An Overview of Asset–Price Models

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) optionpricing 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 Lévy processes and their applications to financial modelling.