ISRN Mahemaical Analysis Volume 213, Aricle ID 56771, 5 pages hp://dx.doi.org/1.1155/213/56771 Research Aricle On Opion Pricing in Illiquid Markes wih Jumps Youssef El-Khaib 1 and Abdulnasser Haemi-J 2 1 Deparmen of Mahemaical Sciences, UAE Universiy, P.O. Box 17551, Al-Ain, UAE 2 Deparmen of Economics and Finance, UAE Universiy, P.O. Box 17555, Al-Ain, UAE Correspondence should be addressed o Abdulnasser Haemi-J; Ahaemi@uaeu.ac.ae Received 3 April 213; Acceped 26 May 213 Academic Ediors: G. Gripenberg, M. Winer, and C. Zhu Copyrigh 213 Y. El-Khaib and A. Haemi-J. his is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. One of he shorcomings of he Black-Scholes model on opion pricing is he assumpion ha rading of he underlying asse does no affec he price of ha asse. his assumpion can be fulfilled only in perfecly liquid markes. Since mos markes are illiquid, his assumpion migh be oo resricive. hus, aking ino accoun he price impac on opion pricing is an imporan issue. his issue has been deal wih, o some exen, for illiquid markes by assuming a coninuous process, mainly based on he Brownian moion. However, he recen financial crisis and is effecs on he global sock markes have propagaed he urgen need for more realisic models where he sochasic process describing he price rajecories involves random jumps. Noneheless, works relaed o markes wih jumps are scan compared o he coninuous ones. In addiion, hose previous sudies do no deal wih illiquid markes. he conribuion of his paper is o ackle he pricing problem for opions in illiquid markes wih jumps as well as he hedging sraegy wihin his conex, which is he firs of is kind o he auhors bes knowledge. 1. Inroducion Financial derivaives are imporan ools for dealing wih financial risks. An opion is an example of such derivaives, which gives he righ, bu no he obligaion, o engage in a fuure ransacion on some underlying financial asse. For insance, a European call opion on an asse wih he price (S ) [,] is a conrac beween wo agens (buyer and seller), which gives he holder he righ o buy he asse a a prespecified fuure ime (he expiraion dae) for an amoun K (called he srike). he buyer of he opion is no obliged o exercise he opion. When he conrac is, issued he buyer of he opion needs o pay a cerain amoun of money called hepremium.hepayoffforhisopionisdefinedash(s )= Max(S K,) = (S K) +.hewrierofheopionreceives a premium ha is invesed in he combinaion of he risky and risk-free asses. he pricing problem is hen o deermine he premium, ha is, he price ha he seller should charge for his opion. he pricing problem has been solved in he pioneer work of Black and Scholes [1]. One of he shorcomings of he Black-Scholes model is he assumpion ha an opion rader canno affec he underlying asse price. However, i is well known ha, in a marke wih imperfec liquidiy, rading does affec he underlying asse price (see, e.g., Chan and Lakonishok [2], Keim and Madhavan [3], and Sharpe e al. [4]). In Liu and Yong [5], he auhors sudy he effec of he replicaion of a European opion on he underlying asse price. hey obain a generalizaion of he Black-Scholes pricing parial differenial equaion (PDE) as he following: V (S, ) + σ 2 S 2 2 V 2(1 λ (S, ) S( 2 V/ S 2 ) (S, )) 2 (S, ) S2 +r V S (S, ) r V (S, ) =, for (S, ) ], + [ ], ], (1) V (S, ) =f(s), <S<, (2) where λ(s, ) is he price impac funcion of he rader. he classical Black-Scholes PDE is a special case of (2) when λ(s, ) =. here are also several oher papers ha have sudied he financial markes wih jumps (among ohers are Meron [6],
