# Volatility swap

In finance, a volatility swap is a forward contract on the future realised volatility of a given underlying asset. Volatility swaps allow investors to trade the volatility of an asset directly, much as they would trade a price index. Its payoff at expiration is equal to

$(\sigma _{\text{realised}}-K_{\text{vol}})N_{\text{vol}}$ where:

• $\sigma _{\text{realised}}$ is the annualised realised volatility,
• $K_{\text{vol}}$ is the volatility strike, and
• $N_{\text{vol}}$ is a preagreed notional amount.

that is, the holder of a volatility swap receives $N_{\text{vol}}$ for every point by which the underlying's annualised realised volatility $\sigma _{\text{realised}}$ exceeded the delivery price of $\sigma _{\text{strike}}$ , and conversely, pays $N_{\text{vol}}$ for every point the realised volatility falls short of the strike.

The underlying is usually a financial instrument with an active or liquid options market, such as foreign exchange, stock indices, or single stocks. Unlike an investment in options, whose volatility exposure is contaminated by its price dependence, these swaps provide pure exposure to volatility alone. This is truly the case only for forward starting volatility swaps. However, once the swap has its asset fixings its mark-to-market value also depends on the current asset price. One can use these instruments to speculate on future volatility levels, to trade the spread between realized and implied volatility, or to hedge the volatility exposure of other positions or businesses.

Volatility swaps are more commonly quoted and traded than the very similar but simpler variance swaps, which can be replicated with a linear combination of options and a dynamic position in futures. The difference between the two is convexity: The payoff of a variance swap is linear with variance but convex with volatility. That means, inevitably, a static replication (a buy-and-hold strategy) of a volatility swap is impossible. However, using the variance swap ($\Sigma _{T}^{2}$ ) as a hedging instrument and targeting volatility ($\Sigma _{T}$ ), volatility can be written as a function of variance:

$\Sigma _{T}=a\Sigma _{T}^{2}+b$ and $a$ and $b$ chosen to minimise the expect expected squared deviation of the two sides:

${\text{min}}E[(\Sigma _{T}-a\Sigma _{T}^{2}-b)^{2}]$ then, if the probability of negative realised volatilities is negligible, future volatilities could be assumed to be normal with mean ${\bar {\Sigma }}$ and standard deviation $\sigma _{\Sigma }$ :

$\Sigma _{T}\sim N({\bar {\Sigma }},\sigma _{\Sigma })$ then the hedging coefficients are:

$a={\frac {1}{2{\bar {\Sigma }}+{\frac {\sigma _{\Sigma }^{2}}{\bar {\Sigma }}}}}$ $b={\frac {\bar {\Sigma }}{2+{\frac {\sigma _{\Sigma }^{2}}{{\bar {\Sigma }}^{2}}}}}$ 