Recovering the probability density function of asset prices using garch as diffusion approximations

This paper uses Garch models to estimate the objective and risk-neutral density functions of financial asset prices and by comparing their shapes, recover detailed information on economic agents' attitudes toward risk. It differs from recent papers investigating analogous issues because it uses Nelson's result that Garch schemes are approximations of the kind of differential equations typically employed in finance to describe the evolution of asset prices. This feature of Garch schemes usually has been overshadowed by their well-known role as simple econometric tools providing reliable estimates of unobserved conditional variances. We show instead that the diffusion approximation property of Garch gives good results and can be extended to situations with (i) non-standard distributions for the innovations of a conditional mean equation of asset price changes and (ii) volatility concepts different from the variance. The objective PDF of the asset price is recovered from the estimation of a nonlinear Garch fitted to the historical path of the asset price. The risk-neutral PDF is extracted from cross-sections of bond option prices, after introducing a volatility risk premium function. The direct comparison of the shapes of the two PDF_s reveals the price attached by economic agents to the different states of nature. Applications are carried out with regard to the futures written on the Italian 10-year bond.