A Study on Asymptotic Expansion of the Risk-Neutral Pricing Density

Modern nancial markets contain a rich variety of liquidly traded vanilla and exotic contracts, contingent
on a large number of underlying. A key requirement in such dense markets is the consistent valuation
of novel and existing derivative contracts to rule out arbitrage opportunities. A new method for
pricing contingent claims based on an asymptotic expansion of the dynamics of the pricing density is
introduced. The expansion is conducted in a preferred coordinate frame, in which the pricing density
looks stationary. The resulting asymptotic Kolmogorov-backward-equation is approximated by using
a complete set of orthogonal Hermite-polynomials. The derived model is calibrated and tested on
a collection of 1075 European-style ‘Deutscher Aktienindex’ (DAX) index options and is shown to
generate very precise option prices and a more accurate implied volatility surface than conventional
methods. Considering all advantages and drawbacks, the suggested method is very promising and
well-suited for option pricing, even in difficult markets with exceptional conditions.