Jump and Sampling Errors in Option Replication of Variance
Summary
The document examines the limits of replicating variance exposure with a weighted strip of out-of-the-money calls and puts, a construction associated with log-contract replication and the VIX calculation. The cited explanation says the continuous static replication result relies on a pure diffusion price process and continuously sampled realized variance, equivalent to annualized quadratic variation of log prices under those assumptions.
Practical implementations depart from that ideal in three ways: the underlying may jump, variance may be sampled discretely, and the option strip is represented by a finite set of contracts. The discussion points to research on the effects of jumps and discrete sampling in a particular stochastic-volatility jump model. It does not quantify a general cutoff for when jumps are small enough to ignore or provide a jump-robust replication method; those conclusions depend on model and contract details.
Key ideas
- Continuous static replication of variance with options assumes a pure diffusion process.
- The result also assumes continuously sampled realized variance.
- Jumps, discrete sampling, and a finite option strip each introduce potential replication error.
- The cited research studies these effects within a specified stochastic-volatility jump model.
- The document supplies no universal threshold for negligible jumps or robust solution.
Tags
Full text
# Replicating Log Contract - Errors Introduced by Jumps # 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? ## Answer by Quantuple (score 3) https://quant.stackexchange.com/a/35094 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.
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.