We consider closed-form expansions for European put option prices within several stochastic volatility frameworks with time-dependent parameters. Our methodology involves writing the put option price as an expectation of a Black-Scholes formula and performing a second-order Taylor expansion around the mean of its argument. The difficulties then faced are computing a number of expectations induced by the Taylor expansion in a closed-form manner. We establish a fast calibration scheme under the assumption that the parameters are piecewise-constant. Furthermore, we perform a sensitivity analysis to investigate the quality of our approximation and show that the errors are well within the acceptable range for application purposes. Lastly, we derive bounds on the remainder term due to the Taylor expansion.
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