Explicit approximations of option prices via Malliavin calculus in a general stochastic volatility framework

Abstract: 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 error generated by the expansion procedure, and bound it in terms of moments of functionals of the underlying volatility process. 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.