2 ISRN Mahemaical Analysis Drischel and Proer [7], El-Khaib and Privaul [8], and El- Khaib and Al-Mdallal [9]). However, none of he previous sudies based on he jump-diffusion approach deals wih illiquid markes, o he auhors bes knowledge. his paper exends he model of Liu and Yong [5] byincludingajumpdiffusion srucure in he underlying opion pricing model. his appears o be an imporan issue because he model ha is suggesed in his paper allows for he possibiliy o accoun for sudden and random significan changes in he marke ha migh no be capured by he exising models in he lieraure such as he coninuous model suggesed by Liu and Yong [5]. Hence, he approach ha is developed in his paper is expeced o be more useful in financial risk managemen, especially in he cases in which he financial markes are under sress. he disposiion of he res of he paper is he following. Secion 2 inroduces he jump-diffusion model for an illiquid marke. Secion 3 deals wih he pricing problem of an opion wihin he conex of a jump-diffusion model along wih he proof for he suggesed soluion. Secion 4 concludes he paper. 2. A Jump-Diffusion Model for Illiquid Markes We sar wih presening some necessary denoaions. Le (N ) [,] be a Poisson process wih deerminisic inensiy ρ. LealsoM = N ρbe is associaed compensaed process. he process (W ) [,] denoes a Brownian moion. he probabiliy space of ineres is (Ω, F,P)wih (M ) [,] and (W ) [,] being independen. Le (F ) [,] signify he filraion generaed by (N ) [,] and (W ) [,].hemarke is assumed o have wo asses: a risky asse (S ) [,] and a risk-free asse denoed by (A ) [,].hemauriyis, he srike is K, andhepayoffish(s )=(S K) + Max{S K, }.AsinLiuandYong[5],hereurnonherisk-freeasse indirecly depends on S, and he opion rader s rading in he sock marke has a direc impac on he sock price. his price impac, which an invesor can cause by rading on an asse, funcions in such way ha i increases he price when buying he asse and i decreases he price when selling he asse. he price of he risk-free asse is given by da =r(, S ) A d, [, ], (3) where r > denoes he ineres rae. he price of he risky asse is generaed by he following sochasic differenial equaion: ds S =μ(,s )d+σ(,s )(dw +adm ) +λ(,s )dθ, [, ], S =x>, where μ and σ represen he expeced reurn and volailiy, respecively, he erm a is a real consan, and λ(s, ) denoes he price impac facor creaed by he rader via selling or buying he underlying asse. θ ishenumberofsharesha he rader has in he sock a ime.hence,λ(s, )dθ capures (4) he price impac of rading. Before dealing wih he pricing of a European opion in a jump-diffusion illiquid marke, we need o observe he following remark. Remark 1. he parameer a in (4) deermines he direcion of he jumps (i also affecs he jumps size). In fac, he following can be saed. (i) If a<, hen he jumps are pushing he sock price down;hais,hesockpriceisdecreasingaeach jump. (ii) If a=, hen here are no jumps, and herefore model (4) is reduced o he model in Liu and Yong [5]. (iii) If a>, hen he jumps are pushing he sock up; ha is, he sock price is increasing a each jump. 3. Pricing of a European Opion in Jump-Diffusion Illiquid Markes Le (V ) [,] be he wealh process for he rader. Le also (ψ ) [,] denoe he number of unis invesed in he risk-free asse. hen, he value of he porfolio is given by V =ψ A +θ S, [, ]. (5) Assume ha he number of shares of he risky asse saisfies he following condiion: dθ =η d + ζ (dw +bdm ), [, ]. (6) Le us consider a European call opion wih he payoff defined as h(s ):=(S K) +. In order o replicae he opion for a perfec hedge, we search for a sraegy (ψ,θ ) [,] which, a he expiraion dae of he opion, leads o having a value of heunderlyingwealhobeequalohepayoff;hais,v = h(s ). hen we can sae he following Proposiion. Proposiion 2. he wealh process for he rader of he jumpdiffusion model (4) saisfies he following sochasic differenial equaion: dv ={r(,s )V +[μ(,s ) r(,s )+λ(,s )η ]θ S }d +θ S [λ (, S )ζ +σ(,s )] dw +θ S [aσ (, S )+bλ(,s )ζ ]dm. Proof. By using (3), (4), (5), and (6), we have he following: dv =ψ da +θ ds = V θ S da A +θ S [μ(,s )d+σ(,s )(dw +adm )+λ(,s )dθ ] (7)
