Option pricing under stochastic volatility models with latent volatility

Abstract

An important challenge regarding the pricing of derivatives is related to the latent nature of volatility. Most studies disregard the uncertain nature of volatility when pricing options; the few authors who account for it typically consider the risk-neutral posterior distribution of the latent volatility. As the latter distribution differs from its physical measure counterpart, this leads to at least two issues: (1) it generates some unwanted path dependence and (2) it oftentimes requires to simultaneously track the physical and risk-neutral distributions of the latent volatility. This article presents pricing approaches purging such a path-dependence issue. This is achieved by modifying conventional pricing approaches (e.g. the Girsanov transform) to formally recognize the uncertainty about the latent volatility during the pricing procedure. The two proposed risk-neutral measures circumventing the aforementioned undesired path-dependence feature are based on the extended Girsanov principle and the Esscher transform. We also show that such pricing approaches are feasible, and we provide numerical implementation schemes.