The Effect of Exercise Price

The Effect of Exercise Price

Consider two European calls 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 -

   C_e(S₀, T, X₁) or C_e(S₀, T, X₂).

To cut a long story short, the price of a European call must be at least as high as the price of an otherwise identical European call with a higher exercise price:

                                    C_e(S₀, T, X₁) ≥ C_e(S₀, T, X₂)

The above result can be proved with basic arbitrage principles. The same relationship holds for American calls. The price of an American call must be at least as high as the price of another otherwise identical American call with a higher exercise price:

                                    C_a(S₀, T, X₁) ≥ C_a(S₀, T, X₂)

It follows from this result that the difference in prices of two European calls 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 ≥ C_e(S₀, T, X₁) - C_e(S₀, T, X₂)

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

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