U.U.D.M. Projec Repor 21:7 Opion pricing and hedging in jump diffusion models Yu Zhou Examensarbee i maemaik, 3 hp Handledare och examinaor: Johan ysk Maj 21 Deparmen of Mahemaics Uppsala Universiy
Maser hesis of Mahemaics Specializaion in Financial Mahemaics Opion Pricing and hedging in Jump-diffusion Models
Yu Zhou Advising professor: Johan ysk Deparmen of Mahemaics, Uppsala Universiy May, 21 1
Absrac he aim of his aricle is o solve European Opion pricing and hedging in a jump-diffusion framework. o beer describe he realiy, some major evens, for example, may lead o dramaic change in sock price; we impose a jump model o classic Black Scholes model. Under he assumpion ha he underlying asse is driven by a simple jump-diffusion process, along he fac ha risk free rae and volailiy are deerminisic funcions of ime, by changing he probabiliy measure o risk-neural measure Q, we obain he pricing formula of European opions. he hedging sraegy is defined in order o minimize he risk of hedging under a risk neural measure. Key Words: Black-Scholes model Jump-diffusion model opion pricing 2
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Conens 1. Inroducion 1.1 Jump-Diffusion Model 3 1.2 Incomplee Marke......3 1.3 Mahemaical ools..4 1.4 Abou his aricle.4 2. Opion Pricing and Hedging in he Black-Scholes Model 2.1 Inroducion o Opion Pricing....5 2.2 Opion Pricing in Black-Scholes Model.6 2.3 Pricing..7 2.4 Hedging....8 3. Opion Pricing and Hedging in he Jump-Diffusion Model 3.1Dynamics of he Underlying Asse 9 4
3.2 Condiions for S o be a Maringale.1 3.3 European Opions Pricing.12 3.3.1 Admissible Porfolios...12 3.3.2 European Opions Pricing 14 3.4 Hedging European Opions...16 4. Reference.2 5
Chaper 1 Inroducion 1.1 Jump-Diffusion models In he Black-Scholes Model, he sock price is driven by Brownian moion and is based on a coninuous funcion of ime. However, in realiy his is no ofen he case, since cerain imporan evens can lead o dramaic change in he sock price. o model such phenomenon, many sudies inroduced disconinuous sochasic 6
processes by adding jump-diffusion o he classic model. o price opions, we mus noice ha a jump diffusion model leads o an incomplee marke. herefore, he classic hedging mehods are no applicable here. here are raher exensive sudies in he jump-diffusion model. As early as 1976, Rober Meron noed ha when major evens happen, sock price change disconinuously, or, jump. He also noed ha he sock price is driven by Brownian moion (1976) as well as Poisson process. Having aken ha ino consideraion, Meron creaed he Jump-diffusion model. [4] 1.2 Incomplee marke In an arbirage-free marke, here is an equivalen probabiliy measure such ha he underlying asse is a maringale under risk neural measure, which implies ha he asse is a semi-maringale under he objecive measure P. Mahemaically, a complee marke means ha any coningen claim can be replicaed as a sochasic inegral of a sequence of semi-maringales. he inegran in such replicaions provides a sequence of hedging sraegies which are self-financing or, super hedging. Whils in Jump-diffusion models, more random variables are added o he marke, making i Incomplee. In his case, a general claim is no necessarily a sochasic inegral of he underlying asse. Economically speaking, his claim has an 7
