Interpreting the Jump Intensity in Merton’s Jump-Diffusion Model

Summary

The document clarifies the meaning of the Poisson rate in Merton’s jump-diffusion model. The rate λ is an intensity measured as expected jumps per unit of time. Over a short interval of length dt, the expected number of jumps is λdt, which corresponds to the approximate jump probability in that interval under the stated model.

This distinguishes the time-based intensity from the probability of a jump across one simulation step: the latter depends on both λ and the step length. The discussion provides a definition and an interpretation of the rate, but no calibration procedure, parameter estimate, or option-pricing comparison. Applying the intensity therefore requires specifying the model’s time units consistently with the interval used in a simulation.

Key ideas

• The Poisson intensity λ represents the expected number of jumps per unit of time.
• For a short interval dt, the expected jump count is λdt.
• The jump probability per simulation step depends on the step duration as well as the intensity.
• Use time units consistently when specifying jump intensity in a simulation.

Tags

# Merton's Jump diffusion model: Specify poisson rate


# 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 $\lambda$)?

## Answer by Ezy (score 2, accepted)

https://quant.stackexchange.com/a/43277

$\lambda$ 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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.