserman (997a, henceforh BG997a. For a general-purpose mplemenaon of he mehod, Avramds and Hyden (999 observed severe bas n he mesh esmaors and developed a bas-reduced mesh esmaor ha drascally mproves he accuracy, measured as he nverse of he mean square error. For a more complee reamen of he bas-reduced esmaor, see Avramds (2000. Ths work was funded by he Fnancal Indusry Soluons Cener, a on venure of Slcon Graphcs Inernaonal and Cornell Unversy based n New York Cy. 2 Asssan Professor, School of Operaons Research and Indusral Engneerng, Cornell Unversy, Ihaca, NY School of Operaons Research and Indusral Engneerng, Cornell Unversy, Ihaca, NY Drecor, Cornell Theory Cener, Ihaca, NY 4853 and Drecor, Fnancal Indusry Soluons Cener, 55 Broad S New York, NY. 5 Cornell Theory Cener and Fnancal Indusry Soluons Cener

2 In hs paper, we repor on a parallel mplemenaon of mesh-ype esmaors. The paper s organzed as follows. For compleeness, we descrbe he esmaors n BG97 and A00 n Secon 2. Secon 3 conans he algorhm srucure and mplemenaon deals. In Secon 4 we repor compuaonal resuls on mngs and he effcency of he parallel mplemenaon. 2. Background: Amercan Opon Prcng and Mesh Esmaon e S (,..., n = S S denoe he vecor of secures underlyng he opon, modeled as a Markov process on R n wh dscree me-parameer = 0,,, T. The argumen = 0,,, T ndexes he se of me pons (n ncreasng order when he opon s exercsable, also called exercse opporunes or smply sages. e h( s be he payoff from exercse a me n sae s, dscouned o me 0 wh he (possbly sochasc dscoun facor recorded n S. Snce S s Markovan, he opon value a (me, sae par (s s obaned by dynamc programmng: h( T, s q( s = max{ h( s, c( s} for = T and all s for < T and all s where c( s s he dscouned value of he opon assocaed wh he decson o connue,.e., no exercse he opon a ( s, hereby holdng unl a leas sage +: c( s = E[ q( +, S + S s] ( = The quany c( s s called he connuaon value a ( s. Arbrage-prcng heory suggess ha he arbrage-free prce of he opon s obaned when he condonal expecaon n ( s wh respec o he Equvalen Marngale Measure (EMM. Under he EMM, he value of any radeable secury, dscouned o me 0, s a marngale. The problem s o compue he opon value a me 0, q(0, s 0, where s 0 s he known sae of underlyng asses a me 0. Example. In a smple applcaon, S s a vecor of d sock prces. A max call opon has d + payoff h( S = (max ( S,.., S K ; a geomerc average call opon has payoff d k / d ( S + k= h( S = K, where K s he srke prce and x + sands for max (x, 0. In revewng he mesh mehod, we follow BG997a. The mehod generaes a mesh of randomly sampled saes (also called pons S, =,,b for each =,,T. For convenence, we defne nonrandom mesh pons a sage 0 equal o he sae of underlyng secures a me 0, S 0, = s 0, =,,b. For =,,T, le g (. denoe he probably densy from whch he pons {S : =,,b} are sampled (o be specfed laer, and le f (x, denoe he condonal EMM densy of S + gven S = x. I s assumed ha f (x, exss for all x and s known n closed form or can be evaluaed numercally a neglgble cos. The hgh mesh esmaor of he opon value s defned recursvely: 2

3 , qh ( T, ST, = h( T, ST, =,..., b; (2a for =T-, T-2,,0, he hgh mesh esmaor s q H ( S = max( h( S, c( S, =,..., b, (2b where he esmaed connuaon value of each pon sampled a sage depends on he prevously esmaed connuaon values of all pons sampled a sage +: c ( S (3 b = q H ( +, S+, w( +, S, S+, b = where he weghs w(, are f ( S, S+, w( +, S, S+, =. (4 g ( S + +, The weghng of he combnaon of pons (S, S + above s necessary n lgh of he fac ha he pons a sage + were sampled from he densy g + ( nsead of he densy f (S, approprae for samplng a pah o esmae he connuaon value of pon S. The choce of denses g ( s crucal. For he res of he paper, we assume he mesh s generaed by samplng b ndependen and dencally dsrbued pahs of S : {S, = 0,, T}, =,, b are..d. pahs of S. (5 We call he par S, S + a paren and chld, respecvely, o ndcae he sochasc (, dependence. BG997a provde evdence ha a good choce s o sample b pahs as n (5 and vew he pons a sage + as a sample of dencally dsrbued pons from he average condonal EMM densy assocaed wh her parens: b g+ (S, u = f ( S, u, (6 b = where S = (,,,..., b. Noe ha g + (S, depends on all parens of pons sampled S = a sage and corresponds o forgeng he paren-chld relaonshp. We connue by defnng he mesh esmaors n Avramds (2000. A frs sep s o consruc he mesh low esmaor q (, whch s a based-low esmaor for he S, opon value a ( S obaned from whn he mesh. The dea behnd he consrucon s o use dson ses of pons for esmaon of he opmal exercse polcy and he esmaon of connuaon values (n case he esmaed opmal polcy s o connue. Unlke he BG997a mesh hgh esmaor, we remember he paren of each pon. For smplcy, assume b s even, and defne 3

