27 Continuous time financial models: Statistical applications of stochastic processes

Abstract

Publisher Summary This chapter focuses on the continuous time financial models. There are two principal justifications for the use of continuous time formulations in finance, the first theoretical and the second due to the richness of stochastic calculus. The convergence of discrete time asset models to a limiting form continuous time model involves the convergence of the information structure governing the evolution of asset prices and the convergence of the stochastic process describing asset prices. Nelson (1990) has shown that certain empirically rich discrete stochastic processes, such as the ARCH process, can be approximated by continuous time stochastic processes. These contributions strengthen the contention that continuous time financial models are indeed proper approximations to a wide class of discrete time formulations. While the theoretical justifications for continuous time processes in finance are strong, there is arguably a more compelling reason for their use, because of the richness of ItO's stochastic calculus. The chapter reviews some of the critical applications of continuous time modeling in finance; in particular the estimation of ItO processes for asset pricing distributions and the use of ItO processes in pricing contingent claims.