Swap Rate Variance Swaps

Abstract

We study the hedging and valuation of variance swaps defined on a swap interest rate. Our motivation is the recognition of the fundamental role of variance swaps in the transfer of variance risk, and the extensive empirical evidence documenting that the variance realized by interest rates is stochastic. Working in a diffusion setting, we identify a replication rule as the difference between a static European contract and the gains of a dynamic position on interest rate swaps. Two distinguishing features arise in the context of interest rates: the nonlinear and multidimensional relationship between the values of the dynamically traded contracts and the underlying rate, and the possible stochasticity of the interest rate used for reinvesting dynamic gains. The combination of these two features leads to additional terms in the cumulative dynamic trading gains, terms which depend on realized variance and are taken into consideration in the determination of the appropriate static hedge. We characterize the static payoff function as the solution of an ordinary differential equation, and derive explicitly the associated dynamic strategy. We use daily interest rate data between 1997 and 2007 to test the effectiveness of our hedging methodology and verify that the replication error is small compared to the bid-ask spread in swaption prices.