Computation of Greeks Using Binomial Tree

Transcription

1 Journal of Mathmatcal Fnanc, 07, 7, ISSN Onln: 6-44 ISSN Prnt: Computaton of Grks Usng Bnomal Tr Yoshfum Muro, Shntaro Suda Graduat School of conomcs and Managmnt, Tohoku Unvrsty, Snda, Japan Mtsubsh UFJ Trust Invstmnt Tchnology Insttut Co., Ltd. (MTC, Tokyo, Japan How to ct ths papr: Muro, Y. and Suda, S. (07 Computaton of Grks Usng Bnomal Tr. Journal of Mathmatcal Fnanc, 7, Rcvd: Aprl, 07 Accptd: July 4, 07 Publshd: July 7, 07 Copyrght 07 by authors and Scntfc Rsarch Publshng Inc. Ths work s lcnsd undr th Cratv Commons Attrbuton Intrnatonal Lcns (CC BY Opn Accss Abstract Ths papr proposs a nw ffcnt algorthm for th computaton of Grks for optons usng th bnomal tr. W also show that Grks for uropan optons ntroducd n ths artcl ar asymptotcally quvalnt to th dscrt vrson of Mallavn Grks. Ths fact nabls us to show that our Grks convrg to Mallavn Grks n th contnuous tm modl. Th computaton algorthm of Grks for Amrcan optons usng th bnomal tr s also gvn n ths artcl. Thr ar thr advantagous ponts to us bnomal tr approach for th computaton of Grks. Frst, mathmatcs s much smplr than usng th contnuous tm Mallavn calculus approach. Scond, w can construct a smpl algorthm to obtan th Grks for Amrcan optons. Thrd, ths algorthm s vry ffcnt bcaus on can comput th prc and Grks (dlta, gamma, vga, and rho at onc. In spt of ts mportanc, only a fw prvous studs on th computaton of Grks for Amrcan optons xst, bcaus prformng snstvty analyss for th optmal stoppng problm s dffcult. W blv that our mthod wll bcom on of th popular ways to comput Grks for optons. Kywords Optons, Grks, Bnomal Tr. Introducton Grks ar quantts that rprsnt th snstvty of th prc of drvatv scurts wth rspct to changs n th prc of undrlyng assts or paramtrs. Thy ar dfnd by drvatvs of th opton prc functon wth rspct to paramtrs such as th prc of undrlyng assts, volatlty lvl, and spot ntrst rat. Th computaton of ths snstvts s vry mportant for rsk managmnt, for an xampl on ths and smlar topcs, rfr to Hull []. Th rcnt dvlopmnt of th Mallavn calculus approach n fnancal mathmatcs nabls us to comput Grks for a varous knds of contngnt clams, for a DOI: 0.436/jmf July 7, 07

2 prvous rsarch on ths topc, s Fourné t al. [] and Kohatsu-Hga and Montro [3] among othrs. S also Brns t al. [4] for th computatonal mthods of Grks for xotc optons such as knock-out optons and lookback optons. In ths studs, Grks wr usually rprsntd by xpctaton formulas drvd from th Mallavn calculus; ths xpctatons ar computd usng Mont Carlo smulatons. Many prvous studs hav focusd on th computatonal mthods of Grks for uropan optons and xotc optons such as Asan optons. In rcnt tms, svral studs on th computaton of Grks for Amrcan optons hav bn rportd. S Gobt [5] and Bally t al. [6], for xampl. On can obtan Grks by smply takng th dffrntal of th opton prcng functon f th closd-form prcng formula s known. Howvr, on gnrally cannot comput Grks for Amrcan optons usng ths drct mthod bcaus th xplct formula for th prc of Amrcan optons s not known. Th most drct approxmaton mthod to drv Grks for Amrcan optons s th fnt dffrnc mthod. Howvr, t s wll known that th fnt dffrnc mthod somtms lads to unstabl rsults. In ordr to ovrcom ths shortcomng, Mallavn calculus approachs combnd wth Mont Carlo mthods ar wdly usd. Dspt rcnt progrss of th Mont Carlo smulaton approach to th prcng of Amrcan optons, such as Longstaff and Schwartz [7], t s not asy to drv th prc and Grks for Amrcan optons usng ths approach. Although Gobt [5], Bally t al. [6] consdrd th Mont Carlo smulaton an approach to comput Grks for Amrcan optons, th approach s mathmatcally dffcult to undrstand. Thrfor, th fnt dffrnc approach s stll wdly usd n practcal purposs to comput vga and rho. Thrfor, w ntroduc a nw approach n ths papr. Muro and Suda [8] [9] proposd nw mthods for th computaton of Grks for uropan optons usng th bnomal tr mthods of Cox t al. [0] and th dscrt Mallavn calculus ntroducd by Ltz-Martn [] and Prvault [] [3]. S also Prvault and Schoutns [4]. In that artcl, w took drvatvs of th xpctaton form-formula for uropan optons drctly and computd t furthr usng rlatonshps btwn th dscrt Mallavn calculus and th dscrt Skorohod ntgrals. Although ths s an lgant way to drv Grks for uropan optons, ths mthod stll rqurs a closd-form formula for th opton prc to drv opton Grks. Muro and Suda [8] [9] took drvatvs of th prcng formula for uropan optons, howvr, n ths artcl w tak drvatv at ach nod on th bnomal tr to drv Grks for Amrcan optons. In othr words, w mployd a stp-by-stp approach. In ths artcl, w also show Grks for uropan optons ar asymptotcally quvalnt to dscrt vrson of th Mallavn Grks and th bnomal Grks ar convrgng to Grks n th contnuous tm modl. Ths modl s also xtndd to th jumpdffuson modl n Suda and Muro [5]. Chung t al. [6] also provd th bnomal dlta for plan vanlla uropan optons s convrgng to dlta n th contnuous tm modl. Th rmandr of ths papr s organzd as follows. In Scton, w gv brf 598