inrinsic risk ha we only hope o reduce o his minimal componen. hus he problem is o find and characerize hese sraegies which minimize he risk. 1.3 Mahemaical ools he main mahemaical ools o apply o his field are (1) Sochasic Differenial Equaions, (2) Mone Carlo (simulaion) and (3) Maringale mehods. As he firs wo ools are raher ofen used in previous papers in our deparmen, his aricle would ry o derive he same resuls wih Maringale mehods. In he more general maringale approach, one specifies a sochasic process for he underlying asse. hen one can choose an equivalen probabiliy measure urning he discouned underlying ino a (possibly local) maringale and compues he derivaive's value as he condiional expecaion of is discouned payoff under his risk-neural measure. If he model has a Markovian srucure, hen his value urns ou o be some funcion u, say, of he sae variables. In he PDE approach, one can describe he sae variables by a sochasic differenial equaion (SDE) and hen derives for he funcion u based on he underlying maringale valuaion a PDE involving he coefficiens of he given SDE. [5] 8
1.4 Abou his aricle Our sudy will be divided ino hree pars. (1) he firs chaper is he inroducion, in which we presen he background of he jump-diffusion models and hedging in incomplee marke. (2) In he second chaper we begin wih he basic heory of opion pricing, Black-Scholes framework and hedging, where r is consan. (3) he hird chaper we assume ha he jump follows a Poisson disribuion, and ha he ineres rae is ime deerminisic. We firsly find he sufficien and necessary condiions for he underlying asse o be a maringale. Under such condiions, by changing he measure, we can obain he pricing formula under risk-neural probabiliy measure Q, and hen be able o derive he hedging sraegy by minimizing he risk. Overall, his aricle presens an alernaive way o solve opion pricing and hedging problem, in he Jump-diffusions model and incomplee marke. 9
Chaper 2 Opion Pricing and Hedging in he Black-Scholes Model 2.1 Inroducion o Opion Pricing 2.1.1 Opions An opion is a conrac beween a buyer and a seller ha gives he buyer he righ, (bu no he obligaion), o buy or o sell a paricular asse on or before he opions expiraion ime, a an agreed price (he srike price). In reurn for graning 1
he opion, he seller collecs a paymen (he premium) from he buyer. A call opion gives he buyer he righ o buy he underlying asse and a pu opion gives he buyer of he opion he righ o sell he underlying asse. If he buyer chooses o exercise his righ, he seller is obliged o sell or buy he asse a he agreed price. he buyer of a call opion wans he price of he underlying insrumen o rise in he fuure; he seller eiher expecs ha i will no, or is willing o give up some of he upside (profi) from a price rise in reurn for he premium and reaining he opporuniy o make a gain up o he srike price. In case of a pu opion, buyer acquires a shor posiion by purchasing he righ o sell he underlying asse o he seller of he opion for specified price during a specified period of ime. If he opion buyer exercises heir righ, he seller is obligaed o buy he underlying insrumen from hem a he agreed upon srike price, regardless of he curren marke price. [Wiki] 2.1.2 Opion Pricing In exchange for having an opion, he buyer pays he seller or opion wrier a fee, known as he opion premium. he opion premium is he maximum loss in an opion exchange, of he pary aking long posiion of he underlying asse, which depend on 11
he price change of he underlying asse. Since he underlying asse is usually a risky asse (e.g. a sock), he price of underlying asse is usually a sochasic process. Bu when he underlying asse price is seled down, he opions wrien on ha is also seled down. Mahemaically, for opion price F, here exiss a binomial funcion F(, S) such ha he price of he opion F = F(, S). A he examining dae, F is deerminisic. F = (S K) + (call) (K S ) + (pu) Opion pricing problem is abou o derive he above F, hence i is an inverse problem. 2.2 Black Scholes Model opion pricing and hedging 2.2.1 he Model Black Scholes is a model based on coninuous ime. here are wo differen ypes of asses on he marke: he risk free asse, which has price S a ime, and saisfies he equaion: ds = rs d 12
