Method of Moments for Merton Jump Diffusion Parameters

Summary

The document asks how to estimate parameters of the Merton jump diffusion model from historical returns using the method of moments. It proposes expressions for return variance, skewness, and kurtosis in terms of diffusion volatility and jump intensity and jump-size distribution parameters. It also considers estimating jump intensity by separating jumps from regular returns with a fixed threshold.

No answer or derivation is included, so the proposed moments are not verified and no estimation procedure is established. The question itself flags a potential modeling choice: a fixed threshold determines which observations count as jumps, but the document provides no evidence about how to select it or quantify resulting errors. It also gives no references or empirical results. Readers should treat the formulas and threshold idea as an open estimation question rather than validated guidance for calibrating the model.

Key ideas

• The document proposes variance, skewness, and kurtosis expressions for method-of-moments estimation in a jump diffusion model.
• The proposed moments depend on diffusion volatility and jump intensity and size parameters.
• A fixed return threshold is suggested as a way to distinguish jumps from ordinary returns.
• The document provides no derivation or answer confirming the formulas.
• Threshold selection and its effect on parameter estimates remain unresolved.

Tags

# Moments of the Merton Jump Diffusion model for method of moments estimation


# Moments of the Merton Jump Diffusion model for method of moments estimation












For performing a parameter estimation for the MJD model ($d(\ln S_t) = \left(\mu - \lambda k - \frac{1}{2} \sigma^2\right) dt + \sigma \, dW_t + \ln Y \, dN_t$) using the method of moments based on historic data, are the following the correct theoretical moments?



- Variance:$\text{Var}(r_t) = \left(\sigma^2 + \lambda (\sigma_j^2 + \mu_j^2)\right) \Delta t.$

- Skewness:$M_3 = \left(\mu_j^3 + 3\sigma_j^2 \mu_j\right) \lambda \Delta t.$

- Kurtosis:$ M_4 = \left(3\sigma^4 + 6\lambda \mu_j^2 \sigma_j^2 + \lambda \mu_j^4 + 3\lambda \sigma_j^4\right) \Delta t^2.$

My plan is to estimate lambda via a fix threshold separating jumps from regular returns.

Are there any potential sources of error?

Literature regarding the MoM in the context of the MJD model is also very welcome!

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