Investigations in financial time series: Model selection, option pricing, and density estimatio

The focus of this dissertation is two-fold. The first objective is to establish a means of pricing European call 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-linear path other than geometric 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 processes. These pricing procedures are implemented after a specific model, such as an ARCH-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 examined. The parametric option valuation discussion concludes with a comparison 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 divisible 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 nonparametric estimate for the Kolmogorov canonical measure is suggested based on the empirical characteristic function (e.c.f.). Certain probability properties of this measure are investigated. For t in a neighbourhood of the origin, the weak convergence of nf'' nt-f ''t to a Gaussian complex process is proven, where f (t) and fn (t) are the cumulant generating functions of a distribution function F(x) and the associated empirical distribution function Fn( x), respectively. Using this result, the weak convergence of the empirical canonical measure to the true Kolmogorov canonical measure is then studied. Implementation of the proposed 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.