Stochastic volatility models are the new generation of option pricing models. These models can explain how certain pricing patterns emerged in the option markets, such as "smile" and "skew". In the world of stochastic volatility models, time-dependent parameters are indispensable to price complicated derivatives, such as exotics. This is so because exotics do not only depend on the distribution of the underlying at a particular point in time, but on its dynamics through time.

In this note, we work out closed forms solutions to effectively calibrate time dependent stochastic volatility models. We show how to adapt these to a dynamic version of the SABR model. Indeed, for a model with time-dependent parameters, calibration with numerical algorithms such as partial differential equations (PDE) and Monte Carlo can be too time-consuming.

This paper explains how to effectively calibrate stochastic volatility models leveraging on a recent work by Piterbarg which suggests the use of "effective" constant parameters such that European option prices match the model with time-dependent parameters. In this paper, we first work on a concrete example, square root model. In section 3, we describe the square root model. After solving the effective constant parameters in section 4, we give a standard result by Lewis in section 5 to treat the constant parameter stochastic volatility model. Then we compare the result with the result of the standard PDE method. In the appendix, we give an application on another stochastic model, SABR.

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