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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. notA.
3. Show that notA permits an arbitrage opportunity.1
4. Hence, notA 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
riskfree 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 riskfree payoffs with zero initial capital outlay.
Formally, an arbitrage opportunity is a portfolio that requires zero initial capital, and results in a nonnegative
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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 Spring '12
 R.E.Bailey
 Economics

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