1 Introducton Opton prcng teory as been te core of modern matematcal nance snce te dervaton of te famous Blac-Scoles (19) formula wc provdes a partal

Transcription

1 CIT-CDS - Dstrbuton-Based Opton Prcng on Lattce Asset Dynamcs Models Yuj YAMADA y James A PRIMBS z Control and Dynamcal Systems 1-81 Calforna Insttute of Tecnology Pasadena, CA 911, USA August, Abstract In ts paper, we propose a new opton prcng formula based on an arbtrarly gven stoc dstrbuton We rst formulate a European call opton prcng problem as an optmal edgng problem by usng a lattce based ncomplete maret model A dynamc programmng tecnque s ten appled to solve te mean square optmal edgng problem for te dscrete tme mult-perod case by assgnng sutable probabltes on te lattce, were te underlyng stoc prce dstrbuton s derved drectly from emprcal stoc prce data wc may posses \eavy tals" We sow tat tese probabltes are obtaned from a networ ow optmzaton wc can be solved ef- cently by lnear programmng A computatonal complexty analyss demonstrates tat te number of teratons for dynamc programmng and te number of parameters n te networ ow optmzaton are bot of square order wt respect to te number of perods umercal experments llustrate tat our formula generates te mpled volatlty smle n contrast wt te standard Blac-Scoles case In all correspondence, contact te rst autor, Y Yamada y E-mal: yuj@cdscaltecedu, Pone: , Fax: z E-mal: jprmbs@cdscaltecedu 1

2 1 Introducton Opton prcng teory as been te core of modern matematcal nance snce te dervaton of te famous Blac-Scoles (19) formula wc provdes a partal derental equaton governng te prces of dervatves and a teoretcal value for European call/put optons At te eart of te Blac-Scoles dervaton are te followng two ey assumptons Te rst s tat te underlyng stoc prce follows a geometrc Brownan moton, mplyng tat ts log return dstrbutons are Gaussan Te second s te assumpton tat tradng may tae place n contnuous tme However, tese assumptons fal to old n practce, because emprcal stoc prces are not Gaussan and ter margnal dstrbuton usually possesses eavy tals (eg, Mandelbrot (199) and Samorodnts (199)) Moreover, trades can not be placed contnuously n real marets Ts dscrepancy may results n derences between te Blac-Scoles prce and te true opton's prce, and appears as te \smle eect" or \mpled volatlty smle" n real marets (see Capter 1 of te boo by Hull (1999) and references teren) Based on emprcal maret beavor, opton valuaton tecnques ave been extended to more realstc assumptons for te underlyng stoc process (eg, Cox and Ross (19), Hull (198), Jarrow and Rudd (198), Merton (19)) and marets (eg, Hammarld (1998), Scal (199), and Scwezer (199)) In partcular, researc on prcng models wc tae nto account te smle eect and/or eavy tal penomena as become actve n recent years (eg, Derman and Kan (199), Karandar and Racev (199), Prner et al (1999), Potters and Boucaud (1998), and Rubnsten (199)) wt emprcal evdence tat mpled volatlty ncreases for n-te-money or out-of-te-money optons In ts body of lterature, Hull and Wte (198) developed a stocastc volatlty model and sowed tat te prce of a call opton s gven by te Blac-Scoles prce ntegrated over te dstrbuton of te mean volatlty under te assumpton tat te underlyng asset prce and ts volatlty are uncorrelated By usng ts prcng formula, one can generate an mpled volatlty smle under certan condtons, and te stocastc volatlty model produces eaver tals tan tose for Gaussan dstrbutons On te oter and, Jarrow and Rudd (198) sowed tat te prce of an opton depends on te urtoss parameter f te dstrbuton s eavy taled, by approxmatng te rs neutral probablty densty functon by te \true" dstrbuton A smlar formula s also presented by Potters and Boucaud (1998), were a cumulant expanson s appled to explan te relaton between te volatlty smle and te urtoss parameter Altoug bot results substantate te smle, tere s stll room for mprovement In te stocastc volatlty case, te volatlty smle s not as large as n real marets and te prcng model almost becomes Blac-Scoles for sort maturty optons In te latter case, te real dstrbuton may not be a good approxmaton to te rs neutral dstrbuton for long maturty optons, and ence, te prcng formula may not

