Int. J. Business Performance and Supply Chain Modelling, Vol. 1, No. 1,

Transcription

1 In. J. Busine Performance and Supply Chain Modelling, Vol., No., A non-linear Black-Scholes equaion Yan Qiu and Jens Lorenz* Deparmen of Mahemaics and Saisics, Universiy of New Mexico, Albuquerque, NM 873, USA ccsindy@mah.unm.edu lorenz@mah.unm.edu *Corresponding auhor Absrac: We sudy a modificaion of he Black-Scholes equaion allowing for uncerain volailiy. The model leads o a parial differenial equaion wih nonlinear dependence upon he highes derivaive. Under cerain aumpions, we show exisence and uniquene of a soluion o he Cauchy problem. Keywords: Black-Scholes equaion; uncerain volailiy; nonlinear parial differenial equaions. Reference o his paper should be made as follows: Qiu, Y. and Lorenz, J. (009) A non-linear Black-Scholes equaion, In. J. Busine Performance and Supply Chain Modelling, Vol., No., pp Biographical noes: Yan Qiu is currenly a PhD suden in he Deparmen of Mahemaics and Saisics a he Universiy of New Mexico. Her research ineress are PDE approaches o opion pricing and sochasic volailiy. Jens Lorenz received his PhD in 975 from he Universiy of Münser in Germany. He was a Faculy Member in Applied Mahemaics a Calech from 983 o 989 and joined he Universiy of New Mexico in 990. He is currenly Inerim Chair of he Deparmen of Mahemaics and Saisics. His research ineress are numerical analysis and parial differenial equaions. Inroducion The field of mahemaical finance has gained significan aenion since Black and Scholes (973) published heir Nobel Prize work in 973. Using some simplifying economic aumpions, hey derived a linear parial differenial equaion (PDE) of convecion diffusion ype which can be applied o he pricing of opions. The soluion of he linear PDE can be obained analyically. In his paper we are ineresed in a non-linear modificaion of he Black-Scholes equaion where he volailiy σ is no aumed o be consan, bu is aumed o be a known funcion of v Here v is he value of he opion and s is he price of he underlying ae. A simple choice for σ = σ(v ), suggesed for example in Wilmo (000), is he disconinuous funcion Copyrigh 009 Inderscience Enerprises Ld.

2 34 Y. Qiu and J. Lorenz σ d ( v ) + σ if v < σ if v > = 0 0 where σ + and σ are esimaes for he maximal and minimal values of he volailiy, respecively. Since, he resuling PDE becomes non-linear in is highes derivaive, he mahemaical heory of he equaion is by no means rivial. In his paper we consider he case where he volailiy σ is a smooh funcion of v and presen a basic exisence and uniquene resul. The paper begins wih a brief inroducion o he claical Black-Scholes model in Secion. Secion 3 provides he derivaion of he modified Black-Scholes equaion. In Secion 4, we skech an exisence and uniquene proof for he Cauchy problem under periodic boundary condiions. These boundary condiions are no realisic and are used here, only o simplify he mahemaical analysis of he non-linear PDE. In fuure work, we plan o exend our analysis and include more realisic boundary condiions. In he las secion, we commen furher on fuure research ineress. Claical Black-Scholes model An opion is an agreemen ha gives he holder a righ, no an obligaion, o buy from or sell o, he seller of he opion a cerain amoun of an underlying ae a a specified price (he srike price) a a fuure ime (he expiraion dae). Clearly, he value v of an opion is a funcion of various parameers, wrien v(s, ; μ, σ; E, T; r). Here, s is he price of he underlying ae; is he curren ime; μ is he drif of s; σ is he volailiy of s; E is he srike price and T is he expiraion dae of he opion; r is he risk-free ineres rae. The aumpions leading o he claical BSM model are as follows: he risk-free ineres rae r is a known consan for he life of he opion he price s of he underlying ae follows a log-normal random walk and he drif μ and volailiy σ are consans known in advance 3 ransacion coss aociaed wih buying or selling he underlying ae are negleced 4 here are no dividends paid from he underlying ae 5 hedging can be done coninuously 6 he price of he underlying ae is divisible so ha we can rade any fracional share of he ae 7 we have an arbirage-free marke. Le π denoe he value of a porfolio wih long posiion in he opion and shor posiion in some quaniy Δ of he underlying ae,

