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.
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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