3 be accurate for suc optons Te objectve of ts paper s to propose an opton prcng formula based on an arbtrarly gven stoc dstrbuton wc may possess eavy tals We rst formulate te opton prcng problem as an optmal edgng problem (eg, Hammarld (1998), Scal (199), and Scwezer (199)) on a tree-based ncomplete maret model Te mult-perod case s ten solved on a trnomal lattce usng dynamc programmng (see te orgnal lterature by Bellman (19), and te recent wor by Fedotov and Malov (1999) and references teren) In contrast to te Blac-Scoles formula, ts formula allows us to nd te prce of a call opton under any gven probablty dstrbuton for te underlyng stoc return drectly by assgnng sutable probabltes on te lattce We sow tat tese probabltes are assgned based on a networ ow optmzaton wc can be solved ecently by lnear programmng A computatonal complexty analyss sows tat te number of teratons for dynamc programmng and te number of parameters n te networ ow optmzaton are bot of square order wt respect to te number of perods, wc ndcates tat te total computatonal complexty s gly tractable; ts may be a great advantage n usng lattce-based computatonal models We also llustrate tat our formula generates te mpled volatlty smle by numercal experments Altoug one may tn tat our results are related to te so-called mpled bnomal trees developed by Derman and Kan (199) and Rubnsten (199) n te sense tat te probablty assgnment on trees s done based on emprcal data, an mportant derence s tat our model s based on real probabltes wereas mpled bnomal trees are based on rs neutral probabltes Ts paper s organzed as follows: We state te problem formulaton n Secton Secton provdes te man result and conssts of tree subsectons We rst solve te sngle perod trnomal tree mnmum edgng problem n Subsecton 1 In Subsecton, we formulate a networ ow optmzaton to assgn probabltes on a trnomal lattce Te algortm for prcng a call opton on a mult-perod lattce s presented n Subsecton, along wt a computatonal complexty analyss umercal experments are made to llustrate our proposed formula n Secton Secton oers some concludng remars Problem Formulaton We consder a maret consstng of two basc securtes, a rsy asset (or stoc) and a rsless bond, n te tme nterval t [; T ], were traders are allowed to purcase and sell te two basc securtes at dscrete tmes t = ; = ; 1; : : : ; = T = Let S be te prce of te stoc at t = t, were S maps to S +1 under a certan stocastc process, 1 and let B be te 1 We wll specfy a stocastc process for te stoc later

4 prce of te bond wt a constant nterest rate r wc satses B = (1 + r)b?1 ; = 1; : : : ; : (1) Also, we dene a portfolo to be a couple ( ; ) < ndexed by tme = : : : and let := S + B ; = : : : () be te value of te portfolo, were represents te number of sares of te stoc eld between t [t ; t +1 ) and represents te number of bonds eld by te trader Assume tat te portfolo s self-nancng, e,?1 S +?1 B = S + B ; 8 = 1 : : : : () Fnally, let C, = ; 1; : : : ; be te value of a European call opton wt a stre prce K, wc pays C = (S? K) + at maturty t = T Te objectve s to optmally edge te payo of te call opton troug a self-nancng tradng strategy and determne te value of te call opton at t = on te stoc wose value s S at t = If tere exsts a tradng strategy ( ; ) < ; = : : : suc tat te value of te portfolo perfectly replcates te nal payo of te call opton, e, = C, te ntal value of te call opton must be te same as te ntal value of te replcatng portfolo, or n fact, C = must old for all = ; : : : ;? 1 due to an absence of arbtrage argument Ts stuaton occurs wen te prce of te stoc at t = t, S, taes two possble states at t = t +1, e, S canges to eter us wt probablty p or ds wt probablty 1? p, were u and d are speced up and down rates satsfyng d < 1 + r < u and ud = 1 Let q be a number between and 1 dened as q = R? d u? d : () Ten, te ntal value of te call opton at = can be calculated as C = 1 (1 + r) KX? (x = ) S u d (?)? K () = were () s te bnomal dstrbuton dened as (x = ) := n! q (1? q) (n?)

