Covariance and Correlation Swaps in Energy Markets

Abstract

Many valuations of variance and volatility derivatives already exist, especially in discrete time, but continuous-time valuations of covariance and correlation swaps do not currently exist for assets following the Heston stochastic volatility model. Energy commodity markets contain some of the broadest and most liquid options markets. Thus, the development of covariance and correlation swap valuation will be of significant use to investors looking to hedge covariance or correlation. In this paper, we derive approximations for an arbitrage-free valuation of natural gas and crude oil covariance and correlation swaps in the Heston model using a continuous-time regime. The approximations are obtained through the use of successive Talyor approximations on otherwise intractable terms. We find that the first order approximations of covariance and correlation swap fair strikes are reasonably effective. When the approximations are taken to the second order, a significant problem with the ML-ARCH approximation of the GARCH(1,1) process creates a large error in the valuation and its error bounds. Leveraging the properties of the Lagrange error bound, we refine our approximation to avoid the use of a frequently miscalibrated parameter, leading to a better approximation of the fair strikes of the two swaps.