Complementarity and Fixed Point Problems by M.L. Balinski, Richard W. Cottle

By M.L. Balinski, Richard W. Cottle

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Example text

A financial market is complete if and only if every contingent claim is attainable. Harrison and Pliska (1983) proved that a financial market is (arbitrage-free and) complete if and only if there exists a unique equivalent martingale measure. 8 Consequently, the following three results characterize noarbitrage pricing by martingales: 1. The market is arbitrage-free if (and only if) there exists a martingale measure. 2. The market is complete if and only if the martingale measure is unique. 3. 9 We see that a self-replicating strategy must yield the same price as the discounted claim payoff under a risk-neutral measure if and only if there is to be an absence of arbitrage.

What does it tell us about the fair (arbitrage-free) price of the option? 56) To summarize: (1) To prevent arbitrage, the fair price of the option f = f(S, t) must satisfy the Black-Scholes PDE subject to the payoff condition. (2) There exists a unique dynamic replicating strategy {(∆t, Nt), 0 ≤ t ≤ T} with the P/L matching the option’s payoff in all states of the world. The weights in the replicating portfolio are: ∆t = ∆ S −f ∂f (St , t ) and Nt = t t t ∂S At where f = f(S, t) is the fair (arbitrage-free) price of the option—a unique solution to the Black-Scholes PDE subject to the payoff condition.

61) If this relationship does not hold, then arbitrage opportunities may exist depending on transaction costs. As an example, if we assume there are zero transaction costs and C(S, t) + Xe–r τ > P(S, t) + Se–qτ, then we can sell the call short (receiving the call premium C), borrow an amount Xe–r τ, go long one put option, and purchase Se–qτ shares of the underlying security with the amount borrowed. If the call option expires in-the-money, S > X, the put expires worthless, but we give the stock to the call buyer (who exercises the call against us) and receive X, which is used to pay of the loan.

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