3 xplanatons on th bnomal tr mthods of Cox t al. [0]. Th computatonal mthods of Grks ar ntroducd n Scton 3. Computaton algorthms of Grks for Amrcan optons ar dscussd n Scton 4. Th numrcal rsults ar gvn n Scton 5 followd by th concludng rmarks n Scton 6. In th Appndx, w show th drvd Grks convrg to th Mallavn Grks n Black and Schols modl.. Bnomal Tr In ths scton, w brfly xplan th bnomal tr modl of Cox t al. [0], whch has bn xpland n many txtbooks such as Hull []. Th basc da for computaton of Grks for uropan optons s also prsntd. On th bass of ths xplanatons, th computatonal mthods of Grks usng th bnomal tr ar xpland n dtal n th nxt scton. Introduc ndpndntly and dntcally dstrbutd random varabls on th probablty spac ( Ω,,Q, whr s a random varabl { } =,, N wth probablty Q t = = p, Q t = = p. Th tm stp t s fxd as T N t, gvn by W t = N =. W ntroduc a random walk procss, { },, W = + + t. On can rgard th stochastc procss { W t} =,, N wth p = as an ap- proxmaton of th standard Brownan moton. On th othr hand, th stocha- W s gnrally rgardd as an approxmaton of th = p Brownan moton wth drft n th gnral cas ( p. Th bnomal tr s a computatonal mthod for prcng optons on scurts whos prc procss s govrnd by th gomtrc Brownan moton dp = P rdt+ d Z, P = s, ( stc procss { t},, N ( 0 t t t whr { Zt} t 0 s a standard Brownan moton undr th rsk-nutral masur Q. A bnomal tr s constructd n th followng mannr. W consdr a modl wth N prods and assum that th maturty dat of th optons s fxd as T = N t. If th prc of undrlyng assts at tm t( = 0,,, N s gvn by S t, th prc movs to us t or ds t( d < < u n th nxt tm prod ( + t. Th probablty of th prc movng upward to ( us t s p and downward to ( ds t s p. In ordr to construct th bnomal tr so that th xpctaton and varanc ar consstnt wth th gomtrc Brownan moton (, w fx th paramtrs u, d, p as r d u =, d =, p = u d whr w assum an addtonal rlatonshp, u = d. Consdr uropan optons wth a pay-off functon Φ ( and a maturty dat T. Th prc of optons at tm t s dnotd by ( x, f th prc of undrlyng assts s x at tm t. Th prc s gvn by th backward nducton algorthm 599

4 ( r ( x, t = p ( xu, ( + t + ( p ( xd, ( + ( xn, t =Φ( x. ( On can drv th closd-form formulas for Grks for uropan optons usng th dscrt Mallavn calculus (S Muro and Suda [8]. 3. Computaton of Grks for uropan Optons Nw computatonal mthods of Grks for uropan optons ar prsntd n ths scton. Although computatonal mthods of dlta and gamma usng th bnomal tr hav alrady bn proposd by Plssr and Vorst [7], computa- tonal mthods of othr Grks such as vga and rho usng th bnomal tr hav not bn dply studd, xcpt n th fnt dffrnc approach. Bcaus th opton prc s gvn by th wghtd sum of th pay-off functon, w can comput Grks f w assum that th pay-off functon s smooth. Howvr, th pay-off functon for, say, uropan call optons s not smooth. Thrfor, th smoothnss of th pay-off functon s too strong to b assumd for practcal purposs. In ths scton, howvr, w comput Grks undr th followng assumpton. W wll show that th obtand formulas n ths scton convrg to Grks n th contnuous tm modl undr th mldr condtons. It s dscussd n Scton 7.. Assumpton. W assum that th pay-off functon Φ ( s a smooth functon. 3.. Computaton of Dlta In ths subscton, w calculat dlta, whch s usd to masur th snstvty of th opton prc wth rspct to changs n th prc of undrlyng assts. Dlta s gvn by th frst drvatv of th opton valu functon wth rspct to th prc of undrlyng assts. Dlta for uropan optons s computd as r ( s,0 p ( s, t ( p ( s, t = + s s (3 r d d = p ( s, t + ( p ( s,. dx dx z Now, w hav two approachs to comput ( s z, on s gvn by ( s, ( s, z=±. Th frst z ( s, t = + O(, (4 z z=± and th scond on s gvn by z d z ( s, t = ( x, t s. z dx ± t z=± z=± t x= s Ths two formulas yld th approxmaton formula for th drvatv d ( x, : dx ± x= s 600

5 ( s, ( s, Y. Muro, S. Suda d ( x, t = + O(. (5 ± dx ± s x= s Substtutng ths formula n (3 lads to th followng rlaton ( s, t t ( s, t( t r + ( s,0 = + O(. s (6 Taylor xpanson ylds th approxmaton formula for p, and t s gvn by 3 p = + µ t + O( t, (7 µ = r whr µ s a constant gvn by. Lt us comput th xpc- taton ( s, (. Th abov approxmaton formula lads to (, ( s t t (, ( µ ( (, ( µ ( s, t ( s, t O( t. = p s t + p s = + + Not that w hav usd th dntts t t ( ( s, (, t + s t = ( s, t + O( ( s (, t s, t = ( s z, t + O( t = O( z to drv th last qualty. Ths formula allows furthr computaton of (6. It s gvn by z= 0 r ( s,0 ( s, t( µ t O( t = + s r s s s (, ( µ (,0 W call ths dlta, whch s th on-stp vrson of dscrt Mallavn dlta gvn n Muro and Suda [8]. Actually, w can show th strongr rsult, ( s ( s O( t (8 (9,0 =,0 +, (0 f w assum smoothnss to th pay-off functon. Ths s organzd as th followng thorm, bcaus w us th Formula (0 to show th accuracy of Vga. Thorm. W assum that th pay-off functon Φ ( as a smooth functon. W can stmat th accuracy of dlta as ( s ( s O( t,0 =,0 +. [Proof] Frst w comput a hghr ordr trm for th rror trm for dlta. Taylor xpanson, 60

