# Numerical Methods for Merton Model

The stochastic differential equation for an underlying with jumps in Merton model is: $$d{{S}_{t}}=\mu \,{{S}_{t}}dt+\sigma \,{{S}_{t}}\,d{{W}_{t}}^{P}+(J-1){{S}_{t}}d{{q}_{t}}$$ where

$t \quad\,\,\, \quad$ = time

$S \quad\, \quad$ = Underlying stock price

$\mu\,\,\quad\quad$ = Drift rate

$\sigma\quad\,\,\quad$ = Volatility

$dW\,\,\quad$ = Increment of Gauss-Wiener process

$dq\quad\quad$ = Poisson process

$J -1 \,\,\,\,\,$= Impulse function producing a jump from $S$ to $S\lambda$

$K\quad\quad$ = $E(\lambda -1)$ , expected relative jump size

and

define a Poisson process $dq_t$ as follows:

d{{q}_{t}}=\left\{ \begin{align} & 0\,\,\,\,\,\,,\,\,\,\,with\,probability\,\,1-\lambda (t)dt\, \\ & 1\,\,\,\,\,,\,\,\,\,with\,probability\,\,\,\,\,\,\,\,\lambda (t)dt\, \\ \end{align} \right.

where $\lambda$ = Poisson arrival intensity

We assume that Gauss-Wiener process and jumps are independent. Based on the SDE the resulting $PIDE$ for a contingent claim $V(S,t)$ that depends on $S$ is given by (Merton 1976): $$\frac{\partial V}{\partial t}+(r-K \lambda )S\frac{\partial V}{\partial S}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}}-(r+\lambda )V+\lambda \,\int_{0}^{\infty }{g(J)V(J{{S}_{t}},t)\,}dJ=0$$ now I want to solve this $PIDE$ with some numerical methods like "Monte Carlo" , "Binomial Tree " etc. in order to pricing European Option. Would anybody give or teach me some useful and instructive note or some references that I can learn more than 3 numerical methods for Pricing European Options under Merton model ?

I appreciate any help.

• The book 'Financial Modeling with jump processes' by Cont & Tankov provides a chapter about solving Integro-differential equations. They give numerical solution methods (finite difference, finite elements and Markov methods) and analytical methods as well. – Cavents May 16 '16 at 13:53
• hi dear @Siron , Thanks for the tips, it has 2 editions , which one is better ? – Roozbe May 16 '16 at 14:03
• I'm using the 2004 version and I can say that I find it an excellent book if you wish to learn about financial modeling of Lévy processes. Though I don't have much experience with solving PIDE's, I think it can be worth it to take a look at the book. If I were you, I would go for the latest version since this is the most updated one. Good luck! – Cavents May 16 '16 at 15:11

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