Pricing of spread and exchange options in a rough jump–diffusion market

Abstract

Asset dynamics with rough volatility recently received a great deal of attention in finance because they are consistent with empirical observations. This article provides a detailed analysis of the impact of roughness on prices of spread and exchange options. We consider a bivariate extension of the rough Heston model with jumps and derive the joint characteristic functions of asset log-returns under the risk neutral measure and under a measure using a risky asset as numeraire. These characteristic functions are expressed in terms of solutions of fractional differential equations (FDE’s). To infer these FDE’s, we rewrite the rough model as an infinite dimensional Markov process and propose a finite dimensional approximation. Next, we show that characteristic functions of log-returns admit a representation in terms of forward differential equations. FDE’s are obtained by passing to the limit. Spread and exchange options are valued with a two or a one dimensional discrete Fourier Transform. The numerical illustration reveals that considering a rough instead of Brownian volatility does not systematically increase exchange option prices.