where r is he risk free rae; herefores = S e r, from now on we assume S = 1 for convenience. hen, S = e r,. And he risky asse, which has price S a ime, and saisfies he equaion: ds = μs d + σs db where μ and σ > are consan, and ha B is a sandard Brownian moion. Now define he presen value of S as S, i.e., S = e r S, By Io formula, where W = B + θ, θ σ = (μ ). σ ds = re r S d + e r ds = S (μ r)d + σdb = S σdw We define L = exp θ s ds 1 2 θ s 2 ds, dq = L dp hen Q and P are equivalen, hen we know from Girsanov heorem ha W is a Sandard Brownian Moion under Q. By Io formula, S = S exp (σw σ2 2 ), hus S is a maringale under Q. herefore Q is so called he risk neural measure or equivalen maringale measure. 2.2 Self financing Porfolio 13
Consider a porfolio ϕ = (ϕ ) = (h, h ), where h and h represen he number of shares of risk-free asse and risky asse, respecively. hus his porfolio is, a ime, worh Furhermore, is discouned value is V (ϕ) = h S 1 + h S V (ϕ) = e r V (ϕ) = h + h S. where S = e r S is he underlying asse price under measure Q ha we are going o discuss laer. Definiion 2.2.1 An F S adaped porfolio ϕ = (ϕ ) = (h, h ) is called self-financing if he value process V saisfies he condiions dv (ϕ) = h ds 1 + h ds h d + h 2 d < (2.1) where he laer condiion makes he porfolio admissible, which is known as he inegrable condiion. Definiion 2.2.2 Porfolio ϕ = (ϕ ) = (h, h ) is called aainable if i is self-financing, as well as is discouned value V (ϕ) is non-negaive and square-inegrable. 2.3 European opions pricing 14
o price a European opion, we define is coningen claim h = f(s ), where f(s) = (S K) + for a call opion and f(s) = (K S) + for a pu opion. A porfolio is called can be replicaed if for is coningen claim h here exiss a sraegy ϕ such ha he value of he porfolio sraegy is equal o h, i.e. V (ϕ) = h. his porfolio sraegy is hus called he replicae sraegy. Now we denoe F(, S) be he price of he opion a ime. so, F(, S) = E e r( ) h F S = E e r( ) f(s ) F S For European call: C(, S) = SN(d 1 ) Ke r( ) N(d 2 ) where d 2 = ln S K + r+σ2 disribuion 2 ( ) σ, d 2 = d 1 σ, and N(x) is sandard normal For European pu: P(, S) = ke r( ) S + C(, S) where i has he same noaions as in a call opion. Resuls of price expressions in his secion refer o source [1]. 2.4 Hedging Wih he previous definiion for replicaion, suppose our claim h admis an Io 15
represenaion for he form where h saisfies equaliy (2.1). h = h + h ds (2.2) Clearly, sraegy V (ϕ) = h + h ds is admissible. Moreover, i is self-financing. hus, Io represenaion (2.2) leads o a sraegy which produces h from iniial value of h wih no risk involved. Now, we inroduce he sandard arbirage-free assumpions, namely he risk-neural maringale measure, or he equivalen maringale measure. We assume a maringale measure Q such ha and S is a maringale under Q. Q P L2 (Ω, F, P) And h is obained in a Randon-Nikodym derivaive [6] h = F S (, S ) Δ(, S ) where Δ = F s (, s()) is called is Δ hedging. Noe ha in he Black Scholes Model h is he parial derivaive of he value of he opion a ime wih respec o he underlying asse price a ime. So far we have summarized he mahemaical consrucion of hedging in a complee financial marke, where every claim is aainable. In such case, hedging allows oal eliminaion of risk in handling an opion. 16
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Chaper 3 Opion Pricing and Hedging in he Jump-Diffusion Model o inroduce jump, we have o consider a sriking feaure ha disinguishes i from he sandard Black-Scholes model: his makes he marke incomplee, and here is no perfec hedging of opions in his case. I is no longer possible o price opions using a replicaing porfolio, and he se of probabiliy measures under which he discouned sock price is a maringale is infinie. In his chaper, we will discuss he compuaion of European opion prices and examine hedging sraegies ha minimize he quadraic risk under he pricing measure ha we choose. 18
