Friday, July 31, 2015

options - how we can derive PIDE of double exponential Jump-diffusion model (we know as kou model)?


I'm working in double exponential Jump-diffusion model (we know as kou model) with following form , under the physical probability measure P: dS(t)S(t)=μdt+σdW(t)+d(N(t)i=1(Vi1))

‎ where W(t) is a standard Brownian motion, N(t) is a Poisson process with rate λ , and {Vi} is a sequence of independent identically distributed (i.i.d.) non negative random variables such that Y=log(V) has an asymmetric double exponential distribution with the density fY(y)=p.η1eη1yy0+q.η2eη2yy<0,η1>1,η2>0
‎ where p,q0, p+q=1, represent the probabilities of upward and downward jumps.


Solving the stochastic differential equation gives the dynamics of the asset price: S(t)=S(0)exp{(μ12σ2)t+σW(t)}N(t)i=1Vi
and also The stock price process, (St)t0‎‎, driven by these model, is given by: St=S0eLt
where S0 is the stock price at time zero and Lt is defined by: ‎Lt:=γct+σWt+Nii=1Yi
here,γc‎ is a drift term , σ‎ is a volatility, Wt‎‎‎‎‎ is a Brownian motion, Nt‎ is a Possion process with intensity ‎λ‎, Yi is an i.i.d. sequence of random variables.Since σ>0 in up equation, there exists a risk-neutral probability measure Q‎‎‎‎ such that the discounted process ‎{e(rq)St}t0‎ becomes a martingale,where ‎r‎‎is the interest rate and q is the dividend rate.Then under this new measure Q, the risk-neutral Levy triplet of Lt can be described as follows: ‎(γc,σ,ν)
where ‎γc=rq12σ2+R(ex1)ν(dx)=rq12σ2+λη
‎ Here we focus on the case where the Levy measure is associated to the pure-jump component and hence the Levy measure‎ν(dx)‎ can be written as ‎λf(x)dx‎, where the weight function ‎f(x)‎ can take the following form: f(x):=p.η1eη1xx0+(1p).η2eη2xx<0,η1>1,η2>0


Also in ‎η=R(ex1)f(x)dx‎‎ represents the expected relative price change due to a jump. Since we have defined the Levy density function ‎f(x)‎ for double exponential Jump-diffusion model, ‎η‎‏‎ can be computed as: ‎η=pα1α11+(1p)α2α2+11


This is found by integrating ex‎ over the real line by setting ‎α1>1‎ and ‎α2>0.



We let τ=Tt, the time-to-maturity, where T is the maturity of the financial option under consideration and we introduce x=logSt, the underlying asset's log-price. If u(x;τ) denotes the values of some (American and European) contingent claim on St when logSt=x and τ=Tt, then it is well-known, see for example, (Cont and Tankov, 2004) that u satisfies the following PIDE in the non-exercise region: τu(x,τ)=12σ22xu+(rq12σ2λη)xu(r+λ)u+λRu(x+y,τ)f(y)dy

‎‎‎ with initial value ‎u(x,0)=g(x):=G(ex)={max{exk,0},call optionmax{kex,0},put option


my question is how we can derive the above PIDE I've searched a lot of article but most of them only mention the PIDE and we said you can find in Cont and Tankov Book and also I've searched in this book but I could not find the Exactly above PIDE.


thanks for help.



Answer



Let {Ptt0} be a compound Poisson process, where Pt=Nti=1(Vi1),

and Nt is a Poisson process with intensity λ and jump times τi, i=1,,. Let Yi=lnVi and f(x) be the density function. Then PtλtE(V1)=PtλtR(ex1)f(x)dx
is a martingale. We denote by η=R(ex1)f(x)dx. Moreover, we assume that the equity price process {Stt0} satisfies the SDE dStSt=(rqλη)dt+σdWt+dPt,
where {Wtt0} is a standard Brownian motion. Then St=S0exp((rq12σ2λη)t+σWt+Nti=1Yi).
That is, dlnSt=(rq12σ2λη)dt+σdWt+dNti=1Yi.


Let Xt=lnSt, and u(Xt,t) be the option price at time t, where 0tT. Then, by Ito's formula, u(Xt,t)=u(X0,0)+t0tu(Xs,s)ds+t0xu(Xs,s)dXs+12σ2t0xxu(Xs,s)ds+st[u(Xs,s)u(Xs,s)xu(Xs,s)ΔXs](where ΔXs=XsXs)=u(X0,0)+t0tu(Xs,s)ds+t0xu(Xs,s)dXcs+12σ2t0xxu(Xs,s)ds+st[u(Xt,t)u(Xt,t)](where Xct=(rq12σ2λη)t+σWt)=u(X0,0)+t0tu(Xs,s)ds+t0xu(Xs,s)dXcs+12σ2t0xxu(Xs,s)ds+t0R[u(Xs+y,s)u(Xs,s))]μ(ds,dy)(where μ=i=1δτi,Yi)=u(X0,0)+t0tu(Xs,s)ds+t0xu(Xs,s)dXcs+12σ2t0xxu(Xs,s)ds+t0R[u(Xs+y,s)u(Xs,s))](μ(ds,dy)dsv(dy))+t0dsR[u(Xs+y,s)u(Xs,s))]λf(y)dy,

where v(dy)=λf(y)dy. Here Mt=t0R[u(Xs+y,s)u(Xs,s))](μ(ds,dy)dsv(dy))
is a martingale. Since u(Xt,t)ert is a martingale, and d(u(Xt,t)ert)=ert[rudt+du],
we obtain that ru(Xt,t)+tu(Xt,t)+(rq12σ2λη)xu(Xs,s)+12σ2xxu(Xt,t)+R[u(Xt+y,t)u(Xt,t))]λf(y)dy=0.
That is, tu(Xt,t)+(rq12σ2λη)xu(Xs,s)+12σ2xxu(Xt,t)(r+λ)u(Xt,t)+λRu(Xt+y,t)f(y)dy=0.


No comments:

Post a Comment

technique - How credible is wikipedia?

I understand that this question relates more to wikipedia than it does writing but... If I was going to use wikipedia for a source for a res...