Model

An Overview of Asset-Price Models

Peter J. Brockwell

Department of Statistics, Colorado State University, Fort Collins, Colorado pjbrock@stat.colostate.edu

Summary. Discrete-parameter time-series models for financial data have received, and continue to receive, a great deal of attention in the literature. Stochastic volatility models, ARCH and GARCH models and their many generalizations, designed to account for the so-called stylized features of financial time series, have been under development and refinement now for some thirty years. At the same time there has been a rapidly developing interest in continuous-time models, largely as a result of the very successful application of stochastic differential equation models to problems in finance, exemplified by the derivation of the Black-Scholes-Merton (BSM) option-pricing formula and its generalizations. In this overview we start with the BSM option-pricing model in which the asset price is represented by geometric Brownian motion. We then discuss the limitations of the model and survey the various models which have been proposed to provide more realistic representations of empirically observed asset prices. In particular, the observed non-Gaussian distributions of log returns and the appearance of sharp changes in log asset prices which are not consistent with Brownian motion paths have led to an upsurge of interest in L´vy processes and their applications to financial modelling. e

1. Introduction For approximately thirty years now, discrete-time models (including stochastic volatility, ARCH, GARCH and their many generalizations) have been developed to reflect the so-called stylized features of financial time series. These properties, which include tail heaviness, volatility clustering and serial dependence without correlation, cannot be captured with traditional linear time series models. If Sn denotes the price of a stock or other financial asset at time N}, n, n = 0, 1, 2, . . ., then the series of log returns,...