I defined such a model for stock price
(1).... $$dS = \mu\ S\ dt + \sigma\ S\ dW + \rho\ S(dH - \mu) $$
, where $H$ is a so-called "resettable poisson process" defined as
(2).... $$dH(t) = dN_{\lambda}(t) - H(t-)dN_{\eta}(t) $$
, and $\mu := \frac{\lambda}{\eta}$.
Is it possible to derive some analytic results similar to Black-Scholes equation (3)?
(3).... $$ \frac{\partial V}{\partial t} + r\ S \frac{\partial V}{\partial S} + \frac{\sigma^2S^2}{2}\frac{\partial^2 V}{\partial S^2} - r\ V = 0$$
Even better, could we derive something similar to Black-Scholes formula for call/put option prices?
I tried but failed.
In classic GBM model, to get Black-Scholes euqation (3), the essential steps are:
By Ito's lemma,
(4)... $$df = (f_x+\mu f_x+\sigma^2/2\cdot f_{xx})dt + \sigma f_x dW$$
Based on GBM stock price model (5),
(5)... $$ds = \mu S dt + \sigma S dW$$
We have
(6).... $$dV = \left( \frac{\partial V}{\partial t} +\mu S \frac{\partial V}{\partial S}+\frac{\sigma^2 S^2}{2}\frac{\partial^2 V}{\partial S^2}\right)dt + \sigma S \frac{\partial V}{\partial S}dW$$
Putting (5) in (6) again we have
(7).... $$dV - \sigma S \frac{\partial V}{\partial S}dS = \left( \frac{\partial V}{\partial t} +\frac{\sigma^2 S^2}{2}\frac{\partial^2 V}{\partial S^2}\right)dt $$
then we can define (8)... $$\Pi = V - \frac{\partial V}{\partial S} S$$ , so the LHS of (7) is just $d\Pi$ and it's not related to any random effect, so we have
(9)... $$d\Pi = r \Pi dt$$ then we get (3).
After I introduce the "resettable Poisson process" $H_{\lambda, \eta}(t)$ in the model, I couldn't find a way to cancel both the $dW$ and $dN$....
Do you know how to solve this?
Any suggestions are appreciated, I'm stuck here...