6 t ± z ( s, t = ( s, t + ( s, ( z z=± z 3 + ( s, ( t + O (, z ylds th hghr ordr trm for qualty (4. It s gvn by z z ( s, t = z=± z=± ( s, ( s, z ± ( s, t + O(. z z=± Ths lads to th hghr ordr xpanson formula for th dlta gvn by (6: ( s, t + ( s, ( r ( s,0 = s ( r z z + p ( s, ( p ( s, t + O(. s z z z= t z= Th scond trm n ( s calculatd as p s t p s t z z ( t z z (, ( (, z= t z= 3 = µ z z z= 0 z= 0 = O z z ( s, t ( s, t t O( t, whr w hav usd th approxmaton (7. Ths rsult nabls us to comput dlta gvn by (. It s computd as by ( s, t t ( s, t( t r + ( s,0 = + O(. s Applyng th Formula (8, th hghr ordr xpanson for dlta s fnally gvn Ths shows th thorm. ( s ( s O( t,0 =,0 +. ( Dscrt Mallavn Grks for uropan optons usng an N-stps bnomal tr wr obtand by Muro and Suda [8]; thy usd th dscrt Mallavn drvatvs ntroducd by Ltz-Martn []. S also Prvault [] [3]. Dscrt Mallavn dlta s gvn by D rn D = Φ s W N s N WN ( ( N t µ. As dscussd n Appndx, dlta and dscrt Mallavn dlta s actually quvalnt. Bcaus Muro and Suda [8] usd th dscrt vrson of th Mal- lavn calculus approach, on cannot us thr mthod to drv Grks for Am- rcan optons. In our study, w xplot th stpws approach to drv Grks 60

7 for Amrcan optons. S Scton 4 for a dtald dscusson on th computaton of Grks for Amrcan optons. If w xplot anothr approxmaton formula for th drvatv d ( x,, w can obtan anothr approxmaton formula for dlta. If dx ± x= s w us a mor drct formula for th drvatv, t ± ( s, ( s, d ± ( s, t ± dx s s (3 ( s, ( s,, s w gt anothr approxmaton formula for dlta for uropan optons n th Hull on-prod tm modl. Substtutng (3 n (3 ylds, and t s gvn as ( s, ( s, Hull ( s,0. s Ths s th dlta ntroducd by Hull []. Ths fact rvals that th two dffrnt d ± t approxmaton formulas for ( s, t gvn by (5 and (3 yld two dx dffrnt approxmaton formulas for dlta. Our computatonal rsults ndcat that dlta convrgs a lttl bt fastr than dlta ntroducd by Hull []. D Morovr, as shown n th Appndx, th formula, ( s,0 = ( s,0, s actually satsfd. Ths mans that dlta s mor natural rprsntaton of dlta than Hull s dlta. Lastly, w can show th convrgnc of dlta to dlta n Black and Schols modl, vn f w do not assum th smoothnss to th pay-off functon. Ths s shown n Appndx. 3.. Computaton of Gamma In ths subscton, w calculat gamma, whch s usd to masur th snstvty of dlta wth rspct to changs n th prc of undrlyng assts. Gamma s gvn by th scond drvatv of th opton valu functon wth rspct to th prc of undrlyng assts. Th prcng algorthm for uropan optons ( ylds dlta for uropan optons: r ( s,0 = p ( s, t + ( p ( s, x x { ( ( ( } r = p s, t + p s,. Applyng chan rul to ( s z, lads to z ( s, ± ( s, t z =. z s z=± Dlta for uropan optons s gvn by Actually, dlta s dply rlatd to Hull s dlta. W hav a rlaton, ( s,0 ( s,0 O( t = +, bcaus w hav Formulas (8 and (9. r t Hull 603

8 r z z ( s,0 = p ( s, t ( p ( s, t. s z + z= t z z= Takng drvatv wth rspct to th prc of undrlyng assts ylds gamma for uropan optons as r z z ( s,0 ( s,0 p ( ( s, t Γ = + s s z Usng th approxmaton formula z z z ( p ( ( s t +,. z z= z ( s, t z z=± (, (, s s = + O allows furthr calculaton of gamma as z= ( t r ( s,0 Γ = ( s, ( ε s r + s t + O s r = ( s, ( s r + s t + O s r ( s, ( s r + ( s, ( s Γ s,0 ( (, ( µ ( (, ( µ ( (4 W call ths formula gamma. Ths formula s vald only f th pay-off functon Φ ( s a smooth functon. In ths cas, th ordr of th rror trm s O t Γ s,0 =Γ s,0 + O t. Scond qualty n (4 s shown (,.. ( ( ( by usng th Formula (0. W wll also prov that gamma s asymptotcally quvalnt to th Gamma n th Black and Schols modl n th Appndx Computaton of Vga Th computatonal mthod of vga for uropan optons s prsntd n ths subscton. Vga s th snstvty of th opton prcng formula wth rspct to changs n volatlty lvl,. It appars that th computatonal mthods of vga and rho usng th bnomal tr hav not yt bn consdrd srously, xcpt for th fnt dffrnc approach. S Hull [] for computaton of vga usng th fnt dffrnc approach wth th bnomal tr. Lt us assum that th prc for undrlyng assts at tm t s gvn by s t. Th prc and vga for uropan optons at tm t ar dnotd by ( s, ; and 604

9 ( s t, ;, rspctvly. Vga for uropan optons s gvn by takng drvatvs to th prcng Formula (: { ( ( } s, t; = p s, + ; r ( ( t t ( ( + p s, + ; = + +, 3 whr,, ar gvn by 3 r p = ( s t,( + ; ( p + ( s t,( + ; r = p ( s t,( + ; s t x + ( p ( s t,( + ; s t ( x r + ( ( + = s t, + ; s t + + ( ( r 3 = s, + ;. Th drvatv of p wth rspct to s gvn by r ( ( + r 3 p= t = + t + O(. 4 Ths rsult and th approxmaton Formula (8 lad to th approxmaton formula for : r r = + ( s t,( + ; t + ( s t,( + ; ( 3 + O( r t r = + ( s t,( + t; + O(. For furthr computaton of, on nds dlta at tm ( + t. Morovr, t s ncssary to comput vga for uropan optons at tm ( + t to valuat. Ths mpls that on has to us th backward nducton algorthm 3 to comput vga for uropan optons. Vga at th maturty dat s gvn by ( sn t, N t; = 0. Ths rsults ar combnd to form th computatonal formulas for vga: r t r + + ( s t, t; = + ( s t,( + r t ( s t,( + ; s t + (5 r ( s t,( + t; + O(. Dlta and vga at tm ( + t ar substtutd by dlta and vga to gt 605