4 b b b 2, B B = {,2,..., }, B2 = { +, + 2,... b}, and τ ( =. ( , B2 To calculae he low mesh esmaor a ( S, assume he low mesh esmaor of he opon value a each of he sampled pons a sage + has been calculaed. Defne he - h esmae of he opmal-exercse acon usng only he pons n B from sage +: e ( S = { h( S > c( S, B }, =, 2, (8 where { } s he ndcaor funcon of he correspondng even and c ( S, B = q ( +, S+, k w ( S, S+, k, (9 b / 2 where he weghs above are k B w ( S, S +, k = g f ( S +, B ( S, S +, k, S +, k. (0 The weghs n (0 correspond o vewng he pons { S+, k} k B as beng dencally dsrbued from a densy equal o he average condonal EMM densy assocaed wh her parens: g S, u. ( +, B ( =, u f ( S l b / 2 l B The low mesh esmaor of he opon value s defned recursvely:, q ( T, ST, = h( T, ST, =,..., b ; (2a for =T-, T-2,,0, he low mesh esmaor s b f ( S, S, + q ( S = e τ ( ( S h( S + [ e τ ( ( S ] q ( +, S+,, =,..., b = g+ ( S, S+, (2b By consrucon, each forward pon S +, s used n conuncon wh an esmae of he opmal exercse acon e τ (, based on pons oher han S +, (recall (7. ( We are ready o defne he recursvely averaged esmaor of Avramds (2000:, qa ( T, ST, = h( T, ST, =,..., b ; (3a for =T-, T-2,,0, he recursvely averaged esmaor s 4

5 H, A q A ( S ( q S q = H, A(, +, A( S, =,..., b, (3b 2, A H S, S, where q ( S and q ( S dffer from q ( and q (, respecvely, n ha n he former esmaors, we use he values of he recursvely averaged esmaor a sage +, q A ( +,, nsead of he values q H ( +, and q ( +, for he calculaon a sage respecvely. For movaon and resuls on he sascal effcency of hese esmaors, see Avramds ( Algorhm srucure and mplemenaon The followng hree esmaors were mplemened: The hgh mesh esmaor q H of BG997a The low mesh esmaor q of Avramds (2000 The recursvely-averaged esmaor q A of Avramds (2000 An algorhm for he compuaon (ncludng parallel processng of q follows (he code for he oher wo esmaors s smlar and s hus omed. Sep. Generae random mesh pons as n (5 Sep 2 (Backwards recurson: =T; Compue opon values as n (2a. For = T-, T-2,, 0: For =, 2,, b % IN PARAE Compue he densy funcon as n (6 wh u = S, For =,2 For k B % IN PARAE Compue he densy funcon as n ( wh u = + (O(b work S, k + (O(b work For =, 2,, b % IN PARAE For =,2 Compue he wegh as n (0 Compue he connuaon value as n (9 (O(b work Compue he esmae of he opmal exercse acon as n (8 Compue low mesh esmaor as n (2b (O(b work For saemens ha nvolve work ha grows wh b, we ndcaed he work requremen n parenheses n O( noaon. The algorhm complexy as a funcon of T and b s as 5

6 follows. The generaon of mesh pons n Sep requres O(Tb work. The backward compuaon of opon values for all mesh pons n Sep 2 has oal work requremen of order O(Tb 2. Snce he work n Sep s an asympocally neglgble fracon of oal work as b ncreases, we mplemened parallel processng only for Sep 2 (clearly, he generaon of pahs n Sep could also be parallelzed. The For loops labeled % IN PARAE are execued n parallel, as he compuaons for each loop eraon are ndependen of each oher. Ths analyss shows ha he O(Tb 2 work of mesh-ype algorhms can be parallelzed up o b processors wh almos perfec effcency, n whch case he execuon me wll be approxmaely O(Tb, an mpressve speedup. We mplemened all hree esmaors n he C++ programmng language, whch s convenen for dealng wh dynamcal daa srucures. The mplemenaon was done n he obec-orened framework and he followng classes/emplaes were creaed: emplae class Vecor used o sore he nformaon on he underlyng secures and esmaors for each node n he mesh, used o creae a class for mesh pons a a parcular sage and also used as a base class for a whole mesh class TSeed represens a sngle pon of a mesh class TPSeedVecor conaner class for all he mesh pons a sage class TMesh conans a mesh daa srucure self and s able o resample mesh, compue he esmaors ec Here s a dagram showng how hese obecs are nerrelaed: Class TSeed - Vecor<floa> underlyng secures - Vecor<floa> esmaors - Vecor<n> forward lnks Templae <class T> class Vecor Class TPSeedVecor Vecor<TSeed*> Class TMesh: publc Vecor<TPSeedVecor> 3. Parallel Implemenaon For parallel mplemenaon, we used he OpenMP lbrares on he SGI s Orgn 2000 machne. The pragma drecves were used for he auomac parallelzaon of he C++ code. No changes o he orgnal code were necessary. We smply denfed he parallel porons and added he pragma calls before each parallel-execuon loop as shown below. #defne _MAKE_PARAE_ % nalze 6

