# How to simulate a Merton Jump Diffusion process?

I am talking about the Merton Jump Diffusion model, on this page, where they give the following formula:

$$dS_t = \mu S_t dt + \sigma S_t dW_t + (\eta-1) dq$$

where $W_t$ is a standard brownian motion, and $dq$ is an independent Poisson process (its value is 1 with probability $\lambda dt$)

On that page you can find some example code. However, it does not match the formula. I want to simulate stock paths with the MJD model but I do not know how to do it. What formula do they use for their simulation?

I have looked up a lot of papers, but they just give general from of the model and say this is the result, but they do not explain how to simulation goes in detail. What formula I have to use and how to implement the model. I am really frustrated because I do not know how to do this.

I take it you want to do a Monte-Carlo simulation.

You just need to decide of an unit of time $dt$ and then start simulating the path.

$dW_t$ is simulated using a random normal value. In Excel $N\left(\mu, \sigma\right)$ would be simulated by NORMINV(rand(), mu , sigma).

For your Poisson process you just have to simulate random numbers between 0 and 1 and compare against your probability of a jump and create jumps if needed ?

The page you linked used a formula linking $S_t$ and $S_{t+\Delta t}$

• mh, I am not clear what I have to do? @BlueTrin Nov 23 '12 at 13:58
• Ah I got you now ... the code provided use the solution to the equation you have just typed. Nov 23 '12 at 14:07
• no @BlueTrin I mean, how can I do this in R? How can I simulate this in R? I do not know it, because the equation above with all the d's I cannot simulate and I do not understand what formula they use in their code? Nov 23 '12 at 18:22
• Are μ,σ presumed(constant) or are they calculated using historical data of a company? May 27 '14 at 7:51
• @Riste: if your question is for OP, it is better to ask it on the question rather than one of the answer. May 28 '14 at 11:57