Using wavelets to approximate the risk-neutral MGF for options

Options are believed to contain unique information about the risk-neutral probability density function or risk-neutral moment generating function (mgf hereafter). The wavelet approach is appealing in approximating or reconstructing time-varying functions and it provides a natural platform for dealing with the non-stationary properties of real world time-series. This paper applies the wavelet method to aproximate the risk-neutral mgf of the underlying asset from options prices. Monte Carlo simulation experiments are performed to elaborate how the risk-neutral mgf can be obtained using the wavelet method. Option prices are simulated from the Black-Scholes model. The estimation is based on a general option pricing formula derived by Ma (2006) which nests several existing pricing formulae including those derived by Naik and Lee (1990), Merton (1976) and Cox and Ross (1976). We offer a novel method for obtaining the implied risk-neutral mgf for pricing out-of-sample options and other complex or illiquid derivative claims on the underlying asset using information obtained from historical data.