The Value of a Variance Swap - A Question of Interest

In this note we consider the amount by which the fair strike of an equity variance swap will differ in a deterministic interest rate model versus an equity-interest rate hybrid, when both are calibrated to the same vanilla market. We assume that the stock and the bond are Ito diffusions and derive a simple bound for this correction that depends on the interest rate volatility as well as the correlation between the interest rate and the equity. The bound generalises a well known approximation for the effect of stochastic rates that is sometimes used by practitioners. It is shown to perform well on some numerical examples, particularly for long dated contracts. Lastly, we also examine the hedging strategy when the interest rate is stochastic