The value of a variance swap – a question of interest

Pricing equity variance swaps is well understood in the case of deterministic interest rates, but particularly for longer-dated swaps the stochastic nature of the rate cannot be ignored. Here, Per Hörfelt and Olaf Torné derive the fair strike when both the underlying stock and the interest rate are general Itô processes

The floating leg of a variance swap pays the realised variance of the underlying price process. Classical results by Neuberger (1990) and Dupire (1993) show that if the underlying price process is an Itô diffusion with a deterministic short rate, a continuously sampled variance swap can be hedged with a static portfolio of calls and puts, and by keeping a fixed amount in the underlying asset. The cost of keeping a fixed amount in the underlying asset is straightforward to calculate in a model