3 π A non-linear Black-Scholes equaion 35 = v( s, ) Δ s By aumpion, he price s of he underlying ae follows a log-normal random walk, ds = μsd + σ sdx where X is Brownian moion. As ime changes from o + d, he change in he value of he porfolio is due o he change in he value of he opion and he change in he price of he underlying ae, dπ = dv Δ ds By Io s formula, we have dv = v + +. σ s v d v ds s Combining he las wo equaions yields dπ = v + σ s v d + ( v Δ) ds. s Using a dela hedging sraegy, we choose Δ = v s and obain d = v π + σ. s v d By he aumpion of an arbirage-free marke, he change dπ equals he growh of π in a risk-free ineres-bearing accoun, Therefore, dπ = rπd = r v Δ s d. r ( v s ) d = v Δ +. σ s v d Subsiuing Δ = v s one arrives a he Black-Scholes equaion, v + rsv + s v rv = 0 for 0 T s σ The equaion is supplemened by an end-condiion a he expiraion ime T, ( E ) ( ) max s,0 for a call opion; vst (, ) = maxe s,0 for a pu opion; Hs E,0 for a binary call opion; His he Heaviside funcion. () ()

4 36 Y. Qiu and J. Lorenz Denoe he righ hand side in formula () by v 0 (s). If one uses he ransformaion rt ( ) τ = T, x = ln( s) + r σ ( T ), w( x, τ) = e v s, he equaion () ransforms o he hea equaion, w = σ w τ xx and he end-condiion () ransforms o he iniial condiion: x ( 0 ) = = = = w x, v s, T v s v e w x. The problem has he explici soluion or + ( x y) στ w( x, τ ) = e w ( y) dy 0 πσ τ + ln() s + r ( T ) y / ( T ) rt σ σ y v( s, ) = e e v ( e ) dy. 0 πσ T 3 Modified Black-Scholes model wih variable volailiy We can modify he aumpions leading o he Black-Scholes model in differen ways regarding differen parameers. Here, we focus on he consan volailiy aumpion. The volailiy is no known in advance as a consan bu is an uncerain sochasic variable. There are wo radiional ways o measure volailiy: implied or hisorical. Anoher poibiliy is o aume a known range for he volailiy σ: + 0< σ σ σ where σ + and σ are (esimaes for) he maximal and minimal values of σ. We hen have σ sv = + ( σ ) ( σ ) sv min + σ σ σ sv if v if v <0 0 which moivaes o define he disconinuous funcion

5 A non-linear Black-Scholes equaion 37 σ d ( v ) + σ if v <0 =. σ if v >0 (3) As oulined in he previous secion, under dela hedging, Δ = v s we have d = v π + σ. s v d Aume he minimum reurn on he porfolio wih volailiy σ varying over he range σ σ σ + equals he risk-free reurn rπ d. We hen obain v + σ ( v ) s v d = rπd = r( v sv d s) d wih σ d (v ) given by (3). One obains he non-linear PDE v + rsv + = s σ v s v rv 0 d (4) In his case, because of he variabiliy of σ = σ d (v ), he ransformaion x = ln s + r σ T applied in previous secion is no useful since i depends via σ on he soluion v. Insead, we apply he much simpler ransformaion leading o rτ τ = T, x = s, u x, τ = e v s, u = σ ( u ) x u + rxu, x >0 τ d xx xx x ( 0) = v( s T) u x,,. (5) 4 Exisence and uniquene analysis The eenial mahemaical difficuly of (4) lies in he non-linear erm ( u ) u. addre his eenial difficuly, we consider an equaion of he form σ d xx xx u = G u u (6) xx xx where G : R (0, ) is a given smooh posiive funcion. To

