  We obtain an explicit approximation formula for European put option prices
within a general stochastic volatility model with time-dependent parameters.
Our methodology involves writing the put option price as an expectation of a
Black-Scholes formula, reparameterising the volatility process and then
performing a number of expansions. The bulk of the work is due to computing a
number of expectations induced by the expansion procedure explicitly, which we
achieve by appealing to techniques from Malliavin calculus. We obtain the
explicit representation of the form of the error generated by the expansion
procedure, and we provide sufficient ingredients in order to obtain a
meaningful bound. Under the assumption of piecewise-constant parameters, our
approximation formulas become closed-form, and moreover we are able to
establish a fast calibration scheme. Furthermore, we perform a numerical
sensitivity analysis to investigate the quality of our approximation formula in
the so-called Stochastic Verhulst model, and show that the errors are well
within the acceptable range for application purposes.