10 vga at tm t, rspctvly. In othr words, vga at tm t s dfnd rcursvly by th backward algorthm, r s + ( s (, ;,( r t t (,( ; + ( ( + + r + + s t t s t + r + s, + ;. (6 W hav th followng thorm on Vga. Thorm Lt us assum that th pay-off functon Φ ( s a smooth functon. Th accuracy of th vga s gvn by ( s = ( s + O(,0;,0;. W nd a lttl bt mor ffort show ths fact and t s shown n Appndx. W also show th convrgnc of vga to th vga n th Black and Schols modl vn f w do not assum th smoothnss of th pay-off functon n th Appndx. It should b notd that n ordr to comput Vga, w formally N assum = 0 f th pay-off functon Φ ( s not smooth on. As an altrnatv approach, th fnt dffrnc approach s usd to obtan vga for prac- tcal purposs. In many cass, t works wll: howvr, n som t dos not, for xampl on cannot obtan a stabl stmator of vga for dgtal optons Computaton of Rho Th computatonal mthod of rho for uropan optons s dscussd n ths subscton. Rho masurs th snstvty of th opton prc wth rspct to changs n th spot ntrst rat lvl, r. It s dfnd by th frst drvatv of th opton valu functon wth rspct to th spot ntrst rat, r. Th prc and rho for uropan optons ar dnotd by ( s, r ; and ( s, tr ; tvly, f th prc of undrlyng assts at tm t s gvn by culaton ylds whr ρ, ρ, ρ ar gvn by 3 ( s t tr = ρ, ; ρ ρ ρ, ρ + ( ( r = t s, + ; r ( ( ( p ρ, rspc- t s. Smpl cal- t ( ( r p ρ = s, + tr ; + s, + r ; r r Th drvatv p r ( ( r + ρ3 = ρ s, + r ;. n ρ s gvn by r p= t = + O r ( 3. Ths rlaton and th Formula (8 lads to th furthr calculaton, 606

11 ρ ( s t,( tr ; r ρ = ( s t,( + tr ; + O( r + 3 = ( s t,( + r ; ( + t + O(. In ordr to comput ρ, on nds to comput rho at tm 3 ( + t,.., + ρ (, ( ; s t + r. Ths mpls that on has to us th backward nducton approach to valuat rho. Ths rsults ar combnd to form th computatonal formulas of rho for uropan optons: r t + µ + ρ( s t, tr ; = t s t,( tr ; ρ( s t,( + tr ; + O(. ( In ordr to comput rho at tm t, rho at tm ( + t s substtutd by rho at tm ( + t. In othr words, rho s dfnd rcursvly by th backward algorthm, ρ µ s, r ; t s, tr ; ρ ( s t,( + r ;. ( + + ( ( r t t t Bcaus rho at th maturty dat s qual to 0, rho gvn by th Formula (7 s furthr calculatd as 3 ( = N r W = ( s, t; r + O( = = ρ,0; +. r ( s,0; r = ( s, t; r + ρ ( s, t; r + O ( ( s r O( t Ths shows th fnal rsult, ρ ( s r = ρ ( s r + O( t,0;,0;. rho s also asymptotcally quvalnt to th rho n th Black and Schols modl. 4. Computaton of Grks for Amrcan Optons W suggst a nw algorthm for computaton of Grks for Amrcan optons. An Amrcan opton s a contngnt clam that ts holdr can xrcs at any tm bfor ts maturty dat. Consdr an Amrcan opton wth a pay-off functon Φ ( and a maturty dat T = N t. If th prc of undrlyng assts at tm t s gvn by x, th prc of ths optons at tm t s dnotd by (7 (8 607

12 ( x,. Th prc of Amrcan optons s gvn by th backward nducton algorthm: { ( ( } r ( ( ( ( ( ( x, t = max p xu, + t + p xd, +, Φ x, whr th prc of optons at th maturty dat s gvn by ( xn, t =Φ( x Introduc nw sts,. = { x x ( 0,, ( x, t =Φ( x } and = ( 0, \ ( = 0,,, N Th contnuous rgon and stoppng rgon for Amrcan optons ar dfnd by N N = = 0 and = = 0, rspctvly. W wll prsnt an ntutv way for th computaton of Grks for Amrcan optons. If th prc of undrlyng assts at th ntal tm satsfs s 0, Grks ar smply th drvatv of th pay-off functon. Snstvty for Amrcan optons at th ntal tm rspct to a paramtr θ, s computd as θ ( s,0 =Φ( s f th functon Φ ( s dffrntabl at s 0. (Scond drvatvs such as gamma s gvn by ss ( s,0 =ssφ( s. If th prc of undrlyng assts at th ntal tm satsfs s 0, Grks ar also gvn by th drvatv of th prc functon of Amrcan optons, and t s gvn by θ r t θ ( s,0 = { p ( su, t + ( p ( sd, t }. θ (9 Notc that th Formula (9 has a sam functonal form to th uropan optons Grks. Dlta, for xampl, s gvn by ( s sφ f s 0 (,0 r t A s = ( s, ( f s 0. s On can comput othr Grks usng sam mthod. Numrcal dmonstra- tons ar shown n Scton 5. Th xtndd bnomal tr of Plssr and Vorst [7] s on of th most sutabl altrnatv mthods to drv dlta and gamma usng bnomal trs. On can ffcntly and accuratly drv Grks ffcntly and accuratly as dscussd n Scton 5. Howvr, on cannot apply ths mthod to drv vga and rho. 5. Numrcal Rsults In ths scton, w dmonstrat th numrcal rsults for th nw computatonal mthods of Grks that wr ntroducd n prvous sctons. In ordr to chck Strctly spakng, on cannot us our computatonal formula for dlta f th ntal prc s on th arly xrcs boundary, and on cannot us our gamma, vga, and rho formulas f th nods on th bnomal tr ar on th arly xrcs boundary. Howvr, numrcal rsults show that our formula works vry wll whn w comput Grks for Amrcan put optons wth th pay-off Φ ( x = ( K x + f th dlta s fxd at ( s,0 = A on th stoppng rgon and th boundary. Ths s bcaus w hav a smooth-ft condton n th contnuous modl, and our modl s an approxmaton of th Black and Schols modl. 608

