Non-Gaussian GARCH Option Pricing Models and Their Diffusion Limits

Forthcoming, European Journal of Operational Research

30 Pages Posted: 2 Nov 2013 Last revised: 19 Jun 2015

Alex Badescu

University of Calgary

Robert J. Elliott

University of Calgary - Haskayne School of Business; University of Alberta - Department of Mathematical and Statistical Sciences

Juan-Pablo Ortega

Universität Sankt Gallen; Centre National de la Recherche Scientifique (CNRS)

Date Written: October 31, 2013

Abstract

This paper investigates the weak convergence of general non-Gaussian GARCH models together with an application to the pricing of European style options determined using an extended Girsanov principle and a conditional Esscher transform as the pricing kernel candidates. Applying these changes of measure to asymmetric GARCH models sampled at increasing frequencies, we obtain two risk neutral families of processes which converge to different bivariate diffusions, which are no longer standard Hull-White stochastic volatility models. Regardless of the innovations used, the GARCH implied diffusion limit based on the Esscher transform can be obtained by applying the minimal martingale measure under the physical measure. However, we further show that for skewed GARCH driving noise, the risk neutral diffusion limit of the extended Girsanov principle exhibits a non-zero market price of volatility risk which is proportional to the market price of the equity risk, where the constant of proportionality depends on the skewness and kurtosis of the underlying distribution. Our theoretical results are further supported by numerical simulations and a calibration exercise to observed market quotes.

Keywords: finance, non-Gaussian GARCH models, extended Girsanov principle, conditional Esscher transform, bivariate diffusion limit, option pricing

JEL Classification: C22, C32, C5

Suggested Citation

Badescu, Alex and Elliott, Robert J. and Ortega, Juan-Pablo, Non-Gaussian GARCH Option Pricing Models and Their Diffusion Limits (October 31, 2013). Forthcoming, European Journal of Operational Research. Available at SSRN: https://ssrn.com/abstract=2348407 or http://dx.doi.org/10.2139/ssrn.2348407

Alex Badescu

University of Calgary ( email )

University of Calgary
Calgary, Alberta
Canada

Robert James Elliott

University of Calgary - Haskayne School of Business ( email )

2500 University Drive, NW
Calgary, Alberta T2N 1N4
Canada

University of Alberta - Department of Mathematical and Statistical Sciences ( email )

Edmonton, Alberta T6G 2G1
Canada
403-492-5811 (Phone)
403-492-6826 (Fax)

Juan-Pablo Ortega (Contact Author)

Universität Sankt Gallen ( email )

Bodanstrasse 6
St. Gallen, St. Gallen CH-9000
Switzerland

HOME PAGE: http://juan-pablo-ortega.com

Centre National de la Recherche Scientifique (CNRS) ( email )

16 route de Gray
Besançon, 25030
France

HOME PAGE: http://juan-pablo-ortega.com

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