Application of the GramCharlier Approximation （GramCharlier近似的应用）.pdfVIP

Application of the Gram-Charlier Approximation for Option Valuation Sorin R. Straja, Ph.D., FRM Montgomery Investment Technology, Inc. 2 Radnor Corporation Center Suite 121 Radnor, PA 19087 Phone: 610.688.8111 Email: sorins@ The Hermite polynomials form the basis of a Hilbert space and may be used to get an expansion of the probability density function. Usually, this series is called Gram-Charlier. For practical purposes, only the first few terms of this expansion are taken into consideration. The resulting truncated series may be viewed as the normal probability density function multiplied by a polynomial that accounts for the effects of departure from normality. The Gram-Charlier series uses the moments of the real distribution. The Edgeworth series is similar to Gram-Charlier but uses cumulants instead of moments. Although the series are equivalent, for computational purposes the Gram-Charlier series seems to perform better than the Edgeworth series (Johnson et al., 1994). This approach was introduced in financial economics by Jarrow and Rudd (1982), and it has been applied by Madan and Milne (1994), Longstaff ( 1995), Abken et al. (1996a ; 1996b), Brenner and Eom (1997), Knight and Satchell (1997), Backus et al. (1997), Corrado and Su (1997). Non-normal skewness and kurtosis in option-implied distributions are found to contribute significantly to the phenomenon of volatility smile. For the SP 500 Index, historical rates of return have a mean in the ra