3.1 Dynamics of he Underlying Asse Le us consider a financial marke wihou any ransacion fee, in which here are wo ypes of asses, he risk free asse and he risky asse; and he ime inerval [, ], wih he mauriy ime. We assume he risk-free asse value is P ; shor rae is a ime-deerminisic funcion r ; risky asse value is S. Risk-free asse value P follows he sochasic derivaive funcion dp = r d, P = 1 he risky asse consiss wo pars, a coninuous par modeled by a geomeric Brownian moion, and a jump par, wih he jump size modeled by a jump funcion U and he jump ime modeled by a Poisson process, i.e., ds = μ S d + σ S dw + U S dn (3.1) where μ and σ are ime deerminisic funcions; W is a sandard Brownian moion; N is a Poisson process wih parameer λ, which is independen of W ; U is he jump funcion; and τ j is he j-h say ime. So, U is he relaive jumping srengh of S, i.e., U j = S τj S τj /Sτj. Here he risky asse price is obviously lef coninuous, hence τ j is used o disinguish ha. Noe ha {U j } is a sequence of independen idenical disribuions. We define he disribuion of {U j } follows a funcion υ. We ake inegraion of (3.1) and ge he dynamics of he asse, S = S + S u [μ u du + σ u dw u ] he las erm can be furher ransformed as, + S u U u dn u 19
S u U u dn u N = S u U u d(u τ j ) = S τj U j = [S τj S τj ] N N Hence, S = S + S u [μ u du + σ u dw u ] N + [S τj S τj ] (3.2) Using he fac ha U is he relaive jumping size, here s anoher version for his equaion N S = S (1 + U j ) exp μ u 1 2 σ u 2 du + σ u dw u (3.3) 3.2 Condiions for S o be a Maringale In his secion we will provide a formal proof o he Sufficien and Necessary Condiions for S o be a maringale under Q, which enables us o change measure. Lemma 3.2.1 For all s >, denoe by GG s he σ-algebras σ U Ns +1, U Ns +2, U Ns +3, are independen o F s. Lemma 3.2.2 Le φ(y, z) be a measurable funcion from R d R o R, such ha for any real number z he funcion y φ(y, z) is coninuous on R d, and le(y ) be a lef-coninuous process, aking values in R d, adaped o he filraion (F ). Assume ha, for all >, 2
hen he process M defined by E ds υ (dz)φ 2 (Y s, z) < N M = φ(y τj, U j ) is a square-inegrable maringale and λ ds υ(dz)φ(y s, z) M 2 λ ds υ(dz)φ(y s, z) is a maringale. Noe ha by convenion = 1. Lemma 3.2.3 We keep he hypoheses and noaions of Lemma 3.2.3. Le (A ) be an adaped process such ha E A 2 s ds < for all >. Le L = A s dsdw s. hen he produc (L M ) is a maringale. Proofs for Lemma 3.2.1, 3.2.2, 3.2.3 refer o source [2]. Consider ha F is a deerminisic funcion ha saisfies he condiion 1 2 F 2 d <. Now le dw = dw + F d, we can find a non-negaive random variable L in he following expression, L = exp [ F u dw u 1 2 F u 2 du]. Given his expression, L is naurally he soluion of he SDE, dl = F L dw By Girsanov heorem we know ha ha W is a sandard Brownian moion under risk neural measure Q. [2] hus, for < s < we have he risk-neural E S F s as N E S F s = E e r du S 1 + U j exp μ u 1 2 σ u 2 du + σ u dw u F s 21
N = E S 1 + U j exp μ u r u 1 2 σ u 2 du + σ u (dw u F u du) F s N = E S 1 + U j exp (μ u r u σ u F u ) N du 1 2 σ u 2 du = S s E 1 + U j exp μ u r u 1 2 σ uf u du j=n s +1 = S s e λeu1( s) exp μ u r u 1 2 σ uf u du s = S s exp μ u r u 1 2 σ uf u + λeu 1 du s s + σ u dw u F s or, or, Hence, S is a maringale under Q if and only if μ u r u 1 2 σ uf u + λeu 1 du s μ u r u 1 σ 2 uf u du s = = λeu 1 ( s) where, 1 s μ u r u 1 σ 2 uf u du = λeu s 1. Consider he las equaion, < s, le s, we have ha μ r σ F = λeu 1 F = μ r + λeu 1 σ (3.4) Clearly, S is a maringale under Q if and only if (3.4) is saisfied, sufficienly and necessarily, given he inegrable condiion of course, Under such condiions, we have F 2 d < 22
