Multivariate higher-order moments in finance

Abstract

We derive a set of results of a statistical nature. We provide closed-form expressions to calculate the multivariate truncated moments of several distributions. The first major contribution of this research is a formula to calculate moments of an arbitrary order of the lower truncated multivariate standard normal (MVSN) distribution. This result is a generalisation of the previous research on truncated first- and second-order moments. The results on the MVSN are extended to the non-standard case for a lower truncated finite mixture of multivariate normal (FMVN) distributions. We derive the moments of the Student's t-distribution, and the lognormal (MVL) distribution. We also develop a toolbox software in MATLAB for the numerical calculations of these formulae. The applications of these results range from the financial area to general statistical theory. We derive a multi-asset option approximation for general stochastic processes, with a new methodology for valuing options of general continuous-time processes. We derive a general formula to value European options under general continuous-time processes, using an approximation of the risk-neutral density. This approximation is based on an extension to the multivariate case of the Jarrow and Rudd (1982) Univariate Generalised Edgeworth Expansion. Our expansion is called the Multivariate Generalised Edgeworth Expansion (MGEE). The general formula is an approximation using the calculated value of the options under a Wiener process, plus corrections of the value of the option based on the moments of second, third, and fourth order. Results show that a calibrated approximation provides a good fit when the differences between the moments of the risk-neutral density and the auxiliary density are small, and the uncalibrated approximation has immediate implications for risk management and hedging theory. Finally, Multivariate truncated closed-form moments are proposed to be used to price options. The set of options that can be priced with this new methodology includes new multivariate options, like multi-asset power, multi-strike forced rainbow and performance options. We use the multivariate riskneutral distribution and truncated moments theory of lognormal distributions for pricing, instead of the univariate distribution of the option contract. The results of the analytical approximations are provided.EThOS - Electronic Theses Online ServiceGBUnited Kingdo