Interpreting Greeks Under Jump Diffusion Models
Summary
The document asks how option Greeks, especially delta, should be interpreted under Merton and Kou jump diffusion models. Its central explanation is that a continuous hedge cannot offset a discrete price jump instantaneously. A riskless-portfolio argument therefore cannot eliminate jump risk at each moment; the cited answer says the resulting pricing relation and its Greeks apply in an expectation-based sense, making them less directly useful for instantaneous risk analysis.
The discussion also notes that Greeks can be obtained by differentiating an option price formula with respect to the underlying price, including for exotic options. It points to PDE and Fourier methods as approaches used to derive option prices and Greeks across models. The explanation is brief and does not provide a derivation, quantitative example, or detailed comparison of jump models, so its claims about risk interpretation should be treated as a conceptual summary rather than a complete treatment.
Key ideas
- Discrete jumps cannot be hedged instantaneously using a continuous trading strategy.
- The cited riskless-portfolio argument leaves jump exposure that can only be handled in expectation.
- Greeks derived from the model's pricing relation inherit this expectation-based interpretation.
- Differentiating an option pricing formula can produce delta, including for exotic options.
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# Implication of the Greeks under jump diffusion model # 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? ## Answer by roym00 (score 4) https://quant.stackexchange.com/a/15597 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.
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