Fundamental Theorem of Asset Pricing and Separating Hyperplane Theorem


Yue-Kuen Kwok

Department of Mathematics
Hong Kong University of Science and Technology
Clear Water Bay,  Hong Kong, China

Abstract :  An arbitrage opportunity in the financial market is defined as a self-financing trading strategy requiring no initial investment, having no probability of negative value at expiration, and yet having some possibility of a positive terminal payoff. In a frictionless and efficient financial market, there should be no arbitrage opportunity.  The "no-arbitrage" principle is a cornerstone in asset pricing theory.  This presentation illustrates how to apply the Separating Hyperplane Theorem and other results in Linear Algebra Theory to establish the Fundamental Theorem of Asset Pricing. The Asset Pricing Theorem states that the absence of arbitrage opportunity is equivalent to the existence of a risk neutral measure where the price of a contingent claim is given by the expectation of the terminal payoff of the contingent claim under a risk neutral measure. The risk neutral measure is related to the state prices but not related to the actual probability of occurrence of various states in the financial market.

Lecture Slide