  We consider closed-form approximations for European put option prices within
the Heston and GARCH diffusion stochastic volatility models 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
simplifying a number of expectations induced by the Taylor expansion. Under the
assumption of piecewise-constant parameters, we derive closed-form pricing
formulas and devise a fast calibration scheme. Furthermore, we perform a
numerical error and 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 generated
by the Taylor expansion.
