In many textbooks and also in the original Merton's paper the solution of the SDE
dSt=Stμdt+StσdWt+St−d(Nt∑j=1Vj−1)
is written as
St=S0exp((μ−12σ2)t+σWt)Nt∏j=0Vj.
Can someone suggest me a textbook or a paper where the solution is explicitly derived? I am pretty confident that this is an application of the Ito lemma for semimartingale.
Answer
Let dSt=μStdt+σStdWt+St−dJt where Jt=Nt∑j=1(Vj−1) is a compound Poisson process, with Vj i.i.d. jump sizes (positive random variables) whose statistical properties are not relevant for what needs to be proven and Nt a standard Poisson process of intensity λ. The processes Wt, Nt and the random jump sizes Vj are all assumed to be independent of each other and defined over the same probability space.
Applying Itô's formula for semi-martingales with jumps to the function f(t,St)=ln(St) yields (see here) ln(St)=ln(S0)+(μ−σ22)t+σWt+∫t0(ln(Su)−ln(Su−))dNu From the SDE we then have that, at a jump time u Su−Su−=Su−(Vj−1)⟺Su=Su−Vj such that ln(Su)−ln(Su−)=ln(SuSu−)=ln(Vj) and therefore ln(St)=ln(S0)+(μ−σ22)t+σWt+Nt∑j=1ln(Vj) Finally, because Nt∑j=1ln(Vj)=ln(Nt∏j=1Vj) we get St=S0exp((μ−σ22)t+σWt)Nt∏j=1Vj=F(0,t)E(σWt)Nt∏j=1Vj with E(Xt):=exp(Xt−1/2⟨X⟩t) denoting the stochastic exponential of a process Xt (Doléans-Dade exponential).
More info on jump processes (and better mathematical treatment because what I wrote is not always rigorous) in this excellent document
Note that because E0[St]=F(0,t)E0[Nt∏j=1Vj]≠F(0,t) the above dynamics cannot be used for risk-neutral pricing purpose.
To obtain a proper risk-neutral framework, the compound Poisson process needs to get compensated by a drift term (so that the whole emerges as a martingale). The resulting SDE writes
dSt=(μ−k)Stdt+σStdWt+St−dJt
where one can show that k=λ(E(V1)−1)
and where the solution in that case reads St=F(0,t)E(σWt)e−ktNt∏j=1Vj
No comments:
Post a Comment