= Solution
{c}
The difference between the high-strike and low-strike put payoffs satisfies
$$
0\leq(K_1-S_T)^+-(K_2-S_T)^+\leq K_1-K_2.
$$
If its price were negative, buying the nonnegative-payoff spread would generate an arbitrage. If its price exceeded the discounted maximum payoff, selling it and buying a bond for that maximum would generate an arbitrage. Thus the sharper inequality is
$$
\boxed{0\leq P(K_1)-P(K_2)\leq
(K_1-K_2)e^{-r(T-t)}.}
$$
For the positive-rate convention of the paper this implies the printed upper bound $K_1-K_2$. If rates were allowed to be negative, the undiscounted upper bound need not hold; the discounted bound is the correct rate-independent statement. This distinction is about the stated financial assumptions, not a change to the transcribed inequality.
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