Abstract
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.
Highlights
Modern financial markets contain a rich variety of liquidly traded vanilla and exotic contracts, contingent on a large number of underlyings
In order to compute the implied volatility surface, Black–Scholes implied volatilities were calculated for all out-of-the-money plain vanilla calls and puts, because they contain the most information about the volatility structure
This leaves 210 observations of the original low spread sample of 501 options, used for model calibration. This sample is used for estimation of the stochastic volatility inspired (SVI) and SABR parameters in order to fit all models with identical information
Summary
Modern financial markets contain a rich variety of liquidly traded vanilla and exotic contracts, contingent on a large number of underlyings. The key idea in the approach suggested here is to generate modified dynamics of the arbitrage-free pricing density by asymptotic expansion around the classical Black-Scholes dynamics of complete markets. Ait-Sahalia (2002) advanced the Gram–Charlier-series of type A, utilizing Hermite-polynomials orthogonal to the weighting function e−x2/2 to represent an unknown probability density This approach has the advantage that the expansion has a leading Gaussian term and the coefficients are proportional to the cumulants of the approximated density. It derives an asymptotic version of the Kolmogorov-backward-PDE from first principles This equation contains unknown functions to be represented in terms of an orthogonal series expansion using Hermite-polynomials, orthogonal with respect to the weighting function e−x2.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