7 #fdef _MAKE_PARAE_ % ypcal parallel For loop #pragma omp parallel for #endf For =, 2,, b % IN PARAE 4. Compuaonal Resuls Expermens were run for wo opon ypes (max and geomerc average payoff funcon for dfferen mesh szes and varyng dmensonaly of he opon (for he case of he geomerc average opon and number of exercse opporunes (for he case of he max opon. The machne used was an SGI Orgn 2000 cc-numa mulprocessor server. Tmng resuls were obaned wh up o 32 processors. Snce all hree esmaors have smlar compuaonal complexy, we repor execuon mes only for q A n Table. As expeced, gven a fxed number of processors engaged, he execuon me grows approxmaely lnearly wh he number of exercse opporunes T and he number of underlyng secures n, and grows quadracally wh he number of mesh pons per sage b. For large mesh szes (b =2048 we acheved almos perfec parallel effcency, wh an approxmae speedup by a facor of 28 on 32 processors. The speedup mproves wh he mesh sze. Table. Execuon me (seconds on SGI Orgn Opon Type Geomerc Average Max # asses d Mesh sze b # processors -> b = b = b = b = b = b = # sages T 3 b = b = b = b = b = b = b = b = b =

8 4. Parallel Effcency Resuls We defne: Speedup = seral me/parallel me. Parallel effcency = Speedup/P, where P s he number of processors Speedup Char for varable mesh szes (Geo - 7 case SPEEDUP Number of processors 4.2 Emprcal sudy of seral and parallel componens va regresson To separae he seral and parallel componens of he code execuon me, we used he followng formulas o represen parallel and seral execuon mes: seral = c*(s+s2*n+s3*n^2 parallel = c*(p+p2*n+p3*n^2. Here c s a consan N = b/52 measures he normalzed mesh sze, and s, s2, s3 and p, p2, p3 are he coeffcens o be deermned. The regresson model we esmaed s Toalme = seral + parallel/p + error, where P s he number of processors. The esmaed coeffcens were: s= 0.20, s2 = 0.50, s3 = , p = , p2 = 0.240, p3 = Ths confrms our expecaon ha he quadrac componen of execuon me s domnan. The char below compares he acual and regresson-fed execuon mes; noe ha regresson resuls very accuraely mach he daa. 8

9 2500 regresson daa real daa(mes 2000 TIME (n seconds DATA POINTS 5. Concluson In hs paper, we descrbe a parallel mplemenaon of Mone Carlo smulaon algorhms for he esmaon of Amercan-syle opon prces. We focus on he sochasc mesh mehod developed n [3] and [] as a general-purpose Mone Carlo algorhm for addressng he esmaon problem. Three mporan mesh-ype esmaors were mplemened usng C++ on SGI Orgn 2000 cc-numa mulprocessor server. A drec algorhm analyss and an emprcal sudy demonsrae ha almos perfec parallel effcency can be acheved. For moderaely large mesh szes we acheved almos perfec parallel effcency,.e., execuon me declnes almos lnearly wh he number of processors used, and almos perfec processor ulzaon s acheved. The resulng benefs are: much faser esmaon s feasble for compuaonally-nensve and/or crcal applcaons; as Amercan opon prcng n dmenson greaer han 3-4 s sll a hard problem compuaonally, fas esmaon by parallel processng may faclae furher advances n research. 9

10 References. Avramds, A. N Effcency mprovemens for prcng Amercan Opons wh a sochasc mesh. Workng paper, School of ORIE, Cornell Unversy, Ihaca, NY Avramds, A. N., and P. Hyden Effcency mprovemens for prcng Amercan opons wh a sochasc mesh. Proceedngs of he 999 Wner Smulaon Conference, Broade, M., and P. Glasserman. 997a. A sochasc mesh mehod for prcng hghdmensonal Amercan opons. Unpublshed manuscrp. 4. Broade, M., and P. Glasserman. 997b. Prcng Amercan-syle secures usng smulaon. Journal of Economc Dynamcs and Conrol, 2, Hull, J Opons, Fuures, and oher dervaves, 3 rd Edon, Prence-Hall. 0