ISRN Mahemaical Analysis 3 ={r(,s )V +(μ(,s ) r(,s )) θ S }d +θ S {σ (, S )(dw +adm ) +λ(,s )[η d + ζ (dw +bdm )]} ={r(,s )V +[μ(,s ) r(,s )+λ(,s )η ]θ S }d +θ S [λ (, S )ζ +σ(,s )] dw +θ S [aσ (, S )+bλ(,s )ζ ]dm, (8) For any funcion G C 1,2 ([, ] ], [),onehas G(,X )=G(,X ) + ( s G(s,X s )+(g s k s ρ) x G(s,X s ) + 1 2 l2 s 2 xx G(s,X s )) ds + l s x G(s,X s )dw s (12) which ends he proof. Our aim in his paper is o price he European opion wih payoff h(s ) where S is given by (4). We replicae he European opion by searching a wealh (V ) [,] which leads o he erminal value V =h(s ).hus,asinliuandyong[5], we need o solve he following sysem of sochasic differenial equaions: dθ =η d + ζ (dw +bdm ), + (G (s, X s ) G(s,X s )). s Equaion (12) can be wrien in he following forma: G(,X )=G(,X ) + [ s G(s,X s )+(g s k s ρ) x G(s,X s ) + 1 2 l2 s 2 xx G(s,X s ) ds S =[μ(,s )+λ(,s )η ]d +ρ (G (s, X s +k s ) G(s,X s )) ] ds +[σ(,s )+λ(,s )ζ ]dw +[aσ(,s )+bλ(,s )ζ ]dm, dv ={r(,s )V +[μ(,s ) r(,s )+λ(,s )η ]θ S }d (9) + [G (s, X s +k s ) G(s,X s )] dm s + l s x G(s,X s )dw s. (13) +θ S [λ (, S )ζ +σ(,s )] dw +θ S [aσ (, S )+bλ(,s )ζ ]dm, θ >, S >, V =h(s ). he above sysem is called FBSDEs (forward-backward sochasic differenial equaions) sysem. In order o derive he PDE for he European opion price, we need he Iô formula which is given by he following lemma (see Proer [1]). Lemma 3. Le g, l,andk be hree adaped processes such ha g s ds <, l s 2 ds <, ρ k s ds <. Le X=(X ) [,] be he process defined by (1) dx =g d + l dw +k dm. (11) he following proposiion provides he PDE for he price of he European opion in he jump-diffusion illiquid marke presened in Secion2. Proposiion 4. Le f(, S ) denoe he price of he European opion a ime [,]for he model presened in Secion 2. hen he corresponding P.D.E. for he underlying opion price is given by r(,s )V +[μ(,s ) r(,s )+λ(,s )η ]θ S = f(,s ) + (μ (, S )+λ(,s )η ρ [aσ (, S )+bλ(,s )ζ ]) S S f(,s ) + 1 2 [σ (, S )+λ(,s )ζ ] 2 S 2 2 SS f(,s ) + ρ (f (, S (1 + aσ (, S )+ bλ(,s )ζ )) f (, S )), (14) wih he erminal condiion f(, S ) = h(s ).Moreover, he marke is incomplee, and here is no sraegy leading o
4 ISRN Mahemaical Analysis he erminal wealh V = h(s ) := f(,s ).However,he number of shares θ ha minimizes he variance is given by o find he number of shares θ invesed in S,weneedosolve he following problem: θ = (σ+λζ) 2 S 2 S f (σ+λζ) 2 S 2 +ρs 2 (aσ + bλζ) 2 + ρs (aσ + bλζ) (f (, S (1+aσ+bλζ)) f) (σ+λζ) 2 S 2 +ρs 2 (aσ + bλζ) 2. (15) Minimize θ E [Π 2 (θ)], (19) where Π(θ) := (h(s ) V ).Byusing(7), (16), and (17), we have E[Π 2 (θ)] Proof. Le (θ, S, V) be an adaped soluion of he FBSDE (9), and assume ha here exiss a smooh funcion f C 3,1 (], [ [,]) such ha f(, S ) represens he price ofheeuropeanopionaime [,]. Since he price of he opion a mauriy is equal o he payoff, hen f(, S )= h(s ).Now,usingheIôformula(13), we obain df (, S ) = {(μ (, S )+λ(,s )η ρ [aσ (, S )+bλ(,s )ζ ]) S S f(,s ) + 1 2 [σ (, S )+λ(,s )ζ ] 2 S 2 2 SS f(,s )+ f(,s ) + ρ (f (, S (1 + aσ (, S )+bλ(,s )ζ )) f (, S ))} d +[σ(,s )+λ(,s )ζ ]S S f(,s )dw +[f(,s (1 + aσ (, S )+bλ(,s )ζ )) f (, S )] dm. (16) By comparing (7) and(16), one can deduce ha i is impossible o find a sraegy (η,ζ ) [,] ha resuls in he erminal wealh V =h(s ):=f(,s ).hus,wepuheermbelonging o d equaions (7)and(16)equaloeachoher,which gives he PDE of he opion price, and hen we minimize he disance beween he wealh V and he price f(, S )= h(s ) over