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(Vi−1))
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.η1e−η1y↿y≥0+q.η2eη2y↿y<0,η1>1,η2>0
where
p,q≥0,
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{(μ−12σ2)t+σW(t)}∏N(t)i=1Vi
and also The stock price process,
(St)t≥0, 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+Ni∑i=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−(r−q)St}t≥0 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=r−q−12σ2+∫R(ex−1)ν(dx)=r−q−12σ2+λη
Here we focus on the case where the Levy measure is associated to the pure-jump component and hence the Levy measure
ν(dx) can be written as
λf(x)dx, where the weight function
f(x) can take the following form:
f(x):=p.η1e−η1x↿x≥0+(1−p).η2eη2x↿x<0,η1>1,η2>0
Also in η=∫R(ex−1)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=pα1α1−1+(1−p)α2α2+1−1
This is found by integrating ex over the real line by setting α1>1 and α2>0.
We let τ=T−t, 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=T−t, 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)=12σ2∂2xu+(r−q−12σ2−λη)∂xu−(r+λ)u+λ∫Ru(x+y,τ)f(y)dy
with initial value
u(x,0)=g(x):=G(ex)={max{ex−k,0},call optionmax{k−ex,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.
Let {Pt∣t≥0} be a compound Poisson process, where Pt=Nt∑i=1(Vi−1),
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
Pt−λtE(V1)=Pt−λt∫R(ex−1)f(x)dx
is a martingale. We denote by
η=∫R(ex−1)f(x)dx. Moreover, we assume that the equity price process
{St∣t≥0} satisfies the SDE
dStSt=(r−q−λη)dt+σdWt+dPt,
where
{Wt∣t≥0} is a standard Brownian motion. Then
St=S0exp((r−q−12σ2−λη)t+σWt+Nt∑i=1Yi).
That is,
dlnSt=(r−q−12σ2−λη)dt+σdWt+dNt∑i=1Yi.
Let Xt=lnSt, and u(Xt,t) be the option price at time t, where 0≤t≤T. Then, by Ito's formula, u(Xt,t)=u(X0,0)+∫t0∂tu(Xs,s)ds+∫t0∂xu(Xs−,s)dXs+12σ2∫t0∂xxu(Xs,s)ds+∑s≤t[u(Xs,s)−u(Xs−,s)−∂xu(Xs−,s)ΔXs](where ΔXs=Xs−Xs−)=u(X0,0)+∫t0∂tu(Xs,s)ds+∫t0∂xu(Xs,s)dXcs+12σ2∫t0∂xxu(Xs,s)ds+∑s≤t[u(Xt,t)−u(Xt−,t)](where Xct=(r−q−12σ2−λη)t+σWt)=u(X0,0)+∫t0∂tu(Xs,s)ds+∫t0∂xu(Xs,s)dXcs+12σ2∫t0∂xxu(Xs,s)ds+∫t0∫R[u(Xs−+y,s)−u(Xs−,s))]μ(ds,dy)(where μ=∞∑i=1δτi,Yi)=u(X0,0)+∫t0∂tu(Xs,s)ds+∫t0∂xu(Xs,s)dXcs+12σ2∫t0∂xxu(Xs,s)ds+∫t0∫R[u(Xs−+y,s)−u(Xs−,s))](μ(ds,dy)−dsv(dy))+∫t0ds∫R[u(Xs+y,s)−u(Xs,s))]λf(y)dy,
where
v(dy)=λf(y)dy. Here
Mt=∫t0∫R[u(Xs−+y,s)−u(Xs−,s))](μ(ds,dy)−dsv(dy))
is a martingale. Since
u(Xt,t)e−rt is a martingale, and
d(u(Xt,t)e−rt)=e−rt[−rudt+du],
we obtain that
−ru(Xt,t)+∂tu(Xt,t)+(r−q−12σ2−λη)∂xu(Xs,s)+12σ2∂xxu(Xt,t)+∫R[u(Xt+y,t)−u(Xt,t))]λf(y)dy=0.
That is,
∂tu(Xt,t)+(r−q−12σ2−λη)∂xu(Xs,s)+12σ2∂xxu(Xt,t)−(r+λ)u(Xt,t)+λ∫Ru(Xt+y,t)f(y)dy=0.
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