Option Pricing under Ornstein-uhlenbeck Stochastic Volatility

We consider the problem of option pricing under stochastic volatility models, focusing on the two processes known as exponential Ornstein-Uhlenbeck and Stein-Stein. We show they admit the same limit dynamics in the regime of low fluctuations of the volatility process, under which we derive the expressions of the characteristic function and the first four cumulants for the risk neutral probability measure. This allows us to obtain a semi-closed form for European option prices, based on Lewis’ approach. We deeply analyze the case of Plain Vanilla calls, being liquid instruments for which reliable implied volatility surfaces are available. We implement a conceptually simple two steps calibration procedure which considerably reduces the computational burden and we test it against a data set of options traded on the Milan Stock Exchange. Our results show a good agreement with the market data for all the considered models. In particular, the fitted parameters suggest the risk neutral dynamics is in a low volatility fluctuation regime, which supports the reliability of the linear approximation.