### Fundamental Theorem of Asset
Pricing and Separating Hyperplane Theorem

by

**Yue-Kuen Kwok**

Department of Mathematics

Hong Kong University of Science and Technology

Clear Water Bay, Hong Kong, China

Email: maykwok@ust.hk

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.