Black Scholes Option Pricing with Stochastic Returns on Hedge Portfolio


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1 EJTP 3, No Elecronic Journal of Theoreical Physics Black Scholes Opion Pricing wih Sochasic Reurns on Hedge Porfolio J. P. Singh and S. Prabakaran Deparmen of Managemen Sudies Indian Insiue of Technology Roorkee Roorkee 47667, India Received Sepember 006, Acceped 9 Sepember 006, Published 0 December 006 Absrac: The Black Scholes model of opion pricing consiues he cornersone of conemporary valuaion heory. However, he model presupposes he exisence of several unrealisic and rigid assumpions including, in paricular, he consancy of he reurn on he hedge porfolio. There, now, subsiss ample jusificaion o he effec ha his is no he case. Consequenly, several generalisaions of he basic model have been aemped. In his paper, we aemp one such generalisaion based on he assumpion ha he reurn process on he hedge porfolio follows a sochasic process similar o he Vasicek model of shorerm ineres raes. c Elecronic Journal of Theoreical Physics. All righs reserved. Keywords: Econophysics, Financial derivaives, Opion pricing, Black Scholes Model, Vasicek Model PACS 006: Gh, s, 0.50.r, 0.50.Cw, 0.50.Ey Wih he rapid advancemens in he evoluion of financial markes across he globe, he imporance of generalisaions of he exan mahemaical apparaus o enhance is domain of applicabiliy o he pricing of financial producs can hardly be overemphasized for furher progress and developmen of he financial microsrucure. Though a an embryonic sage, he unificaion of physics, mahemaics and finance is unmisakably discernible wih several fundamenal premises of physics and mahemaics like quanum mechanics, classical & quanum field heory and relaed ools of noncommuaive probabiliy, funcional inegraion ec. being adoped for pricing of exan financial producs and for elucidaing on various occurrences of financial markes like sock price paerns, criical crashes ec ]. The Black Scholes formula for he pricing of financial asses 37] coninues o be he subsraum of conemporary valuaion heory. However, he model, alhough of Jainder
2 0 Elecronic Journal of Theoreical Physics 3, No immense pracical uiliy is based on several assumpions ha lack empirical suppor. The academic fraerniy has aemped several generalisaions of he original Black Scholes formula hrough easing of one or oher assumpion, in an endeavour o augmen is specrum of applicabiliy. In his paper, we aemp one such generalisaion based on he assumpion ha he reurn process on he hedge porfolio follows a sochasic process similar o he Vasicek model of shorerm ineres raes. Secion liss ou he derivaion of he BlackScholes formula hrough he parial differenial equaion based on he consrucion of he complee hedge porfolio. Sec 3, which forms he essence of his paper, aemps a generalisaion of he sandard Black Scholes pricing formula on he lines aforesaid. Secion 4 concludes.. The Black Scholes Model In order o faciliae coninuiy, we summarize below he original derivaion of he Black Scholes model for he pricing of a European call opion 37 and references herein]. The European call opion is defined as a financial coningen claim ha enables a righ o he holder hereof bu no an obligaion o buy one uni of he underlying asse a a fuure dae called he exercise dae or mauriy dae a a price called he exercise price. Hence, he opion conrac, has a payoff of max S T E, 0 = S T E + on he mauriy dae where S T is he sock price on he mauriy dae and E is he exercise price. We consider a nondividend paying sock, he price process of which follows he geomeric Brownian moion wih drif S = e µ+σw. The logarihm of he sock price Y = In S follows he sochasic differenial equaion dy = µd + σdw where W is a regular Brownian moion represening Gaussian whie noise wih zero mean and δ correlaion in ime i.e. E dw dw = dd δ on some filered probabiliy space Ω, F, P and µ and σ are consans represening he long erm drif and he noisiness diffusion respecively in he sock price. Applicaion of Io s formula yields he following SDE for he sock price process ds = µ + σ S d + σs dw Le C S, denoe he insananeous price of a call opion wih exercise price E a any ime before mauriy when he price per uni of he underlying is S. I is assumed ha C S, does no depend on he pas price hisory of he underlying. Applying he Io formula o C S, yields dc = µs C S + σ S C S + C + σ S C C d + σsdw, 3 S S The original opionpricing model propounded by Fischer Black and Myron Scholes envisaged he consrucion of a hedge porfolio,, consising of he call opion and