5 n terms of q (; 1), and K s te mnmum nteger suc tat K = mn >? S u d (?)? K > : Here q s te so-called rs-neutral probablty or an equvalent martngale measure, and ts s te well-nown bnomal lattce opton prcng formula developed by Cox et al (199) In ts formula, te real probablty dsappears and te unque rs-neutral probablty domnates te prcng model Wen te stoc model s governed by a contnuous-tme geometrc Brownan moton process, t s nown tat tere exsts an analytc soluton for a European call opton based on te Blac-Scoles opton prcng formula It as been sown by Cox et al (199) tat te Blac- Scoles formula can also be derved by tang te lmt as! 1 n () under approprate up and down rates and tat perfect replcaton s possble wt a self-nancng portfolo by tradng contnuously based on dynamc edgng derved from te Blac-Scoles (19) formula Terefore we see tat marets are complete for bot te bnomal lattce and te Blac-Scoles case, were perfect replcaton s possble and prcng depends solely on te unque rs neutral probabltes However, emprcal marets are not complete n general, and perfect replcaton s never possble Moreover, emprcal stoc prces are not Gaussan n general and ter margnal dstrbutons usually possess eavy tals, wereas complete maret prcng models are usually based on te assumpton tat te underlyng stoc dstrbuton s Gaussan Ts dscrepancy may result n derences between te Blac-Scoles prce and te true opton's prce, and appears as te \smle eect" or \mpled volatlty smle" n real marets In ts paper, we wll develop anoter opton prcng formula based on ncomplete maret models to tae nto account ts emprcal maret beavor Consder te case were te prce of te stoc at t = t as more tan two possble states to tae at t = t +1, say us, ms and ds In ts case, perfect replcaton wt a self-nancng portfolo s not possble anymore because we ave more tan two constrants wt respect to te states, eg, us, ms and ds, for two varables, and As a result, te maret becomes ncomplete Wat we can do nstead for te ncomplete maret models s to solve te mean square optmal edgng problem as follows: Mean Square Optmal Hedgng (MSOH) Mnmze Subject to ( ; ) < satsfyng () for = ; : : : ;? 1: E (C? ) S ()

6 Here te expectaton n () s now dened over te real probabltes Once we nd te tradng strategy to solve te MSOH problem, te ntal value of te call opton sould be gven by C = = S + B : We also tae te value of te call opton at tme as C = ; = 1 : : :? 1: () ote tat, for te bnomal case, tere exsts a tradng strategy satsfyng C =, e, te objectve functon of te MSOH problem taes te mnmum and tat te real probabltes do not matter E (C? ) S = ; An mportant feature of MSOH for an ncomplete maret model s tat te prce of a call opton depends on ts real probabltes In ts paper, we develop an MSOH opton prcng tecnque based on an ncomplete maret model, and sow tat, by tang nto account te eect of te real probablty dstrbuton, our prcng formula generates an mpled volatlty smle on eavy taled dstrbutons, n contrast to te complete maret case were te underlyng stoc dstrbutons are Gaussan For smplcty, we treat te trnomal tree case as an ncomplete maret model; owever te extenson to larger numbers of trees s possble and stragtforward Dstrbuton based Opton Prcng for Incomplete Maret Models 1 Trnomal Tree Mnmum Hedgng To explan te dea, we rst consder te optmal edgng problem for te sngle-perod trnomal tree sown n Fg 1, were p ; = 1; ; are gven probabltes satsfyng p 1 +p +p = 1 Te tree resultng nodes represent multplcaton of te stoc value by u, m, and d, respectvely, were < d < m < u and m = ud Suppose tat te ntal prce of te stoc s S and tat S taes te value us, ms, or ds wt probablty p 1, p and p at te end of te perod Suppose tat te ntal value of te rs free asset s gven by B wt nterest rate R := 1 + r Te problem s to mnmze E (C? V ) = p 1 (C u? (us + RB)) + p (C m? (ms + RB)) + p (C d? (ds + RB)) (8)

7 over and, for a gven (p 1 ; p ; p ), (u; d; m) and stre prce K, were C u, C m and C d are dened as n Fg, respectvely Wt tese and, te ntal prce of te call opton s obtaned as C = S + B: Ts mnmzaton problem s solved as a quadratc programmng problem as gven n Lemma 1 below p 1 us C u = max ( us - K, ) p 1 S p ms C p C m = max ( ms - K, ) p p ds C d = max ( ds - K, ) Fg 1: Trnomal tree Fg : Prce of call opton Lemma 1 Let Q = Q T < nn be parttoned as Q = " # Q 11 Q 1 ; Q Q T 11 = Q T 11 < n 1n 1 ; Q = Q T < n n ; Q 1 < n 1n 1 Q were n 1 + n = n Suppose Q 11 >, and consder te followng quadratc mnmzaton problem: mnmze x T Qx subject to x = " # x 1 < n ; x 1 < n 1 x were x < n and Q < nn are gven Ten te quadratc mnmzaton problem (9) aceves te optmal value opt wt x 1 = x opt, were (9) opt = x T? Q? Q T 1 Q?1 11 Q 1 x x opt =?Q?1 11 Q 1x : (1)

