Why a capped put can match an uncapped call

Take a European call and a European put with the same expiry, both struck at the forward price of the stock. They have the same value, even though the call's payoff has no upper limit while the put's payoff can never exceed the strike. How can that be?

Put-call parity forces equality, and the distribution is skewed. Call minus put pays $S_T-F$, a forward struck at $F$, which is worth zero, so the two prices match. The call's large payoffs come from a long but thin right tail, while the put's payoffs, though capped at $F$, occur on a fat block of downside outcomes; under the lognormal law the expected excesses above and below $F$ balance exactly.