Price limits for a call spread

Question

A bull spread is built by buying a European call with strike $K_1$ and selling a European call with strike $K_2>K_1$ , same expiry $T$ . Between what bounds must its price lie?

Short answer

$0\le C(K_1)-C(K_2)\le(K_2-K_1)e^{-rT}$ . The payoff is never negative and never exceeds $K_2-K_1$ , so by no-arbitrage the spread costs at least zero and at most the present value of the maximum payoff.

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 6.3 Option Portfolios and Exotic Options, 'Bull spread', p.158; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Options, Payoffs and No-Arbitrage Bounds

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