Volatility swaps are forward contracts on future realized stock volatility. Variance swaps are similar contracts on vari- ance, the square of future volatility. Both of these instruments provide an easy way for investors to gain exposure to the future level of volatility.

Unlike a stock option, whose volatility exposure is contami- nated by its stock-price dependence, these swaps provide pure exposure to volatility alone. You can use these instruments to speculate on future volatility levels, to trade the spread between realized and implied volatility, or to hedge the vola- tility exposure of other positions or businesses.

In this report we explain the properties and the theory of both variance and volatility swaps, first from an intuitive point of view and then more rigorously. The theory of variance swaps is more straightforward. We show how a variance swap can be theoretically replicated by a hedged portfolio of standard options with suitably chosen strikes, as long as stock prices evolve without jumps. The fair value of the variance swap is the cost of the replicating portfolio. We derive analytic formu- las for theoretical fair value in the presence of realistic vola- tility skews. These formulas can be used to estimate swap values quickly as the skew changes.