3 Elecronic Journal of Theoreical Physics 3, No a shor sale of he underlying such ha he randomness in one cancels ou ha in he oher. For his purpose, we make use of a call opion ogeher wih C/ S unis of he underlying sock. We hen have, on applying Io s formula o he hedge porfolio,,: d = d ] C S, dc S, C S, S = C d d S d S.dS 4 d where he erm involving d C d S has been assumed zero since i envisages a change in he porfolio composiion. On subsiuing from eqs. & 3 in 4, we obain d d = dc S, d µ + C S, σ S σs C dw S S d = C S, + σ S C S, S 5 We noe, here, ha he randomness in he value of he call price emanaing from he sochasic erm in he sock price process has been eliminaed compleely by choosing he porfolio = C S, S CS,. Hence, he porfolio is free from any sochasic S noise and he consequenial risk aribued o he sock price process. Now d is nohing bu he rae of change of he price of he socalled riskless bond d porfolio i.e. he reurn on he riskless bond porfolio since he equiy relaed risk is assumed o be eliminaed by consrucion, as explained above and mus, herefore, equal he shorerm ineres rae r i.e. d = r 6 d In he original Black Scholes model, his ineres rae was assumed as he risk free ineres rae r, furher, assumed o be consan, leading o he following parial differenial equaion for he call price: d d or equivalenly = r ] C S, = r C S, S = S C S, C S, + σ S C S, S + σ S C S, C S, + rs rc S, = 0 7 S S which is he famous Black Scholes PDE for opion pricing wih he soluion: C S, = SN d Ee rt N d 8 where d = log S E +r+ σ T σ T, d = d σ T = log S E +r σ T σ T N y = π y e x dx and
4 Elecronic Journal of Theoreical Physics 3, No The Black Scholes Model wih Sochasic Reurns on he Hedge Porfolio As menioned earlier, in he above analysis, he ineres rae r, which is essenially a proxy for he reurn on a porfolio ha is devoid of any risk emanaing from any variables ha cause flucuaions and hence risk in sock price process, is aken as consan and equal o he risk free rae. However, his reurn would, neverheless, be subjec o uncerainies ha influence reurns on he fixed income securiies. I is, now, convenional o model hese shor erm ineres raes ha are represenaive of shor erm reurns on fixed income securiies hrough a sochasic differenial equaion of he form 8] dr = ψ r, ] d + η r, ] du 9 where r is he shor erm ineres rae a ime, ψ and η are deerminisic funcions of r, and U is a Wiener Process. In our furher analysis, we shall assume ha his shorerm ineres rae is represened by he Vasicek model 9] viz. dr d where η is a whie noise sochasic process + Ar + B η = 0 0 η = 0, η η = δ The call price process now becomes a funcion of wo sochasic variables, he sock price process S and he bond reurn process ineres rae process r. Hence, applicaion of Io s formula o C S, r, gives dc = C C C d + ds + S r dr + σ S C S d + C d r where ds is given by eq. and dr by eq. 0 respecively. As in Secion, we formulae a hedge porfolio Π consising of a call opion C S, r, and a shor sale of C unis of sock S i.e. = C S C. We hen have, repeaing S S he same seps as in Secion hereof d = dc d d C ds S d = C + C r.dr d + σ S C S + C 3 r Now, using d = r, we obain d C + σ S C C + r S S S r C + C r + C dr = 0 4 r d This equaion defies closed form soluion wih he exan mahemaical apparaus. We can, however, obain explici expressions for he call price C S, averaged over he sochasic par of he ineres rae process, as follows:
