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option_price_bounds_summary - U NIVERSITY OF E SSEX D...

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Unformatted text preview: U NIVERSITY OF E SSEX D EPARTMENT OF E CONOMICS EC372 Economics of Bond and Derivatives Markets Bounds on Option Prices A Common Method of Proof 1. Make a proposition, A. 2. Suppose the contrary, i.e. not-A. 3. Show that not-A permits an arbitrage opportunity.1 4. Hence, not-A is incompatible with market equilibrium in frictionless markets.2 5. Therefore (frictionless) market equilibrium, implies that A holds. A Simple Example 1. Proposition: p X/R (see chapter 18 for notation). In words: the premium for a European put option is never greater than the net present value of its exercise price. 2. Suppose the contrary: p > X/R. (Equivalently, Rp > X .) 3. Strategy: write one put option for p and make a loan of the proceeds (i.e. buy a risk-free bond or put the money in the bank). At date T (expiry date for the option), the loan is worth Rp (the deposit plus interest). Also at date T , either ST X or ST < X . If ST X the option dies, unexercised, and the payoff equals Rp > X > 0. (From the hypothesis of step 2, Rp > X .) If ST < X , the option is exercised, with a loss of X − ST to the writer. The maximum loss equals X (if the underlying asset is worthless, i.e. ST = 0). By hypothesis (step 2), Rp > X . Hence, Rp − X > 0. 4. Hence, p > X/R permits an arbitrage opportunity: a zero initial outlay results in a payoff of at least Rp − X > 0 in every possible outcome. 5. Therefore frictionless market equilibrium implies that p X/R. ***** 1 Arbitrage opportunity: an investment strategy that yields risk-free payoffs with zero initial capital outlay. Formally, an arbitrage opportunity is a portfolio that requires zero initial capital, and results in a non-negative payoff in every state with a positive payoff in at least one state. 2 Frictionless markets: zero transaction costs; no institutional restrictions on trades. Market equilibrium: absence of arbitrage opportunities (i.e., the arbitrage principle holds). ...
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