ABSTRACT

§ 7-1. In previous chapters, we have summarized how returns on assets are computed and how assets will be priced so as to provide an expected return compatible with their inherent risks. However, these models of risk and return have little to say about fundamental supply-side issues; in particular, how the variables comprising a firm’s investment opportunity set influence the market value of the firm’s equity. Our purpose in this and the next chapter is to lay the foundations for a much fuller treatment of the impact that supply-side considerations can have on equity values as outlined in the second half (Chapters 9–12) of this book. We begin our treatment by considering the Laplace model of accumulated errors. Albert Einstein based his mathematical treatment of the Brownian motion on a limiting form of the Laplace model of accumulated errors. The Brownian motion is the foundation stone on which virtually all of modern asset pricing theory is built, and so the seminal importance of the Laplace model of accumulated errors to asset pricing theory cannot be overemphasized. We then move on to demonstrate how the Laplace model of accumulated errors can be generalized by merely changing the probabilities that Laplace attributed to the positive and negative errors of his model. In particular, we formulate the probabilities associated with the Ornstein–Uhlenbeck process, which leads to one of the most widely employed stochastic differential equations in asset pricing theory. The Ornstein–Uhlenbeck process is then used to illustrate the application of the Fokker–Planck equation. The Fokker–Planck equation allows one to determine the distributional properties of a variable directly from the stochastic differential equation through which it evolves, even when there is no closed-form solution for the underlying stochastic differential equation. The final sections of this chapter deal with the problem of determining the properties of functions of stochastic variables. The market value of a firm’s equity hinges, for example, on the firm’s profitability as well as the general outlook for the economy in which the firm operates. The firm’s profitability and the general economic outlook are both stochastic variables, and so the market value of the firm’s equity will be a function of these and many other underlying stochastic variables. Hence, in these final sections of the chapter we introduce a procedure, named for Kiyoshi Itô, which enables one to determine the distributional properties of a stochastic variable that is itself a function of a more primitive set of underlying stochastic variables.