6 38 Y. Qiu and J. Lorenz The funcion σ d in (4) is no smooh, of course, bu we can approximae σ d by a smooh funcion like + + = + anh σ σ σ σ σ u ε xx, ε ( uxx ) ε >0 Figure The graphs of σ d (lef) and σ ε (righ) for σ + = 0.3, σ = 0. and ε = 0.3 (see online version for colours) σ (u ) σ d xx ε (u xx ) σ d 0.5 σ ε u xx u xx Differeniae equaion (6) wice wih respec o x and se w = u xx o obain = + where w h w w h w w (7) xx x h w = G w + G w w. I will be convenien o consider he slighly more general equaion w = h w w + g w, w, w x, 0 = f x, (8) xx x where h(w), g(w, w x ) and f(x) are C funcions of heir argumens. To concenrae on he eenial mahemaical difficuly, he non-linear volailiy coefficien, we aume he iniial funcion f(x) o be -periodic; we hen seek a soluion w(x, ), of (8) which is -periodic in x. Oher boundary condiions will be considered in fuure work. Theorem 4. Consider he -periodic iniial value problem (8) under he above aumpions. In addiion, aume ha h(w) k > 0 for all real w and ha h and g and all heir derivaives are bounded funcions. Then here is a unique C soluion w(x, ) which is -periodic in x. The soluions exiss for 0 <.

7 A non-linear Black-Scholes equaion 39 Remark: If h or g or heir derivaives are unbounded, one can use a cu-off argumen and replace h or g by funcions h % or %g which saisfy he condiions of he heorem. For he original problem, one hen obains a resul which is local in ime. To prove he exisence par of he heorem, we define a sequence of funcions w n (x, ) via he ieraion n+ n n+ n n n+ xx x w = h w w + g w, w, w x, 0 = f x, n = 0,,, K (9) saring wih w 0 (x, ) f(x). Since he problem (9) is linear parabolic, here is no difficuly in esablishing exisence, uniquene and smoohne of he funcions w n (x, ) w n (x +, ) for 0 <. I can hen be shown and his is he main mahemaical difficuly, ha he funcions w n (x, ) are uniformly smooh in any finie ime inerval. More precisely, for any fixed 0 < T <, all derivaives are bounded independenly of he ieraion index n: p+ q n sup max max w x, C p, q, T x T p q n 0 x <. General argumens as deailed in Krei and Lorenz (989), for example, hen show convergence of he sequence w n (x, ), along wih all derivaives, o a smooh soluion w(x, ) of he problem (8). To show uniquene of a soluion, le ψ(x, ) denoe he difference of wo soluions. An energy esimae hen yields d (, ) = (, ) C (, ) d ψ ψ ψ ψ L L L and he iniial condiion ψ ( x, 0) 0 implies ψ 0. 5 Fuure research In fuure work we plan o exend he analysis in various direcions: More general boundary condiions han -periodiciy will be considered. The case of a non-smooh volailiy coefficien G(μ xx ) is of ineres in applicaions, such as he jump funcion σ d (μ xx ). One can rea he jump funcion σ d as he limi of smooh funcions σ ε bu he limi proce for he corresponding soluions is non-rivial.

8 40 Y. Qiu and J. Lorenz 3 The case of a non-smooh volailiy σ d (μ xx ) can also be reaed as a free boundary value problem where lines (x(), ) wih μ xx (x(), ) = 0 will be deermined as free boundaries. References Black, F. and Scholes, M. (973) The pricing of opions and corporae liabiliies, J. Poliical Economics, Vol. 8, pp Krei, H-O. and Lorenz, J. (989) Iniial-boundary Value Problems and he Navier-sokes Equaions, Academic Pre, Boson. Wilmo, P. (000) Paul Wilmo on Quaniaive Finance Volume One and Volume Two, John Wiley & Sons Ld, Chicheser.