Finance in Continuous Time: A Primer

Key Findings

This text is intended for Ph.D. students in finance and other students of continuous-time methods with an interest in finance. The theory of continuous-time methods is presented only to the extent that the practitioner can improve intuition and apply the valuation and problem-solving methods to interesting problems in finance. Technical points are explored more thoroughly in advanced texts; the reader is encouraged to study these texts listed in the bibliography. This primer contains many examples and exercises for the beginning practitioner, complete with worked solutions and citations to the finance literature. Continuous-time methods provide a powerful analytical tool for the description and solution of financial problems. These methods have been applied to the valuation of derivative securities, the valuation of cash flows, the equilibrium description of markets, optimal investment strategies, optimal financing strategies, and many other issues. in in A number of brilliant textbooks have been written to teach students of finance (broadly defined) how to use these methods; one of these textbooks should be used as a primary basis of study a continuous-time course. This primer attempts to make the continuous-time intuition and calculus more accessible to students these courses of study. The primer has a "do-it-yourself" orientation that prods the student to value financial assets himself, and not merely shrug his shoulders after he derives the differential equation for the asset value. The viewpoint of this text is quite applied; theory is presented only to the extent that theory can facilitate thoughtful and rigorous application. The text concentrates particularly on the valuation of cash flows as a basis for all financial asset valuation problems.

Abstract

We begin with the assumption that the reader either is familiar with stochastic processes in discrete time or has experience in time series analysis. We use the discrete-time processes to motivate intuition for the continuous-time processes; the reader then discovers how these processes might be applied to variables of financial interest. The properties of arithmetic Brownian motion (random walk), geometric Brownian motion (proportional random walk), and the Ornstein-Uhlenbeck processes (mean-reverting) are presented so that the reader may check the formulation of financial models for economic meaningfulness. An elementary form of the famous Itô's Lemma is presented in Section 3, along with its multivariate extensions and extension to jump processes. This section motivates the intuition behind Itô's Lemma and facilitates student understanding. The chapter ends with financial applications of Itô's Lemma and by showing how to derive some elementary asset prices. The chapter should be read in its entirety, since later sections depend critically upon developments in earlier sections. Students should attempt all problems at the end of the chapter.