N S = S 1 + U j exp λeu j + 1 2 σ u 2 du + σ u dw u We are going o use hese resuls in he following secions. 3.3 European Opion Pricing 3.3.1 Admissible Porfolios In his secion we will define a rading sraegy, as in he Black-Scholes model, by a porfolio ϕ = (h, h ),, ha has amoun h and h in wo asses a ime, respecively, aking values in R 2, represening he amouns of asses held over ime. Bu, o ake he jump ino accoun, we will consrain he processes h and h o be lef-coninuous. he value a ime of he sraegy ϕ is given in he following definiion. Definiion 3.3.1 he F S adaped porfolio ϕ = (h, h ), is called self-financing if, dv = h r e r s ds d + h ds aking ino accoun dv = h r e r s ds s d + h S (μ d + σ dw ) beween he 23
jump imes and, a a jump ime τ j, V jumps by an amoun ΔV τj = h τj S τj U j. Precisely, he self-financing condiion is or, V = V + h s db s V = V + h s r s e r u du ds N + h τj S τj U j + h s S s (μ s ds + σ s dw s ) s N + h τj S τj U j + h s S s (μ s F s σ s )ds + σ s dw s (3.5) Definiion 3.3.2 A porfolio ϕ = (h, h ), is called admissible if i saisfies he equaliy (3.5) and saisfies he condiions, h s ds < aa. ss., EE h 2 s S 2 s ds < Acually, in order o make sure he value of a hedging sraegy square-inegrable, we have o impose he previous condiions, which are sronger han he class of self-financing sraegies. [2] Proposiion 3.3.3 In an admissible porfolio ϕ = (h, h ), wih iniial value V, for any h,, here exiss a unique h ha keeps he porfolio admissible. he discouned value for his porfolio is given by V == V + h s S s σ s dw s + h τj S τj U j N λ ds h s S s υ(dz)z. 24
Proof of Proposiion 3.3.3 he value of he porfolio ϕ a ime can be wrien as he decomposiion V = Y + Z, in which Y = V + h s r s e r u du ds s + h s S s (μ s F s σ s )ds + σ s dw s N Z = h τj S τj U j hen, differeniaing he discouned produc Y, we have dy = de r u du Y = r e r u du Y d + e r u du dy Moreover, i can be wrien as s Y = V + r s e r u du Y s ds + e r u du dy s = V + r s e r u du Y s ds s s s + e r u du h s r s e r u du ds + h s S s (μ s F s σ s )ds + σ s dw s s = V + r s e r u du Y s ds + e r u du h s S s (μ s F s σ s )ds + σ s dw s s (3.6) Coninue by doing he same ransform o Z, we have N Z = e r u du Z = e r u du h τj S τj U j N τ j N τ j = e r u du + e r u du τ j h τj S τj U j = e r u du h τj S τj U j + e r u du τ j r s ds h τj S τj U j N where in he las erm, he inegraion would only ake place on he poins of s τ j, τ j s s hus we have, N τ j r u du Z = e h τj S τj U j + e r u du h τj S τj U j ds s N N τ j = e r u du h τj S τj U j + e r u du r s Z s ds s (3.7) 25
We ge V = Y + Z by combining (3.6) and (3.7). Afer rearrangemen, we obain, V = V + r s (Y s + Z s )ds N + h τj S τj U j + h s r s ds + h s S s (μ s F s σ s )ds + σ s dw s o simplify he upper formula, we subsiue V by h + h S in he righ hand side of he equaliy and he erms wih h cancels ou, which yields V = V + h s S s [(μ s F s σ s )ds + σ s dw s ] N + h τj S τj U j Finally we ake ino accoun he equaliy (3.4) and ge he following one ha we need o prove. We will coninue o use his in he nex secions. V = V λeu 1 h s S s ds + h s S s σ s dw s N + h τj S τj U j = V() + h s S s σ s dw s + h τj S τj U j N λ ds h s S s υ(dz)z (3.8) Remark 3.3.4 By Proposiion 3.3.3 i is clear ha he processes h and h, ogeher wih he iniial value V, we can uniquely deermine one given he oher wo. Remark 3.3.5 Noe ha here is no h erm in he righ hand side of equaliy (3.8), his indicaes ha he admissible porfolio ϕ = (h, h ) wih iniial value V is compleely deermined by he process (h ) represening he amoun of he risky asse. 26
