Encyclopedia of Financial Models III

Conditional Expectation and Change of Measure

SVETLOZAR T. RACHEV, PhD, Dr Sci

Frey Family Foundation Chair-Professor, Department of Applied Mathematics and Statistics, Stony Brook University, and Chief Scientist, FinAnalytica

YOUNG SHIN KIM, PhD

Research Assistant Professor, School of Economics and Business Engineering, Karlsruhe Institute of Technology, Germany

MICHELE LEONARDO BIANCHI, PhD

Research Analyst, Specialized Intermediaries Supervision Department, Bank of Italy

FRANK J. FABOZZI, PhD, CFA, CPA, CFA

Professor of Finance, EDHEC Business School

Abstract: The current price of an option is obtained by the conditional expectation of the payoff function under the risk-neutral measure. The risk-neutral measure is the measure equivalent to the real market measure under which the discounted price process of the underlying stock becomes a martingale. In the Black-Scholes model, the risk-neutral measure can be obtained by the Girsanov theorem. The Esscher transform has been used to find the risk-neutral measure for the continuous Lévy process models. The general theory of the Esscher transform is applied to find the risk-neutral measure under tempered stable Lévy process models.

In this entry, we present some issues in stochastic processes. We begin by defining events of a probability space mathematically, and then discuss the concept of conditional expectation. We then explain two important notions for stochastic processes: martingale properties and Markov properties. The ...