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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...

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migrated from math.stackexchange.com Jun 8 '12 at 21:44

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do you means that you would like to price under semi-martingale assumptions (and especially jumps)? you should look at Financial modelling with Jump Processes ( Chapman & Hall / CRC Press) by Rama CONT and Peter TANKOV cmap.polytechnique.fr/~rama/Jumps . –  lehalle Jun 10 '12 at 8:06
It is possible, do you know how to calculate the generator of (S,H)? The answer will have a local term - i.e. A term involving integral, you will get a PIDE rather than a PDE. –  Lost1 Dec 29 '13 at 11:05
What is this model called? –  Lost1 Dec 29 '13 at 11:07
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