8 Proof: See te appendx and Let Q = " # Q 11 Q 1 := Q T 1 Q x = E(S ) RBE(S )?p 1 us?p ms?p ds RBE(S ) RB?p 1 RB?p RB?p RB p 1 p p x 1 x := C u C m C d T < < (11) (1) were E(S ) = p 1 (us) + p (ms) + p (ds) E(S ) = p 1 us + p ms + p ds: Snce Q s symmetrc and Q 11 = Q T 11 >, te edged portfolo optmzaton problem wt (8) can be solved as te quadratc programmng problem (9) n Lemma 1 From (1), te optmal soluton [; ] T s calculated as follows, " # " =?Q?1 11 Q 1x = = = 1 RB Var(S ) 1 RB Var(S ) " " #?1 " E(S) RBE(S ) RBE(S ) RB RB?RBE(S )?RBE(S ) E(S ) #" RBE(S C )? RBE(S )E(C )?E(S )E(S C ) + E(S )E(C ) p 1 us p ms p ds p 1 RB p RB p RB E(S C ) RBE(C ) # # # C u C m C d : (1) Consequently we ave and = Cov(S ; C ) Var(S ) = E(S )E(C )? E(S )E(S C ) RB Var(S ) (1) C = S + B = Cov(S ; C ) Var(S ) S + E(S )E(C )? E(S )E(S C ) : (1) R Var(S ) 8

9 We see tat te optmal and are calculated drectly from gven parameters and tat C s obtaned n closed form Terefore, n terms of computatonal tractablty, ts opton prcng formula s as ecent as te bnomal prcng formula for te sngle-perod case ote tat a smlar condton can be derved for te cases of four or more trees by expandng Q and x n (11) and (1) Real Probablty Assgnment by etwor Flow For te mult-perod case, te stoc process s descrbed by usng a trnomal lattce as sown n te left-and sde of Fg, were te stoc prce S goes eter up (us ), mddle (ms ), or down (us ) at eac step In ts case te soluton metod to multperod optons can be obtaned by worng bacward based on dynamc programmng developed by Bellman (19) (see also a recent wor by Fedotov and Malov (1999) and references teren) In contrast to te bnomal lattce case were te real probabltes are not necessary, we need to specfy real probabltes on trees to solve te MSOH problem Ts s te essental derence between te bnomal and trnomal models and comes from te fact tat te maret s ncomplete, e, perfect replcaton s not possble on a trnomal lattce; ts fact enables us to develop an opton prcng formula from te real stoc prce dstrbuton e, te maret s not complete umber of samples σ µ σ Log return (%) Fg : Probablty assgnment from te real dstrbuton on trnomal lattce To specfy te probablty at eac node, we need a dstrbuton of te log stoc return as sown n Fg, were te gure on te rgt-and sde sows a stogram of te log return for te underlyng stoc durng a speced perod (eg, one year) If we dvde by te total 9

10 number of samples, ten te stogram can be consdered as a probablty dstrbuton for te log return Suppose ud = 1 and S = 1 Ten te stoc prce S at te n-t node from te top s S (n)?(n?1) = u wc s lnear n n on a log scale Snce te dstrbuton s a log return on te stoc and S = 1 (terefore ln(s ) s actually a log return), wat we do s matc te prce of te stoc wt te probablty dstrbuton at eac node as sown n Fg Ten we obtan te probablty to attan S (n) Let at te end S (n) ; [; ]; n [1; + 1] be te prce of te stoc at te -t perod on te n-t node from te top, were S (1) = S If ud = 1 and S = 1, ten S (n) ; [1; ] s gven by Also let S (n) = u?(n?1) : P (n) ; [; ]; n [1; + 1] be te probablty to attan S (n) (1), were P = 1 and +1 X n=1 P (n) = 1; 8 [; ]: Te prce of te stoc and te probablty to attan tat prce are arranged on te trnomal lattce as sown n Table 1 From te real dstrbuton, we already now P (n) ; n [1; + 1] Let p (n) ;u ; p(n) ;m ; p(n) ;d ; [; ]; n [1; + 1] be probabltes from S (n) to S(n), +1 S(n+1), and S(n+), respectvely Tese probabltes correspond to an upward, even, and downward move from one node to anoter on te trnomal tree, eg, p 1, p and p n Fg 1 We want to calculate probabltes p (n) ;u ; p(n) ;m ; p(n) ;d for all and n for a gven P (n) ; n [1; + 1] Tese probabltes are obtaned by solvng a lnear programmng problem as gven below Consder a networ ow wt a source s and a sn s as sown n Fg, were we ave egt nodes between te source s and te sn s We rst note tat te left and sde of te 1

