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.
We consider the problem of option pricing under stochastic volatility models, focusing on the linear approximation of the two processes known as exponential Ornstein-Uhlenbeck and Stein-Stein. Indeed, we show they admit the same limit dynamics in the regime of low fluctuations of the volatility process, under which we derive the exact expression of the characteristic function associated to the risk neutral probability density. Its knowledge allows us to compute option prices exploiting Lewis and Lipton formula. We analyze in detail the case of Plain Vanilla calls, being liquid instruments for which reliable implied volatility surfaces are available. We also compute the analytical expressions of the first four cumulants, being crucial to implement a simple two steps calibration procedure . It has been tested against a data set of options traded on the Milan Stock Exchange. The analysis we present reveals a good agreement with the market implied surfaces and corroborates the accuracy of the linear approximation.
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obtain a semi-closed form for European option prices , based on Lewis ' approach. We deeply analyze
compute option prices exploiting Lewis and Lipton formula. We analyze in detail
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also compute the analytical expressions of the first four cumulants, being crucial to implement a
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which considerably reduces the computational burden and we test it
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The analysis we present reveals
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