# Replicating Log Contract - Errors Introduced by Jumps

In the GS Research Note about Volatility Swaps, it is shown that you can replicate a pure variance exposure (hedge) with only vanilla calls and puts, primarily thanks to the Carr-Madan formula of payoff reconstruction.

Derman et al. mention how using the log contract replication is a higher order term replication, so there are times where first and second order terms dominate (and vice versa). This is where the error comes into play.

This is the same formula used to calculate the VIX index - just in discrete form instead of continuous. What are the necessary implications of the discrete VIX calculation and this error term, and is there any method that is robust to jumps?

On page 30 of the report, it is mentioned that if the jump is "small enough" it can be considered part of the diffusion and has no impact on the variance. Can "small enough" be quantified a little further?

• This paper has an overview and comparison of recent proposed statistical methods for detecting whether there are jumps. public.econ.duke.edu/~get/browse/courses/202/… – noob2 Jul 11 '17 at 19:14
• I should be able to see if my payoff replicated correctly at expiration or not - no detection needed. Why does a jump (which is still very ambiguous) impact this payoff? – Jared Jul 12 '17 at 1:55

The static replication result, i.e. a continuous strip of adequately weighted OTM vanillas can be used to replicate a variance swap, only holds:

• under a pure diffusion assumption
• if one considers continously sampled realised variance which is the same as the annualised quadratic variation of the logarithm of the price process.

In practice these assumptions are not met due to

• presence of jumps in the underlying price process (error 1)
• discretely sampled realised variance in the specification of variance swap contracts (error 2)

Anyway, because a continuous strip of options is not practical from a pure trading perspective, it is replaced that by a finite sum (error 3). This is exactly what the VIX formula does.

It seems from your question like you are interested in both errors 1 and 3.

The paper "The Effect of Jumps and Discrete Sampling on Volatility and Variance Swaps" by Broadie and Jain discuss these, along with error 2, in the context of some particular stochastic volatility with jump model, see section 6.1 Effect of jumps on fair variance strikes. The references cited may also be useful. An electronic copy is available here for download.

• Thank you for your answer. In a practical sense - for traders - can we formulate a bound for this error or a threshold of re-sampling or jump size where the error becomes relevant? The error from finite sum $\sum \Delta K$ vs integrating over $\int dK$ does not bother me, but (1) seems worth more attention. Is this wrong - can it be safely ignored in practice? Especially being able to hedge the log contract to any specified time period, a jump still seems very ambiguous to me. – Jared Jul 12 '17 at 14:17
• And more than modeling a result (changing the diffusion) I'd like to know the impact of errors in hedging on real variance swaps pulled from real market results. – Jared Jul 12 '17 at 14:20
• From what we have just said it is clear that variance swaps are more exotic than what it seems in the first place. As a matter of fact, over the past few years, the variance swap market has almost completely emancipated itself from the vanilla options market. This is why on the sell-side cutting edge models attempt to fit both of these markets simultaneously. In practice jumps cannot be ignored, especially for equities over the short-term (cash dividend payments), so yes these make up for an important part of the final price. – Quantuple Jul 12 '17 at 16:34