11 S (1) S () S (1) S () S (1) S () S (1) S () S () S () S () S () S (?1) S ( ) ( +1) S P (1) P () P (1) P () P (1) P () P (1) P () P () P () P () P () P (?1) P ( ) ( +1) P Table 1: Stoc prce and te correspondng probablty dstrbuton networ conssts of a two perod trnomal lattce Eac ow travels from te left to te rgt and nally nto te sn s Let f (1) ;u (1) (1) >, f ;m > and f ;d > be te capactes of eac edge (n) > and f 1;d > be tose from te n-t node at te from te source and let f (n) (n) 1;u >, f 1;m rst perod as sown n Fg Also let f (n) > be capactes nto te sn from te n-t node at te end of te lattce We assume tat te total ow out of te source s one; terefore te total ow nto te sn s also one, e, X n=1 f (n) = 1: (1) We are nterested n ndng admssble capactes f ;u, (1) f ;m, (1) f (1), ;d f (n) 1;u, f (n) 1;m and f (n) 1;d for a gven f (n), satsfyng (1) for te capactated networ of Fg Te problem can be solved as a lnear programmng problem as follows: Snce te total ow s one, we ave f (1) ;u + f (1) ;m + f (1) ;d = 1: Te ow wt capacty f (1) ;u separates to f 1;u, (1) f 1;m, (1) and f (1), respectvely, and terefore Smlarly, 1;d f (1) 1;u + f (1) 1;m + f (1) 1;d = f (1) ;u : f () 1;u + f () 1;m + f () 1;d = f (1) ;m f () 1;u + f () 1;m + f () 1;d = f (1) ;d : 11

12 Moreover we ave ve lnear constrants wt respect to f (n) ; n [1; ]: f 1 = f (1) 1;u f = f (1) 1;m + f () 1;u f = f (1) 1;d + f () 1;m + f () 1;u f = f () 1;d + f () 1;m f = f () 1;d : (1) Fnally (1) s added to tese constrants otng tat all te constrants are lnear, te problem of ndng f ;u, (1) f ;m, (1) f (1), ;d f (n) 1;u, f (n) 1;m and f (n) 1;d can be solved as a lnear programmng problem We now replace f n (1) by te end pont probabltes P (n) ; n [1; ], respectvely, and solve te networ ow problem Ten te probabltes on trees are obtaned as follows: p (1) ;u p (1) ;m p(1) ;d p (n) 1;u p (n) 1;m p(n) 1;d were = u; m; d f n = 1; ;, respectvely = = f (1) ;u f (1) ;m f (1) ;d f (n) 1;u f (n) 1;m f (n) 1;d =f (1) ; P (1) f (1), u f (1) 1, u f (1) 1, m f (1) 1, d P () f (1) f () (1) f, m S S f (1), d P () f () f () P () f () P () Fg : etwor ows on trnomal lattce We now generalze te above results to te case wt any number of perods te result, we need to dene specal matrces denoted by E (n) 1 To state for a gven [;? 1] and

13 n [1; + 1] by E (n) = (n?1) I (?n+1) T < (+) (18) were m < m s a matrx of sze m wose entres are all zero and sgnes tat tere s no matrx, eg, E (1) denotes I T = I : Let f (n) < ; [;? 1] be a vector dened as T f (n) := < ; n [1; + 1]: f (n) ;u f (n) ;m f (n) ;d Also let te end pont probabltes P (n) ; n [1; + 1] be gven Ten (1) for = wt f (n) = P (n) s now rewrtten as P (1) P () Ṇ P ( ) ( +1) P = E (1)?1 E ()?1 E (?1)?1 f (1)?1 f ()?1 (?1) f?1 : (19) Snce te total value of te ow s one, eac value of te ow nto one node s te probablty tat te stoc prce taes tat value at tat node Terefore te followng condtons old: P (1) P () ḳ P () P (+1) = E (1)?1 E ()?1 E (?1)?1 f (1)?1 f ()?1 f (?1)?1 ; [1; ]; () were P (n) s te probablty to attan te value S (n) at te -t perod and n-t node Te left and sde of () s te value of te ow out of eac node, and ence leads to anoter condton: P (1) P () ḳ P () P (+1) = 1 T 1 T 1 T 1 1 T f (1) f () f () f (+1) ; [;? 1] (1)

