EXACT SOLUTION OF DISCRETE HEDGING EQUATION FOR EUROPEAN OPTION

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1 EXACT SOLUTION OF DISCRETE HEDGING EQUATION FOR EUROPEAN OPTION Yaovlv D.E., Zhabi D. N. Dartmt of Highr athmatic ad athmatical Phyic Tom Polytchic Uivrity, Tom, Lia avu 3, 6344, Ruia Th aroach that allow fid Euroa otio ric o th aumtio of hdgig at dicrt tim i rood. Th routi allow fid th otio ric ot for logormal ditributio fuctio of udrlyig at oly but for wid ough cla of ditributio fuctio too. It i how that thr it a ozro oibility that mart aramtr ca ta valu uch that to raliz th hdgig olicy bcom imoibl. Thi fact i ot i cotradictio with Blac-Schol otio ric modl a log a thi oibility td to zro at th limit of cotiuou hdgig. JEL: C6, D4, G Kyword: otio ricig modl, fiac mathmatical modl, dicrt hdgig addr: ENFaua@yad.ru addr: zhabi@hy.tu.ru

2 Itroductio At 973 Amrica citit Blac ad Schol 973 rood th ricig modl for Euroa otio. At th tim big Blac-Schol modl i th commoly ud o for ricig drivativ Hull 999, Wilmott t al Not that thi modl i dvloig u to ow Kor ad Kor, Wilmott 998, art 3. Th modl i bad o aumtio that all baically ay thr thig: ivtor dal o fficit mart; th udrlyig at ric follow th logormal radom wal; tradig of udrlyig at ca ta lac cotiuouly. It hould b otd that thr ar may fault with Blac-Schol modl.g., Wilmott, 998, ctio 9. Som of th ar olvd.g., Sircar ad Paaicolaou 998, Whally ad Wilmott 997, Kraovy 999. But itad of olvd roblm w o ar comig. For itac, a it follow from rviouly ublihd wor.g., Ptr 994 th hyothi of at logormal radom wal i doubtd. orovr, i Wilmott 998, ctio i how that a gralizatio of Blac-Schol modl to th hdgig at dicrt tim i actual ic cotiuou hdgig i imoibl, ad v udirabl, i ractic. I thi ar w roo th tio of Blac-Schol modl to wid ough cla of ditributio fuctio of at rtur i aumtio that tradig of th udrlyig at ca ta lac at dicrt tim oly. I cotrat to Whally ad Wilmott 997 w rt th act way how to fid th otio ric without ay aumtio about gligibility of th cotributio caud by dicrt hdgig. W foud that a th hdgig riod td to zro ad th umbr of hdgig act com

3 to ifiity th rult of Whally ad Wilmott 997 ca b foud with rcribd accuracy. Aroach I th aroach rood w aum that th udrlyig at ric i th mart arov roc Gramr ad Ladbttr, 967. Th at ric { St } t t mart chag togthr with a tim ad th tim i cotiual. Each cro-ctio S t i a tochatic variabl that rrt th at ric at tim t. Sic i a ral mart at ric ar quotd at dicrt tim itrval w aum that a ivtor dal with riod. By dfiitio, ut t T,,, K,, whr i dfid from quality T t. Hr Т i a otio maturity; t i a tartig dat. Thi ma that for th ivtor th at ric i rrtd by a quc of tochatic variabl mart mart S, S S S. Without lo of grality w will aum furthr { } t T that t. I additio, w coidr auiliary variabl uch that S : l S +, i.. S S +. To b dfi, w aum that for a, a, whr a > thr it ctatio of uch that E [ ] <. 3

4 Furthr, at tim t + w cotruct a ortfolio coitig of o log otio oitio ad a hort oitio i om quatity of th udrlyig. Lt u u Π + to dfi th valu of thi ortfolio at tim t +. By cotructio, w gt Π + V + + S +. 3 Now aum that o tim-t wa ad. It i vidt that + i hld fid durig th tim-t. Lt u dot by S + th at ric at tim t + ad by D [A] a variac of th tochatic variabl A. Th for th ortfolio valu at tim t w gt ~ Π V S S V By carfully chooig + w ca limiat th variac of ortfolio 4. It ca b how i th uual way that a oo a + i qual to cov[ V, ] + th ortfolio variac bcom miimal. D[ ] Lt u ot that Blac-Schol ricig olicy may b writt a ~ r E[ Π ] Π +. 5 Hr r i a ri fr itrt rat at tim t. It i ay to chc that quatio 5 i qual to th itgral quatio V f d V +. 6 R 4

5 r m m r Hr f + u ; u d i th ditributio dity of quatiti ; m E[ ]; d D[ ]. Sic th fuctio V i aumd to b ow w may r V for whol tim quc. Lt u rmar that at iry th valu of th call otio ayoff ca b writt a V ma[,]. Taig ito accout th iquality E m V for m m E m < a w ca rov that thr it umbr A m uch that m V A m. I th am way it ca b how that all V ar cotiuou ad thr it umbr W uch that th iquality V iquality ma that for >> E. V W i tru. Th lat To olv 6, lt u rwrit thi quatio with hl of lli traformatio. Rcall that lli traform of a fuctio h i th itgral H h d, 7 if thi itgral it Dotch 954, Churchill 956. Furthr, by H l[ h ] w will dot lli traform; by h l [ H ] w will dot th ivr lli traform. It i clar that th domai of th fuctio H i th t of coml uch that itgral 7 covrg. From rorti of fuctio V dicud abov it follow that for ay uch that a < R < thr it itgral 5

