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Unformatted text preview: Lecture 3 Fundamental Theorems of Asset Pricing 3.1 Arbitrage and risk neutral probability measures Several important concepts were illustrated in the example in Lecture 2: arbitrage; risk neutral probability measures; contingent claims such as call options; two different ways to price a contingent claim. Now begin our general studies on these topics. Lecture 3 contains two fundamental theorems of asset pricing. Theorem 3.1 concerns the equivalence between no arbitrage and existence of risk neutral probability measures; and Theorem 3.2 concerns the equivalence between market completeness and uniqueness of the risk neutral measure. We will demonstrate the valuation of a contingent claim by replicating portfolios or taking conditional expectations with respect to a risk neutral probability measure (or called an equivalent martingale measure ). An arbitrage opportunity is said to exist if there is a self-financing strategy h whose value function satisfies (a) V (0) = 0; (b) V ( T ) 0; (c) P ( V ( T ) > 0) > 0. Although a smart investor may seek and grab such a riskless way of making a profit, it would only be a transient opportunity. Once more investors and traders jump in to share the free lunch, prices of the securities would change immediately. Hence the old equilibrium would break down and be replaced by a new equilibrium, i.e. arbitrage opportunities would vanish. That is why we assume no arbitrage. It is also an implication of the efficient market hypothesis. Example 3.1 In the example in Lecture 2, suppose the constant interest rate is 8%. Then an arbitrage opportunity can be found easily. Just do nothing at t = 0 and t = 1, and short sell a number of shares of the stock (if allowed) at t = 2, deposit the proceeds in the bank account, close the short position at T = 3 (buy back the same number of shares of the stock and return them). This enables the investor to make a net profit at T = 3. (Convince yourself this strategy is self-financing and creates an arbitrage.) Example 3.2 Let the interest rate equal 7%. The situation is similar to but slightly more interesting than Example 3.1. Try to find an arbitrage strategy....
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This note was uploaded on 11/17/2011 for the course STOR 890 taught by Professor Staff during the Spring '08 term at UNC.
- Spring '08