Investigations in financial time series, model selection, option pricing, and density estimation

by Watteel-Sprague, Regina N.

The focus of this dissertation is *O-fold. The first objective is to establish a means of pricing European cal1 and put options. To further this objective' the characteristics of financial time series are examined and current models for price processes are reviewed. Such models include the random walk model, geometric Brownian motion, the autoregressive conditional heteroskedasticity (ARCH) family of models, and the stochastic volatility (SV) models. Although there is a standard option pricing formula, known as the Black-Scholes valuation equation, for prices which follow geometric Brownian motion, there is no standard approach for valuing options for returns which follow an ARCH formulation, the SV framework, or some other type of non-linear structure. Since most price processes follow a non-Iinear path other than geornetric Brownian motion, suitable valuation methods are needed. A risk neutral valuation procedure for ARCH-type processes is proposed along with historical and asymptotic valuation methodology for general non-linear pro- cesses. These pricing procedures are implemented after a specific model, such as an ARCK-type process or an SV model, has been fit. The sensitivity of option prices to the parameter specifications of the SV(1) and various ARCH models is also ex- amined. The parametric option valuation discussion concludes with a cornparison of the different pricing techniques proposed. In addition to the parametric valuation methods discussed herein, a nonparametric Markovian bootstrap technique based on the nonparametric Markovian resampling procedure developed by Paparoditis & Politis (1999) is proposed for pricing ARCH-type processes of finite order. This procedure circumvents the need for model selection and parameter fitting, assuming only a finite ARCH-type structure for the returns process. Distributional assumptions about the returns process plays a key role in pricing options. A review of current literature quickly reveals the role of the infinitely divis- ible family of distributions (i-d-d. 's) for modelling data, including financial returns processes. Associated with each member of this family is a Kolmogorov canonical measure- The estimation of this canonical measure for i.d.d-'s is the second objective of this thesis. A nonpararnetric estirnate for the Kolmogorov canonical measure is suggested based on the empirical characteristic function (e.c.E). Certain probability properties of this measure are investigated. For t in a neighbourhood of the ori- gin, the weak convergence of fi{#x(t) - #"(t)) to a Gaussian cornplex process is proven, where 4(t) and &(t) are the cumulant generating functions of a distribution function F(x) and the associated ernpincal distribution function F, (x) , respectivel. Using this result, the weak convergence of the empirical canonical measure to the true Kolmogorov canonical measure is then studied. Implernentation of the pro- posed estimation procedure is carried out via several numerical examples which, in addition to confirming the veracity of the methodology presented, also suggest the need for smoothing methods for this estimate. Key words and phrases: ARCH processes, Black-Scholes formula, bootstrap, Central Limit Theorem, EGARCH process, empirical characteristic function, FIEG-4RCH process, FIGARCH process, GARCH process, Gaussian processes, geornetric Brownian motion, goodness-of-fit, IGARCH process, infinitely divisible distributions, Kolmogorov canonical meâsure, Markov process, martingale measure, risk-neutral valuation, smoothing, stochastic volatility. To Kevzn and Emily and to my parents