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Payoff range of a long 5 short 10 call spread

Question

A stock is at 77. You buy a call struck at 55 and sell a call struck at 1010 with the same expiry. With MM the largest and mm the smallest possible payoff at expiry, compute M2+m2M^2+m^2.

Short answer

2525. The spread pays nothing below 55, rises one-for-one between 55 and 1010, and is capped at 10−5=510-5=5 above, so M=5M=5, m=0m=0.

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