Why capped puts and uncapped calls cost the same

With zero interest rates, an at-the-money European call can pay off without limit, whereas the matching at-the-money put can pay at most the strike. Show that the two nevertheless have the same price, and explain why this is not a paradox.

Parity with $r = 0$ and $K = S_0$ gives $C - P = S_0 - K = 0$. Pricing uses the risk-neutral mean $\mathbb{E}[S_T] = S_0 = K$, so expected upside above $K$ must equal expected downside below it. The lognormal distribution is right-skewed: the call's large gains come with small probability, while moderate declines below $K$ are more likely, and the two expectations balance exactly.