3.3.2 European Opions Pricing Le us consider a European opion wih mauriy ; is coningen claim be h; i is F -measurable and square-inegrable. he seller of his opion receives V a ime and holds an admissible porfolio beween imes and. If V represens he value of his porfolio a ime, he payoff for he seller a mauriy is given by V h, which is also he hedging mismach. If his quaniy is posiive, he seller of he opion earns money, oherwise loses some. Is presen value is hen V e r u du h In order o beer replicae he coningen claim of European opion, we need o minimize he payoff V h, namely he risk. One way o measure he risk is o define he expeced square, i.e., R = E (V e rdu h) 2 Applying he ideniy E(X 2 ) = E 2 (X) + E(X E(X)) 2 o he variable X = e rdu (V h) we oboain, R = E 2 V e rdu h + E V e rdu h E(V e rdu h) Since, he condiion EE h 2 s S 2 s ds < in Definiion 3.3.2 implies ha he discouned value V is a square-inegrable maringale, we have ha E V = V. Moreover, we have, E V e rdu h = V E e rdu h By applying his ideniy o he R represenaion we obain, 2 27
R = V E e rdu h 2 + E e rdu h E e rdu h + V V 2 According o Remark 3.3.5, he quaniy V V depends only on he process h. hen i is obvious ha when minimizing R, he seller will expec a premiumv = E e rdu h. his appears o be he iniial value of any sraegy designed o minimize he risk a mauriy. For he same argumen, when selling he opion a ime >, one wans o minimize he quaniy R = E (V e rdu h F ) 2 will expec for a premium V() = E(e rdu h F ). Hence, we will use his quaniy o define he price of he opion, i.e., V = E e rdu h is he price of he European opion a ime and V = E(e rdu h F ) is ha a ime. Now, we will give an explici expression for he price of he European call or pu. By he definiion of European opions, he coningen claim h can be specified as follows, h = f(s ), call: f(x) = (x k) +, pu: f(x) = (k x) + V() = E e rdu f(s ) F N N = E e rdu f S 1 + U N +j e μ 1 2 σ2 σf u du + σdw (u) F = F(, S ) Wih Lemma 3.2.1 (U Ns ) s and F are independen, we can rewrie he equaliy as, F(, x) = E e rdu N N f x 1 + U N +j e μ 1 2 σ2 σf(u)du + σdw (u) 28
geing, hen we plug in he opion price for he Black-Scholes model, F (, x) = E e rdu F(, x) = E F, xe λeu j ( ) N( ) f xe μ 1 2 σ2 σf(u)du + σdw 1 + U j n = E F, xe λeu j ( ) e λ( ) λ n ( ) n 1 + U j n! n= (3.9) For differen disribuions of jumping size, we can compue he corresponding EU j numerically and hen can derive he expression for F(, x) explicily. 3.4 Hedging European opions We coninue o use he noaions ha we defined in he las secion. h = f(s ), V = F(, S ), R = E (e r(u)du f(s V )) 2 In secion (3.3.1) we have already reached a conclusion ha given wo ou of he hree variables h, h, and V, we can deermine he hird one explicily. In secion (3.3.2) we have seen ha he iniial value of any admissible sraegy aiming a minimizing he risk R a mauriy is gien by V = E e rdu h. Now we will deermine he process (h ) represening he amoun of he risky asse over he period [, ], so as o minimize R. Specifically, we consider an 29
admissible porfolio ϕ = (h, h ),, wih he value V a ime and ha he iniial value saisfies V = E e rdu h. hen we ry o give he inerpreaion for quadraic risk a mauriy R. Firsly, ake ino accoun he equaliy (3.8): V = F(, S ) + h s S s σ s dw s ) N + h τj S τj U j λ ds h s S s υ(dz)z We also have ha h = e r(u)du f(s ) = e rdu F(, S ). Le us define a funcion F(, x) = e rdu F(, xe rdu ), so ha F, S = e rdu F(, S ) = e rdu E (e Hence, F, S is a maringale under Q. rdu h F ) Le Y = S and φ(y, z) = F, Y (1 + z) F (, Y ) In our case, φ Y τj, U j = F τ, S τj 1 + Uj F τ, S τj = F τ j, S τj F τ j, S τj We denoe N M = φ Y τj, U j λ ds ν(dz) φ(y s, z) N = F τ j, S τj F τ j, S τj N = [F τ j, S τj F τ j, S τj ] λ ds ν(dz) F s, S s (1 + z) F s, S s λ ds ν(dz) F s, S s (1 + z) F s, S s (3.1) in which, according o Lemma (3.2.2), we know ha he process M is a 3
