# Merton's Jump diffusion model: Specify poisson rate

Currently applying the Merton's jump diffusion to test how Option price change as parameters change. However, I am struggling to specify the poisson rate $$\lambda$$. We know that:

$$P(\text{There is a jump})= \lambda dt$$ and $$P(\text{There is not a jump})= 1-\lambda dt$$

I am using the code provided by the following link:

https://www.mathworks.com/matlabcentral/fileexchange/41939-merton-jump-diffusion-option-price-matrixwise

I am confused with the term poisson rate. Do we refer in the jump rate as as percent of the total steps $$T/N$$, which is the cardinality of the partition (e.g $$5\%$$= "5 jumps per 100 steps", which is $$\lambda dt$$) or the arrival rate in the whole interval (e.g 120 jumps in total, which is $$\lambda$$)?

$$\lambda$$ is the intensity of the number of jumps per unit of time.

If you call $$N_t$$ the number of jumps up to time $$t$$ then $$E[dN_t]=\lambda dt$$ is the expected number of jumps in the interval $$(t,t+dt)$$

For more details you can check the wiki page

https://en.m.wikipedia.org/wiki/Poisson_point_process

• So if if $la$ is the percentage jump rate, then as input in the code above should I set $l=la/dt$? Dec 31, 2018 at 14:17
• I have no idea what the function you are calling actually does so i cannnot comment on that specifically.
– Ezy
Dec 31, 2018 at 14:57
• Would be pleased if you ( or someone else) specify which of the 2 should be used. Jan 1, 2019 at 12:24
• @Coxswaiiiin i would suggest you test both on a simple case of european vanilla and you will see based on jump size and intensity which value matches your expectation of total variance since total variance = diffusion variance + jump variance
– Ezy
Jan 1, 2019 at 12:43