Variance Swap Replication and the Impact of Low-Strike Options
Summary
The note explains the theoretical replication of a variance swap using out-of-the-money options across strikes. Its fair variance is expressed as an integral of put prices below the forward and call prices above it, weighted inversely by squared strike. Because that weight grows sharply at low strikes, the downside wing can have substantial influence on the calculated variance level.
It links this replication to delta hedging: option gamma weights squared returns, and aggregating options across strikes produces the variance exposure. In practice, markets quote options at discrete strikes, so traders typically construct a volatility surface and limit the integration range. The appropriate cutoff depends on market conditions and surface quality. Practical replication may also differ from the theoretical log-payoff replication, leaving equity index variance swaps trading at a basis to the replicating portfolio. The note offers no universal definition of a far wing or fixed probability threshold.
Key ideas
- Variance swaps can be theoretically replicated with out-of-the-money puts and calls across strikes.
- The replication weights option prices by the inverse square of strike, increasing the influence of low strikes.
- Option delta hedging connects gamma-weighted squared returns with variance exposure.
- Discrete option quotes require a fitted volatility surface and practical limits on the integration range.
- Replication imperfections can create a basis between traded variance swaps and the theoretical portfolio.
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# Smile wings and varswap pricing
# Smile wings and varswap pricing
Is it true that far wings of the volatility smile have an outsized influence on the price of a variance swap? Is there a mathematical argument demonstrating this idea? What do we generally refer as far wings (5% probability to hit the level or 1% probability etc.)? I could find no paper detailing this dependence.
## Answer by AKdemy (score 4, accepted)
https://quant.stackexchange.com/a/74354
The main concern is usually for the far wing where strikes are low.
Variance swaps have a theoretical replication. The fair variance swap strike $K_{var}$ is computed as
$$ K^2_{var} = \frac{2*e^{rT}}{T} \left[ \int_0^{F(0)} \frac{P(K)}{\boldsymbol{K^2}} dK + \int_{F(0)}^\infty \frac{C(K)}{\boldsymbol{K^2}} dK \right] $$ where $T$ is the contracts maturity, $P(K)$ and $C(K)$ are European option prices with strike $K$ and maturity $T$ and $F(0)$ is the forward price. If strikes are very low, you end up with very small squared numbers, and huge weights.
There are two documents from JP Morgan Variance Swaps and Just what you need to know about Variance Swaps that contain the formula and details, with the latter being more concise. Personally, I recommend reading Towards a Theory of Volatility Trading by Peter Carr et al.
In words: A vanilla option trader, following a delta-hedging strategy, is essentially replicating the payoff of a weighted variance swap where the daily squared returns are weighted by the option’s dollar gamma, which is highest near the strike. Taking this argument one step further, a fair variance swap can be shown to equal the integral of weighted prices of out-of-the-money options over all strikes. These weights are being inversely proportional to squared strikes, an application of the Black Scholes closed-form formula for gamma, which ensures results in constant dollar gamma as shown below.
One obvious problem here is that options markets are composed of a discrete set of option prices for a given maturity. Therefore, it is common to first compute a vol surface. Practically, it is desired to limit the integration region (strike range) to avoid issues with the weights (especially very small strikes are a concern because of the weighting with squared strikes). Where this truncation is done is probably market dependent and depends on the quality of the available vol surface.
On a side note, due to practical difficulties in replicating the actual log payout across strikes, the market for equity index Var swaps usually trades at a basis to the replicating portfolio.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.