square-inegrable maringale. Now le us consider equaliy (3.9), F, S is C 2 [, ) R + on we can apply Io formula in each fragmen beween he jump imes. Apply Io formula o F, S, F, S = F(, S ) + F S s, S s σ 2 s S 2 s ds + F x s, S s [(μ s r s σ s F s )ds + σ s dw s ] + 1 2 2 F x 2 s, S s σ 2 s S 2 s ds + F τ j, S τj F τ j, S τj N ogeher wih he equaliy (3.1), we ge, F, S M = F(, S ) + F x s, S s σ 2 s S 2 s dw s + F x s, S s λeu j S s + F x s, S s + 1 2 F 2 x 2 s, S s σ 2 2 s S s + λ ds ν(dz) F s, S s (1 + z) F s, S s ds We know ha F, S and M are maringales under Q; herefore he process F, S M is also a maringale, which implies ha he hird erm of he righ hand side in he above equaliy should be zero. herefore, hen, F, S M = F(, S ) + F x s, S s σ s 2 S s 2 dw s 31
where, h V = F, S V = F, S M + M V = F(, S ) + F x s, S s σ s S s dw s + V + λ dsh s S s ν(dz)z h s S s σ s dw s h τj S τj U j N + F τ j, S τj F τ j, S τj λ ds ν(dz) F s, S s (1 + z) F s, S s = F x s, S s h s S s σ s dw s + F τ j, S τj F τ j, S τj h τj S τj U j N λ ds ν(dz) F s, S s (1 + z) F s, S s zh s S s M (1) (2) + M (1) F M = x s, S s h s S s σ s dw s N M (2) = F τ j, S τj F τ j, S τj h τj S τj U j λ ds ν(dz) F s, S s (1 + z) F s, S s zh s S s N According o Lemma 3.2.3 M (1) M (2) is a maringale under he risk neural measure Q, and i follows, E M (1) M (2) = M (1) M (2) = Whence, R = E (h V ) 2 = E (1) 2 M + E (2) 2 M where, E (1) 2 M = E ( [ F x s, S s h s ]S s σ s ds) E (2) 2 M = E {λ ds ν(dz) F s, S s (1 + z) F s, S s zh s S s 2 } 32
he risk a he mauriy is hen given by, R = E F x s, S s h s S s σ s ds + E λ ds ν(dz) F s, S s (1 + z) F s, S s zh s S s 2 Using he ideniies (which are obvious), F x s, S s = F x (s, S s), F s, S s (1 + z) = e s r u du F s, S s (1 + z), F s, S s = e s r u du F(s, S s ) = E e s r u du F 2 x (s, S s) h s S 2 2 s σ s + λ ds ν(dz) F s, S s (1 + z) F(s, S s ) zh s S s 2 ds I follows ha he minimal risk R is obained by finding a h s ha gives minimal value for he funcion in brackes [ ] above. ake he derivaive of [ ] wih respec o h s and le i equal o zero, [ ] = 2S 2 h s σ 2 s F s x (s, S s) h s 2λ z S s F s, S s (1 + z) F(s, S s ) zh s S s dν(z) where, Solving his equaion for h s yields, since (h ) is lef-coninuous, Δ(s, x) = h s = Δ(s, S s ) 1 σ 2 s + λ ν (dz)z 2 [σ s 2 F F s, x(1 + z) F(s, x) (s, x) + λ ν (dz)z ] x x By using his h s, we obain a process which deermines an admissible porfolio minimizing he risk a mauriy. Noe ha when here is no jump, i.e. λ =, his recovers Δ hedging of he Black-Scholes Model. Resuls in his secion have been checked wih source [2]. 33
Remark 3.4.1 In he Black-Scholes framework, he hedging is perfec, i.e. R =. However, in he jump-diffusion case, where he marke is incomplee, he minimized risk is generally posiive. 34
Reference [1] omas Björk Arbirage heory in Coninuous ime, second ediion Oxford Universiy Press 24 [2] Damien Lamberon & Bernard Lapeyre Inroducion o Sochasic Calculus Applied o Finance, second ediion Chapman & Hall 27 [3] S. G. Kou A Jump Diffusion Model for Opion Pricing wih hree Properies: Lepokuric Feaure, Volailiy Smile, and Analyical racabiliy Columbia Universiy 1999 35
[4] R. C. Meron Opion Pricing When Underlying Sock Reurns Are Disconinuous Journal of Finance Economics 1976 [5] David Heah Maringale versus PDEs in Finance: An Equivalence Resul wih Examples Univ. of ech., Sydney [6] Hans Föllmer, Marin Schweizer Hedging of Coningen Claims under Incomplee Informaion 199 36