Abstract
This thesis studies two related topics in quantitative finance. The first topic is the valid regions of the Gram-Charlier densities (Chapter 2 and Chapter 3), and the second topic is VXX option pricing by modelling VIX directly (Chapter 4).
In Chapter 2, we focus on the valid regions of Gram-Charlier densities with higher-order cumulants. Based on derivatives of a Gaussian density, the Gram-Charlier series is an infinite expansion. Its truncated series is often used in many fields to approximate probability density functions. Although the expansions are useful, there are constrained regions on the value of the cumulants (or moments) that admit a valid (nonnegative) probability density function. When the truncation order is low (just at the fourth-order), the truncated Gram-Charlier density may be difficult to approximate an implied probability distribution as closely as possible, especially for distributions that are not sufficiently close to a normal distribution. One might increase the order after which the series is truncated until a perfect fit is achieved. However, the series expansion is usually truncated in the existing literature until the fourth-order term because it becomes difficult to find valid regions. Chapter 2 shows how the valid region of higher cumulants can be numerically implemented by the semi-definite algorithm, which ensures that a series truncated at a cumulant of an arbitrary even order represents a valid probability density. We provide examples of two valid regions of the sixth and eighth Gram-Charlier densities (i.e., truncated at the sixth and eighth terms). Our analysis proves that valid regions can be broadened with the higher-order expansions. Furthermore, the impact of higher cumulants on the valid regions has been shown.
Chapter 3 is an extension study of Chapter 2. On the one hand, we first use the representation theorem of such polynomials as sum of squares on Gram-Charlier density to show how to develop the corresponding convex optimization problem for its valid region. On the other hand, we provide the valid skewness-kurtosis regions of Gram-Charlier densities only up to the sixteenth-order because the SDP algorithm fails to calculate these regions when the order is above that. Furthermore, we explore the valid region of the fourth-order Gram-Charlier defined on an arbitrary finite domain [-q, q] but not the field R of real numbers. Our analysis proves that the ranges of skewness and kurtosis can be broadened with the finite domains, which earn a wider application. Furthermore, the impact of the length of finite domains 2q on valid regions has been shown.
In Chapter 4, we first develop a theoretical and model-free VXX formula in terms of VIX futures in both discrete and continuous forms. The discrete form of VXX can quantify the roll yield of VXX, which can be used to explain VXX's underperformance. Using the log-normal Ornstein-Uhlenbeck diffusion model, we show how the number of rolls of VIX futures affects the VXX option pricing formula and its implied volatility. To further verify the non-flat implied volatility of VXX, the VXX option pricing formula under the log-normal Ornstein-Uhlenbeck with stochastic volatility model is also derived. Finally, we analyze their pricing performance and ability to forecast implied volatilities.