6 F V d. orovr, from it follow that th fuctio f l y admit lli traformatio for a R a. By dfiitio, ut [ f l ] U l. 8 It follow that quatio 6 ca b writt with hl of lli traformatio i form F + U F. 9 Obviouly, th olutio of thi diffrc quatio i giv by th followig rio Um m F F. Hr F i lli traform of th ayoff fuctio. Thu w foud th olutio of quatio 6 i trm of lli traform. Thr ar two altrativ way to rcotruct fuctio V. I th firt of lac w ca u th ivr lli traformatio. Thrfor, w gt I thi way w obviouly obtai a+ i h l [ H ] H πi ai d. a + i V F d, whr πi a i a a. I th cod lac w dfi auiliary fuctio G l Um. 3 m 6

7 Uig rorti of lli traformatio, it follow that V G V y dy. 4 y y Hr fuctio G,, y G i Gr fuctio. y y To b rci, i uually mall by valu with rct to th tim till maturity. Thu it i uful to build a aymtotic aio with rct to th mall. A oo a T w ca dot by t ay tim till maturity, hr,,,. Thi ma that V V V t,. Furthr, ad V t, ito a ri i owr of t. Thrfor, w gt l l V t, B t, + B t, + B t, B t,. 5 Erio 5 allow timat th otio ric at tim t for mall hdgig riod. It ca aily b chcd that i ca of th logormal radom wal of th udrlyig th fuctio V t, aroimat th olutio of Whally ad Wilmott 997 with ay rcribd accuracy. Not alo that from aalyi of rio 6 it i follow that for thr it omty t Ξ [, + uch that th fuctio V bcom gativ for Ξ. I othr word thi ma that thr it oibl tat of th mart uch that to raliz th hdgig olicy bcom imoibl. To avoid thi roblm w com to rtrictio o modl aramtr E[ ] l E[ ] r l. 6 E[ ] 7

8 Sic D [ ] E[ ] E [ ] >, it follow that th itrval 6 i ot mty. orovr, a log a w com to th cotiuou hdgig, i.. + whil T, t Ξ dgrat to mty t. If w uo that th udrlyig at i th logormal ditributd tochatic variabl, th th rlatiohi btw Blac- Schol otio ric C BS, t ad otio ric V t, i tru C BS +, t lim V t,. 7 Eaml Lt u coidr th aml of th act rio for th otio ric with hdgig at dicrt tim. Suo that for,, whr i th ormally ditributd tochatic variabl with ma E [] µ ad variac D [ ]. I thi ca, th robability dity fuctio i giv by µ u u. π Evidtly, w hav µ+ µ+ m m, d d. W aum alo that all r ar cotat till otio maturity, i.. togthr, w obtai µ+ U + U. r r. Combiig all 8

9 9 Hr ad ar giv by 3 µ+ µ+ + µ + r r, 3 µ+ µ+ + µ r. So, th olutio of quatio ca b writt i form + + µ F U F F L F C F + µ+. Hc, w obtai l G C L l π µ+ +. It follow that d V G V. Chagig th variabl y, w gt π µ+ + E y dy E y C V l. 8 It ca b how i tadard way that for cotiuou hdgig thi rio of th otio ric i qual to th wll ow Blac-Schol formula.

10 Summary I rt ar w roo th mthod that allow fid th act rio for ric of Euroa otio o aumtio of hdgig at dicrt tim. Aothr advac of th mthod i that thi mthod rovid th tool to ric otio for widr cla of udrlyig at dity fuctio th logormal dity fuctio. W how that th aroach giv th wll ow Blac-Schol formula of otio ric at th limit of cotiuou hdgig ad o th aumtio of logormal dity fuctio of th udrlyig at. orovr, w coidr th aymtotic bhavior of otio ric for mall hdgig riod. It i how that thi aymtotic coicid with rult Whally ad Wilmott 997 with accurat to th firt ordr. Fially, w how that thr it om mart tat uch that to raliz hdgig bcom imoibl. Thi fact i ot i cotradictio with Blac-Schol thory a log a w com to th cotiuou hdgig t Ξ dgrat to mty t. Rfrc: Blac, F. ad Schol,., 973. Th Pricig of Otio ad Cororat Liabiliti. Joural of Political Ecoomy 8, Churchill, R.V., 956. Etio of oratioal mathmatic. Uiv. arylad Boo Stor, Collg Par. Dotch, G., 954. Übr di Sigularität dr lli-traformirt. ath. A 8, 7-76 Gramr, H., Ladbttr,., 967. Statioary ad rlatd tochatic roc. N.Y.

11 Lodo Hull, J., 999. Otio Futur ad othr Drivativ Scuriti. Prtic-Hall, NJ. Kor, R., Kor, E.,. Otio Pricig ad Portfolio otimizatio: odr thod of Fiacial mathmatic. Graduat Studi i athmatic 3, Kraovy, A., 999. Pricig Liquidity ito Drivativ. Ri 65, Ptr, E.E, 994. Fractal art Aalyi. Joh Wily&So Ic, Nw Yor. Sircar, K., Paaicolaou, G., 998. Gral Blac-Schol modl accoutig for icrad mart volatility from hdgig tratgy. Staford Uivrity Worig ar. Whally. A., Wilmott, P A aymtotic aalyi of a otimal hdgig modl for otio ricig with traactio cot. athmatical Fiac 7, Wilmott, P., Howio, S., Dwy, J., 996. Th athmatic of Fiacial Drivativ. Cambridg Uivrity Pr. Wilmott, P., 998. Th thory ad ractic of fiacial girig. Joh Wily&So Ic, Chichtr.