Abstract
This thesis consists of three essays in quantitative finance, with a particular focus on option pricing, higher-order risk-neutral distributions, characteristic functions, and option sensitivities. Essay 1 (Chapter 2) is a theoretical study of option pricing under the Gram-Charlier density. Essay 2 (Chapter 3) is a theoretical and numerical study of option Greeks under the Bakshi and Madan’s (2000) framework. Essay 3 (Chapter 4) is a theoretical and numerical study of the Charlier density and its valid regions.
Chapter 2 derives an explicit finite-sum representation for European option prices under a Gram-Charlier density with a fixed truncation order. The representation makes the exercise probabilities under the share and risk-neutral measures explicit, denoted by Π1 and Π2, respectively. It further proves that the European call delta equals Π1, establishes the equivalence between the proposed formula, existing Gram-Charlier pricing formulas, and the Bakshi and Madan’s (2000) Fourier-integral representation, and derives the mapping of cumulants across measures.
Chapter 3 develops a unified set of Fourier-integral formulas for option Greeks under the Bakshi and Madan’s (2000) option pricing framework. Its contribution is to collect these sensitivities within one characteristic-function-based representation. This chapter studies benchmark applications including the Black-Scholes model, the CEV model, and affine models. Numerical results under jump and stochastic-volatility dynamics demonstrate that the formulas are practically implementable and produce internally consistent model-dependent option sensitivities beyond simple closed-form settings.
Chapter 4 defines the Charlier density through its characteristic function and examines the valid regions of its fourth-order truncation. Unlike the Gram-Charlier and Edgeworth densities, whose parameters and cumulants must be linked through Bell-polynomial relations, the Charlier density has parameters that directly correspond to cumulants. This chapter shows that by truncating the characteristic function to include only skewness and excess kurtosis, the standard normal density, the Gram-Charlier density, and the Edgeworth density can be represented as special cases of the more general Charlier density. In order to ensure the validity (nonnegativity) of the truncated series expansions, the valid regions of the fourth-order Charlier density across approximation orders are explored and compared.