Long memory in option pricing : A fractional discrete-time approach

This article studies the impact of long memory on volatility modelling and option pricing. We propose a general discrete-time pricing framework based on affine multi-component volatility models that admit ARCH(ccc) representations. This not only nests a large variety of option pricing models from the literature, but also allows for the introduction of novel covariance-stationary long-memory affine GARCH models. Using an infinite sum characterization of the log-asset price's cumulant generating function, we derive semi-explicit expressions for the valuation of European-style derivatives under a general variance-dependent stochastic discount factor. Moreover, we carry out an extensive empirical analysis using returns and cross-sections of S&P 500 options over the period 1996 to 2019. Overall, we find that once the informational content from options is incorporated into the parameter estimation process, the inclusion of long-memory dynamics in volatility is beneficial for improving the out-of-sample option pricing performance. The largest improvements in the implied volatility root-mean-square errors generated by our long-memory models occur for options with maturities longer than one year, reaching 33% and 13% when compared to one- and two-component short-memory models, respectively