Option pricing based on mixtures of distributions: Evidence from the Eurex index and interest rate futures options marke
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ractical applications This paper considers several alternatives to the standard Black and Scholes assumption of log-normally distributed security prices in an option pricing framework. Judging from empirical results for a comprehensive database of options transactions, a mixture of two or three log-normal kernels is found to perform well for price forecasting purposes, while the numerical implementation and calibration remain feasible. For non-path-dependent options, the technique could therefore serve as an alternative to complex process assumptions for the underlying, competing distributional shapes or other pricing approaches such as a direct fitting of smile curves.
Abstract
Derivatives Use, Trading & Regulation, Vol. 11 No. 3, 2005, pp. 213–231 䉷 Palgrave Macmillan Ltd 1747–4426/05 $30.00
The use of flexible distributional assumptions in an option pricing framework has been analysed extensively in recent years. Based on a large set of transaction data from 1999 and 2000, this paper readdresses the suitability of flexible distributions for forecasting and hedging purposes of DAX and Euro-Bund-Future options traded on the Eurex. Implied risk-neutral distributions (RND) are derived via the original Black and Scholes model, the series expansion approach of Corrado and Su and models based on mixtures
of two and three log-normal distributions. While, especially for DAX options, the more complex RND prove superior to the log-normality assumption within the Black–Scholes model in terms of forecasting prices, delta-neutral hedge portfolios are best derived from simple Black-Scholes hedge ratios.
INTRODUCTION In addition to the substantial body of theoretical research on the valuation of
Derivatives Use, Trading & Regulation V olume E leven Number Three 2005
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contingent claims, many empirical investigations on the pricing of derivatives have been undertaken in recent decades (see, for example, the comprehensive investigations by Bakshi et al.1 and Dumas et al.2). In spite of the large size and steady growth of worldwide turnover in derivative contracts, both at exchanges and over the counter (OTC), analyses have focused mainly on the US market. Studies of the European Exchange (Eurex), currently the world’s largest futures exchange, are rare. Apart from a few model-free analyses (see Mittnik and Rieken3,4), most studies concentrate on model-specific phenomena such as volatility smile and term structure in the standard Black–Scholes5 model (see, for example, Eberlein et al.,6 Tompkins,7 Hafner and Wallmeier,8 Brunner and Hafner,9 Fengler et al.,10 Fengler and Wang11). The literature, however, offers only a few comprehensive studies on price forecasting and hedging functionality applied to representative option databases. For the Deutsche Terminbo¨ rse, the predecessor of the Eurex, Rachev and Mittnik12 study DAX options between February 1992 and September 1995 with up to 60 days to maturity on the basis of an array of models (eg Laplace, Student and inverse Gaussian distributions). Belledin and Schlag13 investi
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