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?


1 Answer 1


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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