13 th ffctvnss of our approach, Grks for Amrcan put optons ar computd by th nwly proposd approach and th fnt dffrnc approach. W also dmonstrat th xtndd bnomal tr approach of Plssr and Vorst [7] to comput dlta and gamma. It s wll known that th xtndd bnomal tr approach of Plssr and Vorst [7] ylds vry accurat and fast algorthms to comput dlta and gamma for optons. On th othr hand, th fnt dffrnc approach s a vry popular approach for computng vga and rho, as dscussd n Hull []. W also compar to th xstnt othr tr mthods for computatons of Grks. Fgurs -4 and Fgur 6 plot th valus of Grks (dlta, gamma, vga, and rho, rspctvly for Amrcan put optons computd usng our approach and th fnt dffrnc approach. Grks, xtndd bnomal Grks (B Grks calculatd by th xtndd bnomal tr of Plssr and Vorst [7], Grks ntroducd by Hull (Hull s Grks ar also plottd n Fgur and Fgur 3. Th xtndd (N-stp bnomal tr s a N + bnomal tr startng from t, as shown n Fgur. Th B dlta and B gamma ar gvn by Γ = B A whr (, s j s u ( s(,0,0 ( s(,0,0 B A = s(,0 s(,0 ( s(,0,0 ( s( 0,0,0 ( s( 0,0,0 ( s(,0,0 s(,0 s( 0,0 s( 0,0 s(,0 s(,0 s(,0 =. Ths rsults ar computd usng bnomal trs wth N = 4,8,,00 stps for on yar. Th paramtr valus assumd for ths numrcal studs wr ( s = 00, = 0.3, r = 0.05, K = 00, T = yar. Fgur. xtndd bnomal tr. 609

14 Th Grks (dlta, gamma, vga, and rho ar rprsntd by th ral lns and th dottd ln (wthout any mark rprsnts th fnt dffrnc Grks (FD Grks. Two othr knds of dottd lns, dottd lns wth a crcl and squar, rprsnt B Grks and Hull s Grks (dlta and gamma, rspctvly. Th horzontal lns n Fgur and Fgur 3 ar th B dlta and B gamma computd by th xtndd bnomal tr wth 00,000 stps for on yar. Th horzontal lns n Fgur 4 and Fgur 6 ar FD vga and FD rho computd usng th bnomal trs wth 00,000 stps for on yar. Bcaus w us vry fn mshs for th computaton of ths horzontal lns, ths numrcal Fgur. Dlta for Amrcan put optons ( dlta, B dlta, FD dlta, and Hull dlta, K = 00 computd by bnomal trs wth N = 4,8,,00 stps. Fgur 3. Gamma for Amrcan put optons ( gamma, B gamma, FD gamma, and Hull gamma, K = 00 computd by bnomal trs wth N = 4,8,,00 stps. 60

15 rsults ar xpctd to b vry accurat. If th ntal undrlyng asst prc s s n th contnuous rgon,.. s 0, th FD dlta and gamma ar gvn by FD A = FD A ( s+s,0 ( s s,0 s ( s+s,0 ( s,0 + ( s s,0 ( s Γ = and th FD vga and rho ar gvn by FD A ρ ( s,0; FD A ( s,0; r = = ( s,0; + ( s,0; ( s,0; r+r ( s,0; r r r whr s,, and r ar small paramtrs. W tak s = s h, = h, 3 and r = r h, ( h = 0 for our computatons. Fgur shows that dlta convrgs much fastr than th FD dlta. Fgur 3 shows that FD gamma dos not appar n th pctur, and w do not rcommnd th us of th fnt dffrnc approach to comput gamma. Fgur 4 rvals that vga convrgs slowr than FD vga. Howvr, ths s not a unvrsal rsult. vga and FD vga for Amrcan put optons ar plottd n Fgurs 5 wth strk prcs of K = 05 (n-th-mony cas. Th oscllaton phnomnon for FD vga s obsrvabl for th optons wth th strk prc K = 05. Th bhavors of rho and FD rho dmonstratd n Fgur 6 ar almost th sam, and w found ths to b a unvrsal rlatonshp n our numrcal xprnc. It s mportant to not th backward nducton algorthm nds to b usd only onc to obtan rho, whras t has to b usd twc to obtan FD rho. Hnc, th computatonal Fgur 4. Vga for Amrcan put optons ( vga and FD gamma, K = 00 computd by bnomal trs wth N = 4,8,,500 stps. 6

16 Fgur 5. Vga for Amrcan put optons ( vga and FD gamma, K = 05 computd by bnomal trs wth N = 4,8,,500 stps. Fgur 6. Rho for Amrcan put optons ( rho and FD rho, K = 00 computd by bnomal trs wth 4,8,,500 N = stps. tm for rho s xpctd to b shortr than that for FD rho. Tabl lsts th computatonal tm and rsults for rho and FD rho computd by a bnomal tr wth 0,000 stps 3. It was found that th computatonal tm for rho was about 0% shortr vn though th computatonal rsults obtand wr almost th sam. Hnc, computng rho rathr than th FD rho s mor advantagous. Fgurs 7-0 prsnt Grks (dlta, gamma, vga, and rho, rspctvly for 3 It s nough to us bnomal trs wth 00 stps to obtan Grks. Thn, on can comput n an nstant. 6

17 Fgur 7. Dlta for Amrcan put optons as a functon of th undlyng asst prc ( dlta, B dlta, and FD dlta, K = 00. Fgur 8. Gamma for Amrcan put optons as a functon of th undlyng asst prc ( gamma, B gamma, and FD gamma, K = 00. Tabl. Computatonal tm for Rho. rho FD rho Valu Tm 4.77 (s 8.78 (s Amrcan put optons as a functon of th prc of undrlyng assts. Th prc rang of undrlyng assts s from 50 to 00. Othr paramtrs usd for ths numrcal studs ar sam as thos usd n th prvous numrcal studs. Fgur 7 and Fgur 8 plot th, FD, and B dlta and gamma computd usng th 00 stp bnomal trs, rspctvly. Th curvs of all th Grks 63