14 were 1 T m < m s a vector wose entres are all one, eg, 1 = let g := ^E := (f (1) )T (f () )T (f () E (1) E () E () ) T (f (+1) E (+1)? I := dag 1 T ; ; 1T < (+1)(+) P := P (1) P () By usng ts notaton, we ave P ( +1) T < +1 : ^E?1 g?1 = I g ; [1;? 1] ) T T < + < (+)(+) T For [;?1], P = ^E?1 g?1 from (), (1) and (19) Ts togeter wt 1 g = I g = 1 yelds te followng teorem: Teorem 1 Suppose tat te end pont probablty P s gven Ten te probabltes on te edges wc aceve P at te end of trnomal lattce are obtaned by solvng te followng lnear programmng problem: Fnd: g > ; [;? 1] st?i ^E?I 1 ^E 1?I ^E??I? ^E??I?1 g g 1 g g? g? =?1 : () ^E?1 g?1 P Once g > ; p (1) ;u p (1) ;m p(1) ;d p (n) ;u p (n) ;m [;? 1] are found, te probabltes on te edges are gven as follows, p(n) ;d = = f (1) ;u f (1) ;m f (1) ;d f (n) ;u f (n) ;m were P (n) can be calculated by () or (1) f (n) ;d =P (n) ; [1;? 1]; n [1; + 1]; As sown n Teorem 1, te feasblty problem to nd probabltes to attan any gven end pont dstrbuton s solved as a lnear programmng problem However tese probabltes are 1

15 not unque n general To specfy unque probabltes, n te appendx we construct a quadratc objectve functon wc mnmzes te error between assgned probabltes and a set of target probabltes suc tat up, mddle, and down probabltes at eac node are as close to eac oter as possble Wt tese probabltes, we wll sow n Secton tat our formula provdes a good approxmaton to te Blac-Scoles soluton n te Gaussan case, and tat t generates an mpled volatlty smle for eavy taled dstrbutons Algortm and Computatonal Complexty Analyss We are now n a poston to present te algortm to nd te ntal prce of a call opton as follows: Algortm 1 Step 1): For a gven probablty dstrbuton for te log stoc return over a speced perod, coose u, d, m = p ud and te number of perods, and assgn te end pont probabltes P (n) ; n [1; + 1] as n Fg Step ): For speced end pont probabltes P (n) ; n [1; + 1], determne te probabltes on te edges for te perod trnomal lattce by solvng te networ ow optmzaton n () wt te quadratc objectve functon n te appendx Step ): Compute te ntal prce of te call opton C by dynamc programmng on te trnomal lattce wt probabltes obtaned n Step ) ote tat te dynamc programmng n Step ) conssts of a nte number of quadratc mnmzaton problems avng a closed form soluton smlar to (1) Terefore te calculaton to nd C?1 at one step s gly tractable Snce larger gves a more accurate representaton for te dstrbuton of te stoc at te end of te lattce, we need to consder te relaton between te total number of teratons and te number of perods Let () be te total number of teratons, were s te number of perods Snce we calculate C?1 at eac node bacward to nd C, te number of nodes from te ntal pont to te (? 1)-t nodes s () Terefore () s gven by () = X =1 (? 1) = : () Ts mples tat te number of teratons used n dynamc programmng s of square order wt respect to te number of perods We ave + 1 nodes at te end of trnomal lattce 1

16 We also need to consder te computatonal tractablty for te lnear programmng problem () n Step ) Let () be te number of varables g ; g 1 ; ; g?1 n () Snce lnear programmng problems can be solved ecently wt a polynomal tme algortm, () s consdered computatonally tractable f te order of () s polynomal wt respect to From te denton of g < +, we ave () =?1 X = wc s also of square order wt respect to ( + ) = () umercal Experments Te objectve of ts secton s to sow tat te solutons obtaned from trnomal lattce MSOH generate te mpled volatlty smle for eavy tal dstrbutons We also llustrate tat our formula provdes a good approxmaton to Blac-Scoles soluton n te Gaussan case Te orgnal data for ts smulaton s gven as follows: Openng prce S : $ Volatlty : % Rs free rate r f : 1 % Expected rate of return : 1 % Expraton T : umber of perods : Basc tme perod t : monts from now (8 trades a mont) 1/9? = 1= (1 8) We rst solve te mean square optmal edgng problem wt te followng up, mddle, down probabltes and rates (see Appendx ), [p 1 ; p ; p ] := [1=; 1=; 1=] m := exp ( t ) ; u := m exp? p t ; d := m exp?? p t ; were :=? = In ts case, te trnomal tree approxmates te dynamcs of te followng geometrc Brownan moton ds t = S t dt + S t dz: Te log-stoc dstrbuton at te expraton date T s gven by p ln (S T ) ln (S ) + T; T () 1

