Effective stochastic volatility: applications to ZABR-type models

There are numerous models for specifying the uncertainty of future instantaneous volatility or variance, including the Heston, SABR and ZABR models. Often it is observed that a specific stochastic volatility model is chosen not for particular dynamical features, relevant for exotic payoff structures, but instead for convenience and ease of implementation. The SABR model, with its semi-closed form approximate solution for the prices of vanilla options, is a well-known example. In this article, we consider a general approach that includes all practically relevant stochastic volatility models and introduces new variants of the ZABR model. In particular, we consider the mean-reverting ZABR and free ZABR models. We use the method of deriving an effective partial differential equation for the density. This approach leads to the known approximation formula for the SABR model and also provides expressions for arbitrage-free models. Numerical experiments illustrate our approach.