# 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. Mar 12 '17 at 0:01

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.