Stochastic Processes and Models

Abstract

Stochastic processes describe the time series behavior of random variables. Asset prices and financial concepts such as capital, debt, leverage, liquidity, dividends, profits, losses, and numerous others are random variables. If one is interested in their behavior over time, these variables are described as stochastic processes. The simplest stochastic process is the random walk, where the variable next period is the value of the variable currently plus a random shock. Much before economists such as Black, Scholes (1973) and Merton (1973) formalized the idea of a continuous random walk in finance, the nineteenth-century British botanist Robert Brown had discovered this stochastic process by observing the random movement of pollen particles in water and suggested a mathematical description of this motion. When a financial analyst begins with certain stochastic processes and then derives more complex relationships that are themselves stochastic processes, the branch of mathematics that describes this field is called stochastic calculus. For example, if an asset’s price is described by a stochastic process and one is interested in the value of a call option on this asset, the mathematical calculations that need to be performed belong to the field of stochastic calculus. Stochastic calculus is a relatively new branch of mathematics that deals more specifically with the calculus of functions of random processes and which was largely based on the work of Norbert Wiener (1923) and Kiyosi Ito (1951). It was motivated by engineering problems of signaling and noise. Since signaling in finance often is associated with fundamentals or trends, while noise represents randomness, the mathematical methods developed in electrical engineering found applicability in finance. Today, the ideas of stochastic calculus have become the traders’ jargon in London and New York after their successful application in option pricing during the late 1970s. The purpose of this chapter is to present some of the foundational models and properties of these probabilistic methods in order to provide the reader with