Introductory Mathematics and Statistics for Islamic Finance, + Website

Chapter 22Brownian Motion, Risk-Neutral Processes, and the Black-Scholes Model

In this chapter we cover some basic elements and results of continuous time finance. In fact, considerable advances in finance theory have been made in continuous time. Many asset pricing, risk analysis, and rate of return models have been developed in continuous time. There is a close relationship between discrete time and continuous time analysis. Time series are stochastic processes defined on discrete time intervals. More specifically, the observations were made at fixed points in time such as at the close of the market, or end of the month, or every hour. However, modern finance has also used continuous-time stochastic processes with infinitesimal time intervals. The random variable is assumed to be continuous in time. Continuous-time stochastic processes are widely applied in finance theory. Many pricing models such as the Black-Scholes option pricing formula were developed in continuous time. In this chapter, we introduce some basic concepts of continuous-time stochastic models and show their applications in asset pricing theory.

We study a continuous stochastic process in the same way as a time series. We try to characterize the probability law of the random variable, that is, the data generating process; then, we determine the mean and variance of the process. We use the process either to predict the future values of the variable, or to study risk, and price an asset. We examine how to transform ...