Why the Merton Jump Model Uses the Mean Percentage Jump

Summary

The Merton jump-diffusion equation combines continuous price changes with discrete Poisson arrivals that cause stock-price jumps. Its drift includes a compensation term based on the expected percentage change at a jump, expressed as the expected jump multiplier minus one. The question asks why the subtraction is needed instead of using the expected multiplier directly.

The answer turns on how the jump variable is defined: it is a multiplier on the stock price, so a multiplier of one means the price is unchanged. The jump’s percentage move is therefore the multiplier less one. Using that expected percentage move in the drift adjustment accounts for the average impact of jumps. The document provides this core intuition but does not derive the full risk-neutral or physical-measure dynamics, nor does it discuss alternative jump specifications.

Key ideas

• In the Merton model, the jump variable is a multiplicative factor applied to the stock price.
• A jump multiplier of one represents no price change.
• The percentage return caused by a jump is the multiplier minus one, so its expectation uses that subtraction.
• The drift compensation reflects the expected percentage impact of Poisson jumps.

Tags

# Question about the stochastic differential equation in the Merton model


# Question about the stochastic differential equation in the Merton model












in the following stochastic differential equation merton model we have $$\frac{ds}{s}=(\alpha-\lambda k)dt+\sigma dW+dq$$

where $\alpha$ is the instantaneous expected return on the stock; $\sigma^2$is the instantaneous variance of the return, conditional on no arrivals of important new information (i.e., the Poisson event does not occur); $dW$ is a standard Gauss-Wiener process; $q(t)$ is the independent Poisson process ; $dq$ and $dW$ are assumed to be independent; ¸ is the mean number of arrivals per unit time;$ k=E(Y-1)$ where $(Y-1)$ is the random variable percentage change in the stock price if the Poisson event occurs; and $E$ is the expectation operator over the random variable $Y$.

now my question is why we use $E(Y-1)$ and we dont use $E(Y)$ i.e I want to know What is the purpose of -1?

## Answer by Mark Joshi (score 5, accepted)

https://quant.stackexchange.com/a/16202

if $Y=1$ the stock price doesn't change since it's a percentage change not an absolute, so we have to subtract one when drift compensating.

See my book Concepts etc for my discussion.

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