17 and s llustrated n Fg We are nterested n te relaton between te opton prces obtaned from trnomal lattce MSOH and te Blac-Scoles prces To see ts, we compute te opton prces for derent values of stre prces K as sown n Fg, were we also plot te Blac- Scoles prces Snce trnomal lattce MSOH results n Blac-Scoles prces almost exactly, t appears tat tere s only one lne altoug tere are actually two lnes; one for trnomal lattce MSOH and one for te Blac-Scoles formula p(x) log(s T ) Fg : Te log-stoc dstrbuton at te expraton date T for te Gaussan case ext, we solve te trnomal lattce mean square optmal edgng problem under eavy tal dstrbutons We generate eavy tal dstrbutons on a trnomal lattce based on a mxed probablty dstrbuton of a Gaussan dstrbuton p g (x) and a unform dstrbuton p u (x) as p m (x) = p g (x) + p u (x); > ; > n a nterval x [a; b], were and are wegts satsfyng + = 1 If =, p m (x) s gven by a Gaussan dstrbuton p g (x) Assume tat ln(s T ) follows te mxed dstrbuton p m (S T ) and tat p g (S T ) s gven by te Gaussan dstrbuton n () We coose te followng values for, = ; :1; :1; :; 1

18 1 1 1 Opton prce 8 8 Stre prce Fg : Stre prce vs opton prce for te Gaussan case and te mxed dstrbutons gven by tese 's are plotted n Fg wt respect to ln(s T ), were te sold lne denotes te Gaussan case = and tals become fatter as we ncrease We assgn te end pont probabltes based on te mxed dstrbutons and solve te networ ow optmzaton problem wt a quadratc objectve functon to determne md-pont probabltes on te trnomal lattce We ten solve te trnomal lattce MSOH problem for derent values of stre prce K for eac probablty dstrbuton, and computed mpled volatltes Fg 8 sows te mpled volatltes correspondng to te solutons, were te mpled volatltes are plotted vs te log of stre prces for derent values of 's ote tat dentcal lne styles are used n Fg 8 and for te underlyng stoc dstrbutons Clearly, we see tat fatter tals ncrease te smle eect Moreover, we realze tat, for te Gaussan case, te mpled volatlty lne stays at at %, wc concdes wt te Blac-Scoles soluton Concluson In ts paper, we ave presented an opton prcng formula for European style call optons based on an arbtrarly gven probablty dstrbuton We rst formulated an optmal edgng problem and provded a mnmum edgng tecnque for te sngle-perod opton prcng problem based on a trnomal tree Te problem for te mult-perod case was solved on a 18

19 p(s T ) log(s T ) Fg : Te mxed dstrbutons: te sold lne denotes te Gaussan case = and tals become fatter as we ncrease Impled volatlty log(stre prce K) Fg 8: Impled volatltes vs te log of stre prces for derent values of 's 19

20 trnomal lattce by usng te dea of dynamc programmng We ave sown tat te proposed metod allows us to nd te prce of a call opton under any gven probablty dstrbuton for te underlyng stoc return drectly by assgnng sutable probabltes on te trnomal lattce Tese probabltes are obtaned from a networ ow optmzaton wc can be solved ecently by lnear programmng A computatonal complexty analyss sowed tat te number of teratons for dynamc programmng and te number of parameters n te networ ow optmzaton are bot of square order wt respect to te number of perods, wc ndcates tat te total computatonal complexty s gly tractable umercal experments llustrated tat our formula generates an mpled volatlty smle n contrast wt te Blac-Scoles case Acnowledgments We would le to tan Professor Jon Doyle for elpful nsgts about eavy taled dstrbutons, and Professor Peter Bossaerts for dscussons about ncomplete maret models Te autors also would le to tan Professor Davd Luenberger at Stanford Unversty for s valuable comments Suggestons and comments by Professors Hros Konno and Hros Sraawa at Toyo Insttute of Tecnology were greatly apprecated References Bellman, R (19) Dynamcal Programmng, Prnceton Unversty Press, Prnceton, J Blac, F, and M Scoles (19) \Te Prcng of Optons and Corporate Labltes," Journal of Poltcal Economy, 81:{ Cox, JC, and SA Ross (19) \Te valuaton of optons for alternatve stocastc processes," Journal of Fnancal Economcs, :1{1 Cox, JC, SA Ross, and M Rubnsten (199) \Opton prcng: A smpled approac," Journal of Fnancal Economcs, :9{ Derman, E, and I Kan (199) \Rdng on a Smle," Rs, :18{ Fedotov, S, and S Malov (1999) \Stocastc Optmzaton Approac to Optons Prcng," Proc of te 1999 Amercan Control Conference, 1{1 Hammarld, O (1998) \On mnmzng rs n ncomplete marets opton prcng models," Int J Teoretcal and Appled Fnance, 1():{

