Maximum Likelihood Estimation for Merton Jump-Diffusion Parameters

Summary

The document frames a computational question about estimating the parameters of a Merton jump-diffusion model by maximum likelihood. It writes the log-return density as a Poisson-weighted sum of normal densities, where each term represents a possible number of jumps during an observation interval. The likelihood across observations is then formed from those densities and optimized over the model parameters.

The central implementation concern is that the density involves an infinite series, so a direct approach truncates the jump count at a chosen upper bound. The question notes that evaluating this sum for every observation at every optimizer iteration can be costly and introduces approximation error tied to the truncation choice. No answer or proposed algorithm is included, so the document does not establish a faster method or compare alternatives. It is useful as a statement of the likelihood structure and its computational tradeoff, but readers must consult other sources for truncation criteria, numerical safeguards, or alternative estimation techniques.

Key ideas

• The Merton jump-diffusion return density is represented as a Poisson mixture of normal densities.
• Maximum likelihood estimation combines the observation-level densities into a likelihood over the sample.
• Numerically evaluating the infinite mixture typically requires truncating the possible jump count.
• The truncation bound affects both computational cost and approximation accuracy, and no solution is supplied.

Tags

# Efficient way to perform MLE on Merton Jump Diffusion model parameters?


# Efficient way to perform MLE on Merton Jump Diffusion model parameters?












I understand that under Merton Jump Diffusion Model, if we are going to estimate the parameters $ \alpha, \sigma,\mu_J, \delta, \lambda $, we can use maximum likelihood estimation on the probability density of log return $ Y_t = ln(\frac{X_t}{X_0}) $

$ P(Y_t) = \sum_{i=0}^{\infty} \frac{e^{-\lambda t}(\lambda t)^i}{i!} N(Y_t; (\alpha - \frac{\sigma^2}{2} - \lambda k)t +i \mu_J, \sigma^2 t + i \delta^2) $

and we try to maximize the likelihood of

$ L(\theta;Y) = \prod_{t=1}^{T} P(Y_t) $

Here for each calculation of $ P(Y_t) $ we have an infinite series to sum. I can only think of a naive way to just truncate $ i $ at certain large number $ n $. But still every iteration on the optimizer still need to run $ O(T n) $ step assuming every other computation being constant time. It seems slow for large $ T $ and $ n $, and also inaccurate because depending on the arbitrary choice of $ n $. Is there any better algorithm to do this?

Thanks

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