Encyclopedia of Financial Models I

Arbitrage Pricing: Continuous-State, Continuous-Time Models

SERGIO M. FOCARDI, PhD

Partner, The Intertek Group

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

Abstract: The principle of absence of arbitrage is perhaps the most fundamental principle of finance theory. In the presence of arbitrage opportunities, there is no trade-off between risk and returns because it is possible to make unbounded risk-free gains. The principle of absence of arbitrage is fundamental for understanding asset valuation in a competitive market. Arbitrage pricing can be developed in a finite-state, discrete-time setting and a continuous-time, continuous-state setting.

In this entry, we describe arbitrage pricing in the continuous-state, continuous-time setting. There are a number of important conceptual changes in going from a discrete-state, discrete-time setting (as described in the entry “Arbitrage Pricing: Finite-State Models”) to a continuous-state, continuous-time setting. First, each state of the world has probability zero. This precludes the use of standard conditional probabilities for the definition of conditional expectation and requires the use of filtrations (rather than of information structures) to describe the propagation of information. Second, the tools of matrix algebra are inadequate; the more complex tools of calculus and stochastic calculus are required. Third, simple generalizations are rarely possible as many pathological cases appear in connection ...