The Effect of Exercise Price

The Effect of Exercise Price     

                      

Consider two European puts that are identical in all respects except that the exercise price of one is X₁ and the other is X₂, where X₂ > X₁.

We want to know which price is greater: P_e(S₀, T, X₁) or P_e(S₀, T, X₂). To cut a long story short, the price of a European put must be at least as high as the price of a European put must be at least as high as the price of an otherwise identical European call with a lower exercise price:

   P_e(S₀, T, X₂) ≥ P_e(S₀, T, X₁).

The intuition behind why a put with a higher exercise price is more is quite simple. A put is an option to sell an asset at a fixed price. The higher the price the put holder can sell the asset, the more attractive the put.

The same relationship holds for American puts. The price of an American put must be at least as high as the price of another otherwise identical American call with a lower exercise price:

   P_a(S₀, T, X₂) ≥ P_a(S₀, T, X₁).

It follows from this result that the difference in prices of two European puts that differ only by exercise price cannot exceed the present value of the difference in their exercise prices, that is:

   (X₂-X₁)(1+r)^-T ≥ P_e(S₀, T, X₂) - P_e(S₀, T, X₁)

Also, the difference in the prices of two American puts that differ only by exercise price cannot exceed the difference in their exercise prices:

   (X₂-X₁) ≥ P_a(S₀, T, X₂) - P_a(S₀, T, X₁)