Review of Discrete and Continuous Processes in Finance: Theory and Applications

Abstract

We review the main processes used to model financial variables. We emphasize the parallel between discrete-time processes, mainly used by econometricians for risk- and portfolio-management, and their continuous-time counterparts, mainly used by mathematicians to price derivatives. We highlight the relationship of such processes with the building blocks of stochastic dynamics and statistical inference, namely the invariants. Figures and practical examples support intuition. Fully documented code illustrating these processes in practice is available for download

Keywords: invariants, random walk, Levy processes, autocorrelation, ARMA, Ornstein-Uhlenbeck, Heston, CIR, jumps, long memory, fractional integration, fractional Brownian motion, volatility clustering, GARCH, stochastic volatility, subordination, real measure, risk-neutral measure, fat tails

JEL Classification: C1, G11