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Estimating Jump Parameters in a Compound Poisson Model

Article Quant Q&A · Author: rakesh
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Summary

The question asks how to estimate the jump intensity in a Black–Scholes–Merton setting from observed jump counts across three consecutive months. The response points to a compound Poisson process as the model for jumps and directs readers to a statistical treatment of maximum likelihood estimation for that process.

The material identifies a relevant modeling framework and reference, but it does not derive the likelihood, calculate an estimate from the monthly observations, or explain assumptions such as whether jump counts are independent and the intensity is constant over time. The counts alone may inform jump frequency, but estimating the full compound process can also require information about jump sizes. Readers should consult the cited methodological work and define the model and observation interval before interpreting an estimate.

Key ideas

  • Model jump arrivals as a compound Poisson process when that assumption fits the data.
  • Maximum likelihood methods for compound Poisson processes provide a framework for parameter estimation.
  • The response points to a reference but does not work through the likelihood or estimate the parameter.
  • Jump counts describe arrival frequency, while jump-size modeling may require additional observations.

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StatisticsDerivatives pricing
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# How to estimate Black Scholes parameters using Maximum Likelihood estimate method


# How to estimate Black Scholes parameters using Maximum Likelihood estimate method












It might be a naive question but I'm new to finance. I've been trying to get my head around this question from a long time and still totally clueless about this.

Suppose that the observed jumps in three consecutive months are 0, 1, 2 times. How to find the estimated parameters λ̂ of the Black-Scholes-Merton model using maximum likelihood estimate method.

Thanks for your time !

## Answer by Will Gu (score 2)

https://quant.stackexchange.com/a/39905

The jumps are modeled as an independent Compound Poisson Process. Here's a paper (Leopold Simar: Maximum Likelihood Estimation of a Compound Poisson Process in The Annals of Statistics 4(6) November 1976) that describes how to get MLE for such 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.