18 Fgur 9. Vga for Amrcan put optons as a functon of th undlyng asst prc ( vga and FD vga, K = 00. Fgur 0. Rho for Amrcan put optons as a functon of th undlyng asst prc ( rho and FD rho, K = 00. and B Grks ar vry smooth, whras thos of FD dlta and FD gamma ar unstabl. Fgur 9 and Fgur 0 plot th and FD vga and rho usng a 00 stp bnomal tr, rspctvly. As shown n Fgur 9, th shap of vga s vry smooth, whras th oscllaton phnomnon s obsrvd for FD vga. Th oscllaton phnomnon for FD vga s spcally strong whn th strk prc s hghr than th ntal prc of undrlyng assts. Fgur 0 rvals that th numrcal rsults of rho and FD rho ar almost sam. Fnally, w compard our nw mthods wth othr xstng tr mthods. W 64

19 compar dlta wth dlta for uropan optons computd by othr knds of bnomal tr, namly tr mthods ntroducd by Chung and Shacklton [8], Tan [9], and Lsn and Rmr [0]. Ths ar summarzd n Fgur. In ordr to obtan dlta usng th bnomal trs of Chung and Shacklton [8] and Tan [9], w mployd th xtndd bnomal tr approach. On th othr hand, w usd th fnt dffrnc approach to th bnomal tr for Lsn and Rmr, bcaus w wantd to mplmnt smpl calculatons. Lsn and Rmr [0] ntroducd a nw knd of bnomal tr, whch computs th prc of optons ffcntly. Thy construct two knds of trs usng two dffrnt transform formulas. Not that bcaus no sgnfcant dffrnc s obsrvd n two mthods of Lsn and Rmr [0], w usd Mthod- dscrbd n thr artcl. As shown n Fgur, Grks calculatd by trs ntroducd by Chung and Shacklton [8] and Tan [9] convrgs to th ral valu smoothly, howvr, dlta convrgs fastr than ths mthods. Dlta computd by th tr ntroducd by Lsn and Rmr [0] convrgs consdrably fast, f on uss trs wth odd stps.. It should b pontd out that t s not asy to comput vga and rho by Lsn and Rmr s bnomal tr. 6. Concluson Ths papr prsntd nw computatonal mthods of Grks usng th bnomal tr. Thr ar two mportant rsults n ths papr. Frst, w obtan a vry ffcnt algorthm to valuat Grks. It s spcally ffcnt to comput Grks for Amrcan optons. Although many studs hav bn conductd for th computaton of Grks for uropan optons, fw paprs hav xamnd th Fgur. Dlta ( dlta and dlta computd by varous knds of bnomal trs. computd by bnomal trs wth 4,8,,00 N = stps. : dlta. CS: dlta computd by bnomal tr ntroducd by Chung and Shacklton [8]. Tan: dlta computd by bnomal tr ntroducd by Tan [9]. LR: dlta computd by bnomal tr ntroducd by Lsn and Rmr [0]. 65

20 computaton of Grks for Amrcan optons. W ntroduc th bnomal tr approach to ovrcom ths problms and confrm ts ffctvnss for comput- ng Grks for Amrcan optons vry quckly and accuratly. Numrcal rsults ndcat that Grks convrg fastr whn computd usng our mthod than whn computd usng th xtndd bnomal tr approach of Plssr and Vorst [7]. Scond, w show that Grks computd by our algorthm convrg to th Grks n th contnuous tm modl. W also showd th rlaton btwn Grks and dscrt Mallavn Grks. W ar now prparng an artcl on computatons of Grks n th jump dffuson modls usng th bnomal tr approach (Muro and Suda [9] and Suda and Muro [5]. Acknowldgmnts Ths work was supportd by Japan Socty for th Promoton of Scnc. Grant- n-ad for Scntfc Rsarch (C 6K0373.Ths papr was wrttn whl Shntaro Suda was a graduat studnt at Tohoku Unvrsty. Th vws x- prssd n ths papr ar thos of th authors and do not ncssarly rflct th offcal vws of MTC. Rfrncs [] Hull, J. (008 Optons, Futurs and Othr Drvatvs. 7th dton, Prntc Hall, Uppr Saddl Rvr, Nw Jrsy. [] Fourné,., Laszry, J., Lbuchoux, J., Lons, P. and Touz, N. (999 Applcatons of Mallavn Calculus to Mont Carlo Mthods n Fnanc. Fnanc and Stochastcs, 3, [3] Kohatsu-Hga, A. and Montro, M. (003 Mallavn Calculus Appld to Fnanc. Physca A, 30, [4] Brns, G., Gobt. and Kohatsu-Hga, A. (003 Mont Carlo valuaton of Grks for Multdmnsonal Barrr and Lookback Optons. Mathmatcal Fnanc, 3, [5] Gobt,. (004 Rvstng th Grks for uropan and Amrcan Optons. Procdngs of th Intrnatonal Symposum on Stochastc Procsss and Mathmatcal Fnanc at Rtsumkan Unvrsty, Kusatsu, 5-9 March 003, [6] Bally, V., Caramllno, L. and Zantt, A. (005 Prcng and Hdgng Amrcan Optons by Mont Carlo Mthods Usng a Mallavn Calculus Approach. Mont Carlo Mthods and Applcatons,, [7] Longstaff, F. and Schwartz,.S. (00 Valung Amrcan Optons by Smulaton: A Smpl Last Squars Approach. Rvw of Fnancal Studs, 4, [8] Muro, Y. and Suda, S. (03 Dscrt Mallavn Calculus and Computatons of Grks n th Bnomal Tr. uropan Journal of Opratonal Rsarch, 3, [9] Muro, Y. and Suda, S. (07 Computaton of Grks n th Jump-Dffuson Modl usng Dscrt Mallavn Calculus. Mathmatcs and Computrs n Smulaton, 40, [0] Cox, J.C., Ross, S.A. and Rubnstn, M. (979 Opton Prcng: A Smplfd Ap- 66

