# Jim Gatheral's claim on the decay of the effect of jumps on the final return distribution

I got a full answer for my question on Jim Gatheral's book The Volatility Surface. I am going to try my luck again on another question on the same book. In Section The Decay of Skew Due to Jumps on page 63-64 , he claims that some $$T^*$$ that characterizes the decay time of the effect of jumps satisfies $$-(e^{\alpha+\frac{\delta^2}2}-1)\approx \sigma\sqrt{T^*}$$ where the occurrence of jumps is Poisson and the size is lognormally distributed with mean log-jump $$\alpha$$ and standard deviation $$\delta$$, and $$\sigma$$ is the volatility of the diffusion.

Does anyone have a reference to a somewhat rigorous justification of this claim?

@Quantuple made a suggestion in the comment section. While that makes intuitively plausible sense, I have been perplexed by how one exactly and rigorous describe the separate effects on implied volatility from diffusion and jumps that he alludes to and which Gatheral talks about in the paragraph right above that equation in question. Jump adds two effects on the implied volatility at the money in the long time asymptotics. One is the size, the other is the skew.

1) Size.

Is there a simple approximate addition formula to separate out the contributions from diffusion and jump?

2) Skew

Obviously, for time dependent deterministic diffusion volatility, the only cause of skew is from jumps, as there is no contribution to the at-the-money skew from diffusion. Actually, it is the ATM skew Gatheral is concerned about in this section. Since the formula above relates only sizes of jump and diffusion, I do not see what and how this relates to the skew in the long time asymptotics.

For a stochastic volatility model like Heston with jumps, how does the skew from the stochastic volatility relate and compare to that from the jump, in the long time asymptotics?

• Hi Hans. To me this is a mere definition: Gatheral chooses to define the characteristic time scale $T^*$ as the one which makes the average size of a single jump and the standard deviation of the returns (due to the pure GBM component) commensurate. The idea is that for $T >> T*$ it is then expected that the impact of jumps will be negligible compared to the standard deviation of the returns created by the GBM component. Reciprocally, for short T, the standard deviation of the returns will be no match for even a single jump on average. – Quantuple Feb 16 '18 at 19:55
• @Quantuple: I have expanded my question in response to your comment. Please review the question above. Thank you. – Hans Feb 17 '18 at 18:48
• The general principle that is used to distinguish jumps from diffusion is that jumps have an effect $O(T)$ while diffusion has an effect $O(\sqrt{T})$ – noob2 Feb 17 '18 at 22:55
• @noob2: What are you talking about? What do you mean by "effect"? How does that relate to my question? Please be specific and precise. I am not asking for vague, hand waving pronouncements. – Hans Feb 18 '18 at 5:51
• 1. Maybe this paper and references therein can help (princeton.edu/~yacine/jumpvol.pdf). Also I recommend reading Appendix A of Chapter 10 of Bergomi's book "Stochastic Volatility Modeling". 2. The idea of Gatheral example IMO is that by introducing the characteristic time $T^*$ that way we then have that as $T >> T^*$ jumps become negligible vs the diffusion part, hence log-returns pdf tends towards a normal, hence skew decays its short term value (due to jumps only) to zero (asymptotic normal distribution). – Quantuple Feb 19 '18 at 9:07