Abstract
This thesis studies the accuracy of the Bakshi, Kapadia, and Madan (2003) (BKM) risk-neutral moment estimators (Chapters 2 and 3) and the Gram-Charlier model that was used in detail (Chapters 4 and 5).
In Chapters 2 and 3, the BKM estimator is examined with the Gram-Charlier based model and the Duffie, Pan, and Singleton (2000) (DPS) double-jump affine jump-diffusion model, respectively. The BKM estimator is used by many academics and practitioners and often with little consideration for its accuracy or execution. In order to bound the errors of the BKM skewness estimator to be less than 10^{-3}, the interval between strikes must be less than 0.05% of the forward price (ΔK ≤ 0.05%F) and the range of strikes should contain a quarter of the forward price to quadruple the forward price (a = 1/4 = Kmin/F = F/Kmax). If market conditions do not accommodate this directly, constant extrapolation with linear interpolation of the implied volatility curve to satisfy the strike intervals and the range of strikes can be applied to reduce errors. Although linear extrapolation with cubic spline interpolation can yield more accurate results, it is not as stable.
In Chapters 4 and 5, the Gram-Charlier density, an option pricing formula (based on the Gram-Charlier density), its derivation, and its twin, the Edgeworth density, are discussed. Chapter 4 presents existing option pricing formulas with skewness and kurtosis, an area of research that is full of errors. Chapter 5 introduces the Edgeworth density, carefully distinguishes it from the Gram-Charlier density, and develops an option pricing formula which applies to both densities with an arbitrary number of cumulants. Chapter 5 also compares the two models to the BKM skewness estimator.