How Jumps Affect Long-Horizon Implied Volatility Skew
Summary
The document asks for a rigorous basis for a proposed decay-time relationship between diffusion volatility and the effect of lognormally sized Poisson jumps on the final return distribution. It focuses on a claim from a volatility text that relates the characteristic time to jump parameters and diffusion volatility, but provides no derivation or supporting reference.
The author distinguishes two possible jump contributions to long-maturity at-the-money implied volatility: a change in its level and a change in skew. They ask how these effects might be separated from diffusion, why a relation involving jump size and diffusion volatility would explain skew decay, and how jump-driven skew compares with stochastic-volatility skew in a model such as Heston with jumps. The document is a research question rather than an answer; it does not establish the claimed approximation or resolve the asymptotic comparisons.
Key ideas
- The text questions an approximate decay-time relation for jump effects on the return distribution.
- It distinguishes jump contributions to implied-volatility level from contributions to at-the-money skew.
- Deterministic diffusion volatility alone is described as not generating at-the-money skew in the setting discussed.
- The comparison between jump skew and stochastic-volatility skew at long maturities remains unanswered.
Tags
Full text
# Jim Gatheral's claim on the decay of the effect of jumps on the final return distribution
# 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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.