Merton's Jump diffusion model: Specify poisson rate

Question

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 $\lambda$)?

Answer 1

$\lambda$ 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