# Implication of the Greeks under jump diffusion model

Consider jump diffusion model proposed by Merton and Kou.

As far as i know, most paper only dealt the valuation of option under the jump diffusion model.

As i expected, because of the incompleteness of model, implication of the Greeks is somewhat different from that of continuous model.

Am i right?

Why does not the literature on jump diffusion model derive the Greek such as delta?

For example, we have formulas for some exotic option under the jump diffusion.

then we can obtain the delta by differentiating option price with respect to the asset

price. Is it possible to conduct risk analysis using this delta formula?

In the Merton jump diffusion model, the stock price process consists of a continuous part and a discrete part (this one represents the jumps). While deriving the PDE for the riskless portfolio and imposing the riskless evolution, the discrete part can't be instantaneously hedged. In fact, you can assume that the effects of jumps can be nullified on average, but still not instantaneously: this means that the whole PDE can be valid only under expectation. Therefore, also the Greeks will be valid only under expectation since they come from that same PDE, meaning that risk analysis is not instantaneous and is considered less effective.

I didn't study Kou's model, but reading the stock price process from his publication, I see the same feature and the same argument is valid.

Anyway, I think you can find Greeks derivation in literature. If I remember well, A Guide to Quantitative Finance should have them: it derives call option prices and Greeks under many models using the PDE-Fourier method.