5 Elecronic Journal of Theoreical Physics 3, No C S, would, hen, be given by subsiuing T rτ T for he consan risk free ineres rae r in he Black Scholes formula 8. The averaging process happens o be edious wih exensive compuaions so we proceed erm by erm. We have N d = π d e x dx = π H d x e x dx 5 where H x y is he uni sep Heaviside sep funcion defined by 0] 0, x < y H x, y =, x > y On using he inegral represenaion of H x y as H x y = Lim ε 0 πi 0] i.e. we obain H d x = Lim ε 0 dω eiω d x πi ω iε dω eiωx y ω iε 6 N d = Limε 0 π 3 i e x +iω d x ω iε dxdω = Lim ε 0 πi on performing he Gaussian inegraion over x in he second sep. Now where d = log S E + T σ T σ r τ + T σ T d 0 = log S E + σ T σ T = d 0 + T r τ σ T Since he enire sochasic conribuion comes from he expression T N d, we have where I = Proceeding similarly, we have, N d = Limε 0 πi e iω d +i ω ω iε dω r τ in dω eiωd0 ω ω iε I 0 e iω T σ T rτ and ρ denoes he average expecaion of ρ. N d = Limε 0 πi dω eiωd0 ω ω iε I
6 4 Elecronic Journal of Theoreical Physics 3, No d 0 = log S E σ T σ T Similarly he discoun facor e rt will be replaced by e T rτ = I say. To evaluae he expecaion inegrals I, I we make use of he funcional inegral formalism ]. In his formalism, he expecaion I would be given by ]: I = rt Dr exp T r rt Dr exp r drτ T ] + Ar τ + B + iω T σ r τ T drτ ] = P Q 3 + Ar τ + B where Dr = T drτ τ= π is he funcional inegraion measure. We firs evaluae he funcional inegral P. Making he subsiuion x τ = B A r τ, we obain, wih a lile algebra, P = xt Dx exp T x dxτ +Axτ + iω T σ T B A xτ] = { xt Dx exp T x dxτ ] +A x τ A x T x iωbt ] Aσ T iω σ T T } xτ where I 3 = T = T = { xt Dx exp x A x T x ] } iωbt Aσ T I 3 dx τ + A x τ] iω T + σ x τ T dx τ + A x τ + iω ] σ T x τ In order o evaluae I 3, we perform a shif of he funcional variable x τ by some fixed funcion y τ i.e. x τ = y τ+z τ where y τ is a fixed funcional whose explici form shall be defined laer bu wih boundary condiions y = x, y T = x T so ha z τ, hen, has Drichle boundary condiions i.e. z = z T = 0. Subsiuing x τ = y τ + z τ in 5, we obain I 3 = T 4 5 dyτ + dzτ + dyτ dzτ + A y τ + A z τ + A y τ z τ + iω σ iω 6 y τ + T σ z τ T Inegraing he second and hird erm by pars, we ge
7 Elecronic Journal of Theoreical Physics 3, No I 3 = zτ dzτ dyτ + T + T z τ d zτ z τ d yτ + dyτ + A y τ + A z τ + A y τ z τ + iω σ iw y τ + T σ z τ T 7 Now he boundary erms all vanish since z τ has Drichle boundary condiions. Furher, if we define he fixed funcional y τ in erms of he differenial equaion d y τ + A y τ + iω σ T = 0 8 wih boundary condiion y = x, y T = x T we obain I 3 = T { dy τ + A y τ + iω ] σ T y τ + The funcional y τ is fixed and is given by he soluion of eq 8 as ] } z τ d z τ + A z τ 9 y = αe Aτ + βe Aτ γ 30. where γ = iωσ A σ, α = xt eat xe A T e AT e A + γ eat e A e AT e A and β = xt e AT xe A e AT e A Inegraing ou he y τ erms in eq. 9 using eq. 30, we obain + γ e AT e A e AT e A { I 3 = Aα e AT e A β e AT e A Aγ T ]+ ]} T zτ d zτ +A z τ 3 Subsiuing his value of I 3 in eq. 4 we obain, for P, noing ha Dx = Dz since y τ is fixed by eq 8 P =exp A x T x ] iωb T Aσ zt =0 z=0 Dz exp { T On exacly same lines, we obain ] A xt e AT xe A e AT e xt e AT xe A AT e AT e A ω Σ 4 e AT e A A 3 σ T e AT e e AT e A A xt e AT xe A e AT e A iωσ + Aσ e T AT e A xt e AT xe A e AT e A ]} z τ d zτ + A z τ e AT e A e AT e A AT 3 ]
8 6 Elecronic Journal of Theoreical Physics 3, No { ]]} Q = exp A x T x ] A xt e AT xe A e AT e xt e AT xe A AT e AT e A { zt =0 Dz exp ]} T z=0 z τ d zτ + A z τ 33 Hence I = exp iωb T Aσ + ω Σ 4 A 3 σ T iωσ Aσ T e AT e A e AT e e AT e A A xt e AT xe A e AT e A e AT e A xt e AT xe A e AT e A e AT e A ] e AT e A T A 34 which when subsiued in eqs. 0 & shall give he values N d and N d respecively as: and where X = N d = N N d = N e AT e A e AT e e AT e A A log S E + σ σ T B T Σ X A 3 σ T Σ X A 3 σ T Y Aσ Aσ T ] log S E σ σ B T Y T Aσ Aσ T ] e AT e A A T and Y = x T e AT x e A e AT e A x T e AT x e A e AT e A e AT e A e AT e A 37 To evaluae I, we subsiue ω = iσ T in eq. 34 o ge { B T I = exp A Σ 4 A 3 X Σ A ]} Y The closed form soluion for he Black Scholes pricing problem wih sochasic reurn on he hedge porfolio can now be obained by subsiuing he above averages in eq Conclusion In his paper, we have obained closed form expressions for he price of a European call opion by modifying he Black Scholes formulaion o accommodae a sochasic 38
