Matching Mean and Volatility in a Jump Diffusion Simulation

Summary

The document describes a simulation problem: a researcher wants to compare a trading model’s behavior on ordinary price paths and paths with jumps, while isolating the effect of increased skewness. They can match the mean of the jump process to a geometric Brownian motion without jumps, but report that volatility also changes, confounding the comparison.

The author has tried adjusting the jump process with a discount and jump sizes, but finds that this affects both the mean and volatility. The post is a question rather than a worked solution: it gives no calibration method, equations for matching moments, or empirical results. It is useful as a statement of an experimental design issue, but readers will need other material to determine whether and how jump parameters can be chosen to match mean and variance while changing skewness. The author says the series are based on historical stock prices and that option pricing is outside the intended application.

Key ideas

• The researcher wants to isolate skewness effects by comparing jump and non-jump price series.
• Matching the mean alone leaves volatility as a confounding difference.
• Adjusting the jump discount changes both mean and volatility, complicating calibration.
• The post asks for a method but provides no solution or supporting results.

Tags

# 75914


# Is it possible to calibrate Mertons Jump Diffusion Model such that it matches mean and vola from a normal process without jumps?












I'm currently playing around with Mertons version of jump diffusion processes where i'm testing the predicitions of a trading model given a time series with and without jumps to isolate the effects of an increase in skewness of the time series on the outcomes of that model. Until now, the predicitions are in line with what i expected, however, my results are somewhat diluted by the fact that i can match the mean of the original simulated time series (GBM without jumps) but not its vola. So the results of this model are driven by a change in skewness and in vola, which is bad as i want to isolate the effect of an increase in skewness alone.

I tried to substract a "discount" from the observed datapoints of the jump-process and add jumpsize for a given jump probability to get my desired skewness hoping to find a way such that both moments match. I know that if i add more discount, my vola will increase but my mean will shrink, so this is certainly not the correct way of obtaining what i'm looking for. Please note that i'm using historical stock prices as time series and have no interest in option pricing.

What is a better way of archiving what i am looking for? Please let me know in the comments if my question is not entirely clear, i will post code and some of my data here then. As always i appreciate any hints Yours Thomas

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