Suppose the strike prices X₁ and X₂ satisfy X₂ > X₁, show that for European calls on a nondividend paying asset, the difference in the call values satisfies

where B(τ) is the value of a pure discount bond with par value of unity and time to maturity τ . Furthermore, deduce that

In other words, suppose the call price can be expressed as a differentiable function of the strike price, then the derivative must be nonpositive and not greater in absolute value than the price of a pure discount bond of the same maturity. Do the above results also hold for European/American calls on a dividend paying asset?

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-B(T) (X₂X₁) ≤ c(S, t; X₂) - c(S, T; X₁) ≤ 0,