21 proach. Journal of Fnancal conomcs, 7, [] Ltz-Martn, M. (000 A Dscrt Clark-Ocon Formula. Maphysto Rsarch Rport, 9. [] Prvault, N. (008 Stochastc Analyss of Brnoull Procsss. Probablty Survys, 5, [3] Prvault, N. (009 Stochastc Analyss n Dscrt and Contnuous Sttngs. Sprngr, Brln. [4] Prvault, N. and Schoutns, W. (00 Dscrt Chaotc Calculus and Covaranc Idntts. Stochatcs and Stochastc Rports, 7, [5] Suda, S. and Muro, Y. (05 Computaton of Grks Usng Bnomal Trs n a Jump-Dffuson Modl. Journal of conomc Dynamcs and Control, 5, [6] Chung, S.L., Hung, W., L, H.H. and Shh, P.T. (0 On th Rat of Convrgnc of Bnomal Grks. Journal of Futurs Markts, 3, [7] Plssr, A. and Vorst, T. (994 Th Bnomal Modl and th Grks. Journal of Drvatvs,, [8] Chung, S.L. and Shacklton, M. (00 Th Bnomal Black-Schols Modl and th Grks. Journal of Futurs Markts,, [9] Tan, Y. (993 A Modfd Lattc Approach to Opton Prcng. Journal of Futurs Markts, 3, [0] Lsn, D. and Rmr, M. (996 Bnomal Modls for Opton Valuaton-xamnng and Improvng Convrgnc. Appld Mathmatcal Fnanc, 3, [] Hston, S. and Zhou, G. (000 On th Rat of Convrgnc of Dscrt-Tm Contngnt Clams. Mathmatcal Fnanc, 0, [] Walsh, J. (003 Th Rat of Convrgnc of th Bnomal Tr Schm. Fnanc and Stochastcs, 7,

22 Appndx: Closd-Form Formulas for Opton Grks W frst show th rror stmat of vga gvn by (6. Ths s prsntd n Scton A. W also prov that Grks convrg to Grks for contnuous tm Black and Schols modl. Ths s shown n Scton B. Closd-form xpctaton formulas for Grks (dlta, gamma, vga, rho for uropan optons ar nvstgatd n Appndx B. W found that Grks ar approx- matons of th dscrt vrson of th Mallavn Grks n th contnuous tm modl and ths rsults ndcat that Grks convrg to Grks for a contnuous tm modl (Black and Schols modl whn w tak a lmt, 0. A. rror trms for vga W prsnt thorm agan as thorm 3. Thorm 3. Lt us assum that th pay-off functon Φ ( s a smooth functon. Th accuracy of th vga s gvn by [Proof] Insrt ( to th Formula (5 lads to ( s = ( s + O(,0;,0;. r ε s t = + ( s + r t + + (, ;,( t + + (,( ; ( ( ( + + r s t t s t r + s, + t; + O. Ths formula ylds ( s, t; = ( s, t; + ( s, ; r t ( s t ( + t + O( t t t, ;, (0 whr nw varabls and ar dfnd by r + + ( s, t; = + ( s,( + r t ( s t, ; r t ( ( + + s t, t; s t + ( N = + ( s(,( N ; N N r d = Φ ( x s dx x= s N ( N N ( N N ( N If th pay-off functon Φ ( x s not smooth, w formally dfn N ( s(,( N t; = 0, N Φ ( x s smooth n ths subscton. If th pay-off functon ( x although w assum that th pay-off functon vga gvn by (0 s furthr computd as Φ s smooth, 68

23 ( s, t; = ( s, t; + ( s, ; r t ( s t ( + t + O( t t t, ;. Vga for th bnomal tr modl at th ntal tm s gvn by r W ( s,0; = ( s,0; + ( s,0; + O( t + ( s, ; r W = ( s,0; + ( s,0; + O( t + ( s, ; W 3 W + ( s, t; + O( t + ( s, ; = N = 0 r W W (,0; (, ; ( ( s,0; O( t. = s + s t + O = + ( Ths shows th rsult. B. Convrgnc of grks to black and schols modl In Scton 3, w drv Grks undr th Assumpton. As dscussd n Scton 3, th smoothnss of th pay-off functon s too strong to b assumd. Thrfor, n ths scton, w assum that th pay-off functon Φ ( s not a ncssarly smooth functon. W show that Grks obtand n Scton 3 convrg to Grks n th contnuous tm modl undr th followng assumpton. Assumpton. W assum that th pay-off functon, Φ, ( s a functon n th class. s th class of ral-valud functons on R that satsfy th ( C, ( at ach x, th functon followng condtons: ( φ ( s pcws φ ( satsfs φ( x φ( x φ( x ( = + +, and ( φ, φ, and φ ar poly- nomal boundd. W assum that f ( s a functon n th class. For xampl, th pay-off functon for uropan call/put optons s ncludd n class. Thorm 4. W assum that th pay-off functon s n class. W also assum th numbr of stps of th bnomal tr gv by N to b vn. Thn, dlta, gamma vga and rho convrg to dlta, gamma, vga, and rho n Black and Schols modl. [Proof] W show that Grks (dlta, gamma, vga, and rho ar asymptotcally quvalnt to th Mallavn Grks. Mallavna Grks ar Grks calculatd usng th Mallavn calculus. S Kohatsu-Hga and Montro [3] for dtal.. Dlta dlta gvn by th Formula (9 s furthr calculatd as r ( s,0 = ( s, ( s = [ Φ( s ( ] s = Φ( s (, s 69