21 Hull, J (1999) Optons, Futures, and Oter Dervatve Securtes, t edton Englewood Cls: Prentce-Hall Hull, J, and A Wte (198) \Te Prcng of Optons on Assets wt Stocastc Volatltes," Journal of Fnance, :81{ Jarrow, R, and A Rudd (198) \Approxmate opton valuaton for arbtrary stocastc processes," Journal of Fnancal Economcs, 1:{9 Karandar, RL, and ST Racev (199) \A generalzed bnomal model and opton prcng formulae for subordnated stoc-prce processes," Probablty and Matematcal Statcs, 1:{ Mandelbrot, BB (199) Fractals and Scalng n Fnance: Dscontnuty, Concentratons, Rs, Sprnger, ew Yor Merton, RC (19) \Opton prcng wen underlyng stoc returns are dscontnuous," Journal of Fnancal Economcs, :1{1 Prner, CD, AS Wegend and H Zmmermann (1999) \Extractng Rs-eutral Denstes from Opton Prces Usng Mxture Bnomal Trees," Proc of te 1999 IEEE/IAFE/IFORMS Conf on Computatonal Intellgence for Fnancal Engneerng, 1{18 Potters, M, R Cont and J-P Boucaud (1998) \Fnancal marets as adaptve systems," Europys Lett, 1():9{ Rubnsten, M (199) \Impled Bnomal Trees," Journal of Fnance, :1{818 Samorodnts, G, and MS Taqqu (199) Stable non-gaussan random processes: stocastc models wt nnte varance, Capman & Hall, ew Yor Scal, M (199) \On quadratc cost crtera for opton edgng," Mat Oper Res, 19:11{11 Scwezer, M (199) \Varance-optmal edgng n dscrete tme," Mat Oper Res, :1{ 1

22 Appendx Proof of Lemma 1 By expandng te matrx Q, we ave Q = " # T " I n1 Q?1 11 Q 1 Q 11 Q? Q T 1 Q?1 11 Q 1 #" Q? Q T 1 Q?1 11 Q 1? Q? Q T 1 Q?1 # I n1 Q?1 11 Q 1 ; I n were I m < mm s an dentty matrx of sze m Terefore te followng condton olds from Q 11 >, " # T " #" # x T ^x 1 Q 11 ^x 1 Qx = x x were = ^x T 1 Q 11^x 1 + x T x T? Q? Q T 1 Q?1 11 Q 1 x = opt ; 11 Q 1 x ^x 1 := x 1 + Q?1 11 Q 1x : otng tat x T Qx = opt olds wen x 1 = x opt, we conclude tat (9) aceves te optmal value opt wt x 1 = x opt A Quadratc Objectve Functon Here we pose a quadratc optmzaton to assgn up, mddle, and down probabltes at eac node tat are as close to eac oter as possble Te dea s to approxmate target probabltes at eac step suc tat te derences between te target probabltes and assgned probabltes are mnmzed Let ~p u ; ~p m and ~p d satsfyng ~p u + ~p m + ~p d = 1 be target probabltes We would le to mnmze te sum of te square error between te target probabltes and assgned probabltes as gven by X X?1 +1 =1 n=1 under te ow constrants n () capactes on ows, f (n) ;u, f (n) ;m?1 X =1 +1 X n=1 w (n) ;u ~p u? p (n) ;u + ~p m? p (n) ;m + ~p d? p (n) ;d (A1) As condton (A1) s not quadratc wt respect to te (n) and f ;d, we use te followng alternatve form, (n) (n) + w + w ~f (n)? ;u f (n) ;u ;m ~f (n) ;m? f (n) ;m ;d ~f (n)? ;d f (n) ;d ; (A)

23 were w (n), ;u w(n) ;m suc tat and w(n) ;d are wegts and ~ f (n) ;u, ~ f (n) ;m and ~ f (n) ;d are target capactes on ows ~p u = w (n) ;u ~ f (n) ;u ; ~p m = w (n) ;m ~ f (n) ;m ; ~p d = w (n) ;d ~ f (n) ;d : In te Gaussan case were te log of te stoc return follows a normal dstrbuton wt mean T and varance T, e, ln S S (T; T ); a stoc process gven by a geometrc Brownan moton can be approxmated almost exactly by a trnomnal lattce stoc process by coosng te followng parameters for te lattce, m = exp ( t ) ; u = exp t + p t ; d = exp t? p t ; ~p u = 1=; ~p m = =; ~p d = 1=; were t s a basc tme perod Moreover, based on tese target probabltes, we can always nd w (n), ;u w(n) ;m and w(n) ;d suc tat te end pont probabltes approxmate te Gaussan dstrbuton, and te resultng prcng formula provdes a good approxmaton to te Blac-Scoles soluton as llustrated n Secton Even f te stoc dstrbuton s gven by a non-gaussan dstrbuton wt eavy tals, we may stll use te wegts based on te approxmated Gaussan dstrbuton, because, n real marets, stoc dstrbutons usually approxmate Gaussans except n ter tals