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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.

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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. enter image description here

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.

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