Arbitrage Valuation

Session VI: Arbitrage. Risk neutral valuation relationship

Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London

The absence of arbitrage in investment decisions occurs when strategies entailing zero risk have the risk-free rate of return. Therefore zero payoffs are result of zero initial investments. This definition of arbitrage is a generalization of the law of one price (LoOP) that asserts that two assets traded in the same market under the same conditions should have the same price. Hence if two assets have different prices there exists an arbitrage opportunity by buying the asset at the lower price and selling it at the higher price. Individuals adopting this strategy make a profit out of nothing (without taking any risk). This strategy cannot persist long due to the forces driving financial markets. The absence of arbitrage itself does not provide any criteria for investors to choose their portfolios and in turn cannot determine asset prices. It simply places restrictions on asset prices in order to be consistent with market equilibrium. This assumption on market behavior just rules out the existence of prices incompatible with equilibrium market conditions. Arbitrage is defined by actions seeking secure profits in investment decisions without bearing risk. An arbitrage portfolio satisfies the following conditions.

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2 • Wealth of the initial portfolio is zero. Assets can be held in positive and negative amounts but the net value of the portfolio is zero. • It is risk-free and provides positive or at least null payoffs in every state of the nature. An arbitrage opportunity is given by the exploitation of an arbitrage portfolio. The arbitrage principle is defined by the absence of arbitrage opportunities. Thus, the return on an initial investment A under no arbitrage is either R0 (return on a risk-free asset) or positive for some states of nature and negative for others. The...