Modelling financial time series using discrete and continuous paradigms

Abstract

The aim of this dissertation is to look in some depth, at a limited number of statistical models, to describe the fluctuation of prices of financial assets. First, we shall concentrate on the multiplicative binomial model, where, after each succeeding period, the price can either increase or decrease by certain amounts and with certain probabilities. Subsequently, we generalise the binomial tree to a trinomial model. In this case, we assume a three-jump process, either an upward, downward or a level move. These discrete-time models will be based on the principle of no-arbitrage, and hence on the existence of risk-neutral probability measures, for which the discounted asset price process is a martingale. Having modelled the stock price evolution in discrete time, we shall next move on to continuous-time models, namely the Diffusion process, which is described by a stochastic differential equation. Consequently, we will take a look at some of the basic results in stochastic analysis, in particular stochastic integration with respect to Brownian motion. The last part of this study will focus on estimating the parameters of the discrete time financial models introduced above, for two time series, namely the BOV share prices and the NASDAQ Composite index. This analysis will lead to a non-linear optimisation problem, which has as its objective function the minimisation of the residual sum of squares. Eventually, from these parameters, we shall provide an estimation for the instantaneous volatility of the underlying assets, which varies over time.