Swap rate variance swaps

We study the hedging and valuation of generalized variance swaps defined on a forward swap interest rate. Our motivation is 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. We identify a hedging rule involving a static European contract and the gains of a dynamic position on forward 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 swap rate, and the possible stochasticity of the interest rate at which gains are reinvested. The combination of these two features leads to additional terms in the cumulative dynamic trading gains, 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 in arithmetic and geometric variance swaps and verify that the hedging error is small compared with the bid--ask spread in swaption prices.