# 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. May 16 '16 at 13:53
• hi dear @Siron , Thanks for the tips, it has 2 editions , which one is better ? 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! May 16 '16 at 15:11

You should take a look at the BENCHOP project. There we benchmarked around 15 different numerical methods against 6 option pricing problems. One of the problems was the Merton model. The methods were split into 4 families: Monte Carlo, Fourier, Finite Difference, and Radial Basis Function methods.

This is the paper containing the results:
http://dx.doi.org/10.1080/00207160.2015.1072172,
and here is the project page where you can see the implementational details for each of the methods:
http://www.it.uu.se/research/project/compfin/benchop.

BENCHOP – The BENCHmarking project in option pricing
Lina von Sydow, Lars Josef Höök, Elisabeth Larsson, Erik Lindström, Slobodan Milovanović, Jonas Persson, Victor Shcherbakov, Yuri Shpolyanskiy, Samuel Sirén, Jari Toivanen, Johan Waldén, Magnus Wiktorsson, Jeremy Levesley, Juxi Li, Cornelis W. Oosterlee, Maria J. Ruijter, Alexander Toropov, and Yangzhang Zhao
International Journal Of Computer Mathematics Vol. 92 , Iss. 12,2015