9 Elecronic Journal of Theoreical Physics 3, No reurn process for he hedge porfolio reurns. We have modelled his reurn process on he basis of he Vasicek model for he shorerm ineres raes. The need for his exension of he Black Scholes model is manifold. Firsly, he consrucion of he hedge porfolio in he Black Scholes heory implies ha he flucuaions in he price of he derivaive and ha of he underlying exacly and immediaely cancel each oher when combined in a cerain proporion viz. one uni of he derivaive wih a shor sale of C unis of he underlying so ha he hedge porfolio is devoid of any impac of S such flucuaions. This mandaes an infiniely fas reacion mechanism of he underlying marke dynamics whereby any movemen in he price of one asse is insananeously annulled by reacionary response in he oher asse consiuing he hedge porfolio. This is, obviously srongly unrealisic and here may subsis brief periods or aberraions when he no arbirage condiion may cease o hold and hence, reurns on he hedge porfolio may be differen from he risk free rae. One way of aending o his anomaly is o model he reurns on he hedge porfolio as a sochasic process as has been done in his sudy. The parameers defining he process can be obained hrough an empirical sudy of he marke dynamics. Anoher imporan jusificaion for adoping a sochasic framework for he hedge porfolio reurn process is ha he hedge porfolio by is very consrucion, envisages he neuralizaion of he flucuaions of he wo asses iner se i.e. i assumes a perfec correlaion beween he wo asses. In oher words, he hedge porfolio may be consrued as an isolaed sysem ha is such ha insofar as facors ha influence one componen of he sysem, he same facors influence he oher componen o an equivalen exen and, a he same ime, oher facors do no impac he sysem a all. This is anoher anomaly ha disors he Black Scholes model. The fac is ha while he hedge porfolio of he Black Scholes model is immunized agains price flucuaions of he underlying and is derivaive hrough muual ineracion, oher marke facors ha would impac he porfolio as a whole are no accouned for e.g. facors affecing bond yields and ineres raes ec. Consequenly, o assume ha he hedge porfolio is compleely risk free is anoher aberraion i is risk free only o he exen of risk ha emanaes from facors ha impac he underlying and he derivaive in like manner and is sill subjec o risk and uncerainies ha originae from facors ha eiher do no effec he underlying and he derivaive o equivalen exen or impac he porfolio as a uni eniy. Hence, again, i becomes necessary o model he reurn on he hedge porfolio as some shorerm ineres rae model as has been done here.
10 8 Elecronic Journal of Theoreical Physics 3, No References ] V.I. Man ko e al, Phy. Le., A 76, 993, 73; V.I. Man ko and R.Vilea Mendes, J.Phys., A 3, 998, ] W. Paul & J. Nagel, Sochasic Processes, Springer, ] J. Voi, The Saisical Mechanics of Financial Markes, Springer, 00. 4] JeanPhilippe Bouchard & Marc Poers, Theory of Financial Risks, Publicaion by he Press Syndicae of he Universiy of Cambridge, ] J. Maskawa, Hamilonian in Financial Markes, arxiv:condma/0049 v, 9 Nov ] Z. Burda e al, Is Econophysics a Solid Science?, arxiv:condma/ v, 8 Jan ] A. Dragulescu, Applicaion of Physics o Economics and Finance: Money, Income, Wealh and he Sock Marke, arxiv:condma/ v, 6 July ] A. Dragulescu & M. Yakovenko, Saisical Mechanics of Money, arxiv:condma/00043 v4, 4 Mar ] B. Baaquie e al, Quanum Mechanics, Pah Inegraion and Opion Pricing: Reducing he Complexiy of Finance, arxiv:condma/0089v, Aug 00. 0] G. Bonanno e al, Levels of Complexiy in Financial markes, arxiv:condma/ v, 9 Apr 00. ] A. Dragulescu, & M. Yakovenko, Saisical Mechanics of money, income and wealh : A Shor Survey, arxiv:condma/075 v, 9 Nov 00. ] J. Doyne Farmer, Physics Aemp o Scale he Ivory Tower of Finance, adaporg/ Dec ] F. Black & M. Scholes, Journal of Poliical Economy, 8, 973, ] M. Baxer & E. Rennie, Financial Calculus, Cambridge Universiy Press, 99. 5] J. C. Hull & A. Whie, Journal of Finance, 4, 987, 8. 6] J. C. Hull, Opions, Fuures & Oher Derivaives, Prenice Hall, ] R. C. Meron, Journal of Financial Economics, 976, 5. 8] Paul Wilmo, Quaniaive Finance, John Wiley, Chicheser, ] O. A. Vasicek, Journal of Financial Economics, 5, 977, 77. 0] hp://funcions.wolfram.com/generalizedfuncions/unisep/07/0/0/. ] R. P. Feynman & A. R. Hibbs, Quanum Mechanics & Pah Inegrals, McGraw Hill, 965. ] P. C. Marin e al, Phy. Rev. A, 8, 973, 43.
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