Putting Boundaries on Call Option Value

We will use the following ideas to determine the region in which the option price must lie.

1. Definition: Portfolio A is dominant over portfolio B if on some known date in the future, the return on A will exceed the return on B for some possible states of the world, and will be at least as large as the return on B, in all other states of the world.
2. If portfolio A and B are both held in equilibrium, portfolio A must sell for a higher price than portfolio B.
3. Similarly, if two assets offer the same return in all states, then they must sell for the same price.

Boundary results:

1. C(S,X,T)>0
• This follows from the fact that exercise is a right not an obligation. Therefore, the option holder need never pay out a penny.
2. For an American call, if S>X, then C>=S-X
• If this condition is not true there is a riskless arbitrage available.
• Suppose C<S-X purchase the call for C and exercise immediately for (S-X) and receive arbitrage profits of (S-X)-C > 0 by assumption.
3. C(S,X,T)>=max[0,S-X]
• by combining 1. and 2.
4. C(S,X,T)=< S
5. C(S,X,T) > max[0, S-PV(X)]