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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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  • $\begingroup$ I think that there is a latent variable to be modeled as well: the actual number of jumps at each time step, no? But even if that’s not the case: in typical applications you can very easily cut of the model at 5 to 10 jumps (daily returns) and cover virtually the whole jump domain by this... $\endgroup$ – Kermittfrog Dec 31 '20 at 7:21
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    $\begingroup$ You know that the number of jumps per internal is Poisson distributed. Given $\lambda$ and $\Delta t$ you can then compute the upper bound for the sum as the $x$-percent quantile of this distribution, e.g. $x = 99.99$%. This scales well with different underlying assets and you don't need to resort to ad-hoc rules like "not more than 5 per day". $\endgroup$ – LocalVolatility Jan 1 at 13:57
  • $\begingroup$ @LocalVolatility nice - thanks for the hint from my side as well. $\endgroup$ – Kermittfrog Jan 1 at 18:18
  • $\begingroup$ @LocalVolatility thanks that would be a great approach $\endgroup$ – Paul Jan 2 at 14:53

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