A Note on Arbitrage Asset Pricing

Abstract

Arbitrage pricing plays an important role in asset valuation. The most applications of arbitrage asset pricing theories are based on the law of one price or asymptotic arbitrage free markets. We provide some new results on arbitrage and especially the arbitrage pricing theory by distinguishing between the absence of arbitrage, the law of one price and the absence of riskless arbitrage. Then we find the implications of these conditions for arbitrage asset pricing. Since the three concepts of the absence of arbitrage imply that the linear functionals that give the mean and the cost of a portfolio are continuous, hence there exist unique portfolios that represent these functionals. We detect a positive distance between these portfolios and therefore between the functionals. Thus the law of one price and the absence of a riskless arbitrage opportunity lead to systematic mispricing if both the contingent claims and the assets are mispriced. The beta pricing literature usually makes strong assumptions to obtain exact asset pricing. This belongs to a debate over which factors have the best theoretical or empirical justification. In the light of our results it is more advisable to acknowledge that almost only approximate arbitrage asset pricing can be obtained. The introduction of risky arbitrage opportunities in the sense that there might be an arbitrage opportunity with positive probability but not with probability one requires the knowledge of the risk aversion of investors. Therefore exact asset pricing can only be obtained by equilibrium asset pricing models. Our results generalizes to other arbitrage asset pricing theories like the Black and Scholes option valuation model and even the Modigliani-Miller Theorem.