24 whr w usd th fact that,, N s an..d. squnc to dduc th last qualty. Ths ylds N ( s,0 = Φ( s ( N = s = Φ( s ( WT T s T log ( ST s µt = Φ ( ST, st WN t whr S T s gvn by ST s =. Not that ths formula ndcats that th dlta s dntcal to th dscrt Mallavn dlta. S Kohatsu-Hga and Montro [3] about th Mallavn dlta, for xampl. If th pay-off functon Φ ( s smooth, proposton. n Hston and Zhou [] lads to ( log PT s µt ( s,0 = Φ ( PT + O st N = ( PT ZT O s T Φ + N BS = ( s,0 + O N whr th stochastc procss P t s a gomtrc Brownan moton gvn n ( BS s,0 s dlta n a contnuous tm modl (Black and Schols modl. and ( ( vn though dlta dos not approxmat dlta for th bnomal tr modl, f th pay-off functon Φ ( s not smooth, t stll s an approxmaton formula for th contnuous tm dlta. Undr Assumpton, corollary 4. n Walsh [] shows that th opton prc n th bnomal tr modl convrgs to th optons prc n th Black and Schols modl. Not that Walsh show th convrgnc only on th bnomal tr wth vn numbrs. Ths rsult shows 4 ( log PT s µt ( s,0 = Φ ( PT + O st N BS = ( s,0 + O. N As prvously dscussd, dlta dos not approxmat dlta for th bnomal tr modl, f th pay-off functon s not smooth. On th othr hand, vn f th pay-off functon s not smooth, dlta stll s an approxmaton for contnuous dlta.. Gamma gamma for uropan optons s gvn by (4. Furthr computaton of gamma ylds, 4 Th ordr of rror trm s O N lattc pont. Howvr, f all dscontnuts ar on lattc ponts, th ordr of th rror trm s, f th dscontnuty for th pay-off functon Φ ( s not on a O. S Corollary 4. n Walsh [] for dtals. Also not that Chung t al. [6] show that th N rat of rror trms of bnomal dlta for uropan optons s O( N. 60

25 r ( s,0 Γ = ( s, ( s r + ( s, (. s Th frst and scond trms n (3 ar dnotd by G and G,.. Γ ( s,0 = G+ G: r G = ( s, ( s r G = ( s, (. s Th frst trm s gvn by G = Φ ( s ( s N = Φ( s ( s N = Φ ( s ( WT T s T = ( s,0, s and th scond trm s gvn by = r r ( + (, ( µ = ( G s t t t s s rn = Φ ( s ( (. s ( (3 3 Ths formula s dvdd nto thr parts, G = G G + G, whr G, G 3 and G ar G = Φ ( s s = Φ ( s j( s N rt WT ( s = Φ N t s T N = Φ ( s ( WT T + O( N s T W µ T G = Φ ( s µ t = Φ ( s W T s s T 3 W µ T rn G = Φ ( s µ t = Φ ( s. s s Ths rsults ar combnd nto th closd-form formulas for Gamma: W ( W T T µ T Γ ( s,0 = Φ ( s ( WT T + O ( N. s T T As dscussd n th dlta cas, Walsh [] ylds, 6

26 ( s O( N ( ( T ( log S s µ T log ST s µt Γ ( s,0 = Φ ( ST 3 + O( N s T T ( log ( PT s µ T log ( PT s µt = Φ ( PT O N 3 + s T T ZT = Φ( PT ZT + O( N s T T BS =Γ,0 +, vn f Φ ( s not smooth. 3. Vga ( Vga for th bnomal tr modl s gvn by (6. Th xpctaton s gvn by r( N t r t W T µ + = + Φ( s. Undr th condton N, th xpctaton s gvn by r r = ( W t+ + s Formulas lad to W t+ + + (,( ; ( µ ( ( W t+ + s + + t s + + r( N WT WT { ( s + + ( s + µ t} = Φ Φ ( + + ( j( Φ s = Φ s N W T = Φ ( s N N N = = Φ ( s ( WT T N N N ( s W T T T W W ( s ( s Φ W + = Φ = Φ T N = N r( N t W T ( s ( WT T ( s W = Φ Φ T µ ( N T N ( If th pay-off functon Φ ( s a smooth on, th xpctaton r N t N s calculatd as r( N N W( N ( W( N + ( W( N = s t pφ s ( p Φ s W( N ( W( N + z z 3 = s t Φ s + O( z z= 0 ( ( ( ( ( W( N W( N W N W N s t s s s O = Φ +Φ + ( = O N,. 3 ( 6

27 If th pay-off functon Φ ( x s not smooth, th rlaton r( N N = 0( = O( N, stll satsfd, bcaus w formally assumd nd nto N = 0 N + ( s,0; = + Φ( s rt = 0 N (,0;. Ths rsults ar comb- r W T + Φ( s ( WT T Φ( s W T µ = 0 T N ( WT µ T W T = ( WT T Φ ( s + O. T N As dscussd n dlta cas, th approxmaton ( ( T ( log S s µ T log S s µt = Φ ( + T ( log rt ( PT s µ T log ( PT s µt = Φ 3 ( PT + O T N ZT = ZT Φ ( PT + O T N BS = ( s,0; + O N T s S 3 T O N s vald, vn f th pay-off functon Φ ( s not smooth. 4. Rho In ths subscton, w drv th closd-form formula of rho for uropan optons. W also show that rho convrgs to rho n th contnuous tm modl. On th othr hand, th formula W ( (, ; ( t r N t µ t W N = Φ t s t r t s lads th furthr calculaton of rho. Ths formula s pluggd nto th Formula (8 and w hav ρ W T,0; (,0,0;. T ( s r = T Φ ( S = T s ( s ( s r rt T Th last qualty coms from th closd-form formula for dlta gvn by (9 (or (. As s th dscussons n th prvous cass, th formula ρ ( ( ρ ( ( ( ( ( s,0; r = T s s,0 Φ P + O N = s,0 + O N BS rt BS T must b satsfd undr crtan condtons gvn by Walsh []. 63

28 Submt or rcommnd nxt manuscrpt to SCIRP and w wll provd bst srvc for you: Accptng pr-submsson nqurs through mal, Facbook, LnkdIn, Twttr, tc. A wd slcton of journals (nclusv of 9 subjcts, mor than 00 journals Provdng 4-hour hgh-qualty srvc Usr-frndly onln submsson systm Far and swft pr-rvw systm ffcnt typsttng and proofradng procdur Dsplay of th rsult of downloads and vsts, as wll as th numbr of ctd artcls Maxmum dssmnaton of your rsarch work Submt your manuscrpt at: Or contact jmf@scrp.org