he number of shares of he underlying asse, ha is, θ.hepdeofheopionpriceinhiscaseis r(,s )V +[μ(,s ) r(,s )+λ(,s )η ]θ S = f(,s )+(μ(,s )+λ(,s )η ρ[aσ(, S )+bλ(,s )ζ ]) S S f(, S ) + 1 2 [σ (, S )+λ(,s )ζ ] 2 S 2 2 SS f(,s ) +ρ(f(,s (1 + aσ (, S )+bλ(,s )ζ )) f (, S )), wih he erminal condiion (17) f(,s )=h(s ). (18) where =E[( ([σ (, S )+λ(,s )ζ ] 2 S ( S f(,s ) θ )) dw ) ] +E[( (f (, S (1 + aσ (, S )+bλ(,s )ζ )) f(,s ) 2 θ S [aσ(, S ) + bλ(, S )ζ ]) dm ) ] =E[ ([σ(, S ) + λ(, S )ζ ]S ( S f(, S ) θ )) 2 d] +E[ ρ(f (, S (1+aσ (, S )+bλ(,s )ζ )) =E[ l(θ )d], l (x) = (σ+λζ) 2 S 2 ( S f x) 2 f(, S ) θ S [aσ(, S )+bλ(, S )ζ ]) 2 d] (2) + ρ(f (, S (1+aσ+bλζ)) f xs(aσ + bλζ)) 2. (21) he minimum is obained a l (x) =, whichyieldshe following resul: 2(σ+λζ) 2 S 2 ( S f x) θ = 2S(aσ + bλζ) ρ(f(,s (1+aσ+bλζ)) f xs [aσ + bλζ]) =, (σ+λζ) 2 S 2 S f (σ+λζ) 2 S 2 +ρs 2 (aσ + bλζ) 2 + ρs (aσ + bλζ) (f (, S (1+aσ+bλζ)) f) (σ+λζ) 2 S 2 +ρs 2 (aσ + bλζ) 2, which ends he proof. (22)
ISRN Mahemaical Analysis 5 I is worh menioning ha, in he case where here are no jumps, ha is, when a=b=, θ= S f, and he PDE in he previous proposiion is reduced o he PDE ha is obained in Liu and Yong [5], assuming ha here are no dividends. 4. Conclusions Opion pricing is an inegral par of modern risk managemen in increasingly globalized financial markes. he classical Black-Scholes model is regularly used for his purpose. However, one of he main pillars ha makes his model operaional is he underlying assumpion ha he markes are perfecly liquid. his assumpion is, noneheless, no fulfilled in realiy since perfecly liquid markes do no exis. In our opinion, he quesion should no be wheher he markes are illiquid or no; he quesion should be abou he degree of illiquidiy. hus, aking ino accoun he fac ha markes are illiquid can improve he precision of he underlying opion pricing. his paper is he firs aemp, o our bes knowledge, ha exends he exising lieraure on opion pricing by inroducing a jump-diffusion model for illiquid markes. his seems o be a more realisic approach o deal wih a marke ha is incomplee. A soluion for he opion pricing wihin his conex is provided along wih he underlying proof. he suggesed soluion migh be useful o invesors in order o deermine he opimal value of an opion in a marke ha is characerized by illiquidiy. References [1] F. Black and M. Scholes, he pricing of opions and corporae liabiliies, Poliical Economy, vol.81,no.3,pp.637 654, 1973. [2] L.ChanandJ.Lakonishok, hebehaviorofsockpricesaround insiuional rades, Finance, vol. 5, pp. 1147 1174, 1995. [3] D. B. Keim and A. Madhavan, he upsairs marke for largeblock ransacions: analysis and measuremen of price effecs, Review of Financial Sudies,vol.9,no.1,pp.1 36,1996. [4]W.F.Sharpe,G.J.Alexander,andJ.V.Bailey,Invesmens, Prenice Hall, Upper Saddle River, NJ, USA, 1999. [5] H. Liu and J. Yong, Opion pricing wih an illiquid underlying asse marke, Economic Dynamics & Conrol, vol. 29, no. 12, pp. 2125 2156, 25. [6] R. C. Meron, Opion pricing when underlying sock reurns are disconinuous, Financial Economics, vol. 3, no. 1-2, pp. 125 144, 1976. [7] M. Drischel and Ph. Proer, Complee markes wih disconinuous securiy prices, Finance & Sochasics,vol.3,no.2,pp. 23 214, 1999. [8]Y.El-KhaibandN.Privaul, Hedgingincompleemarkes driven by normal maringales, Applicaiones Mahemaicae, vol.3,no.2,pp.147 172,23. [9] Y.El-KhaibandQ.M.Al-Mdallal, Numericalsimulaionsfor he pricing of opions in jump diffusion markes, Arab Journal of Mahemaical Sciences, vol. 18, no. 2, pp. 199 28, 212. [1] P. Proer, Sochasic Inegraion and Differenial Equaions. A New Approach, Springer, Berlin, Germany, 199.
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