Why a Variance Swap Strike Represents Aggregate Implied Volatility
Summary
The document explains why market participants sometimes call a variance swap's fair strike implied volatility. It presents a replication expression in which the fair future variance is derived from prices of options across a range of strikes, with puts below a reference strike and calls above it. The option prices are weighted by inverse squared strike, linking the swap's fair variance to the cost of an option portfolio.
The explanation distinguishes this aggregate measure from Black–Scholes implied volatility quoted for a single option strike. Individual options can have different implied volatilities, for example across an equity volatility skew, whereas a variance swap reflects a weighted combination of strike-level option prices. The document connects that option strip to the market's pricing of future realized variance and notes that the relationship does not require the simple Black–Scholes model to hold. The formula assumes the relevant option prices are available and uses a reference level to divide put and call contributions; the excerpt offers conceptual explanation rather than a worked calculation or discussion of replication frictions.
Key ideas
- A fair variance swap strike can be expressed using prices of options across a range of strikes.
- The option contributions are weighted by inverse squared strike in the replication expression.
- The resulting measure aggregates information across strikes rather than describing one option's implied volatility.
- A volatility skew can make individual strike implied volatilities differ while the variance swap still has one fair strike.
- The interpretation depends on option prices and an idealized replication framework.
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# Why is the fair strike of a variance swap called implied volatility?
# Why is the fair strike of a variance swap called implied volatility?
Probably an easy question for some, but I noticed most of my co-workers call the fair strike of a variance swap implied volatility. Why is that ?
## Answer by zer0hedge (score 2)
https://quant.stackexchange.com/a/33551
As shown in Demeterfi et al. "A Guide to Volatility and Variance Swaps" article the fair value of future variance $\mathbf{K}_{var}$ is:
$$ \mathbf{K}_{var} = \frac{2}{T}\Bigg(rT-\Big(\frac{S_0}{S_T}e^{rT}-1\Big)-\log{\frac{S_*}{S_0}}+e^{rT}\int_{0}^{S_*}\frac{1}{K^2}P(K)dK + e^{rT}\int_{S_*}^{\infty}\frac{1}{K^2}C(K)dK \Bigg) \tag{29} \label{formula}$$
where $S*$ is some fixed reference price that you can think of as the approximate at-the-money forward stock level that marks the boundary between liquid puts and liquid calls, $S_0$ is underlying current spot price, $P(K)$ and $C(K)$ are current prices of put and call both with the strike $K$ respectively, $r$ is risk-free rate and $T$ is time to expiration.
> Equation (29) makes precise the intuitive notion that implied volatilities can be regarded as the market’s expectation of future realized volatilities. It provides a direct connection between the market cost of options and the strategy for capturing future realized volatility, even when there is an implied volatility skew, and the simple Black-Scholes formula is invalid.
## Answer by dm63 (score 1)
https://quant.stackexchange.com/a/41639
The language around implied volatility can be a bit loose. For example, the Black-Scholes implied volatility of an at-the-money call on a stock could be 15%. For a 10% out-of-the-money put option on the same stock, the Black-Scholes implied volatility could be 18%. The variance swap is a measure of the overall implied volatility, not connected with a particular strike price. You could think of it as a weighted average of the implied volatilities of struck options, as given by the integral formulae in @zer0hedge answer.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.