# 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 ?

• Intuitively it is "what the market thinks" the volatility will be in the period ahead. Those who think volatility will be higher would go long the swap and those who think it will be lower would go short. However this is intellectually a bit sloppy because (1) "the market" is not a person (2) there may be a premium or discount between the fair strike and the expected volatility. – Alex C Mar 12 '17 at 0:01

## 2 Answers

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.

• I would very much appreciate if the downvoter of my answer provided his own answer or commented why my answer is wrong from his point of view. – zer0hedge Aug 9 '17 at 13:18

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.