This is a commonly asked question and I have not been able to find a satisfactory answer to it. Let me first phrase it here. Suppose that interest rates are $0$ and consider an at the money put and an at the money call (both European) on the same stock (self-financing, hence no dividends) with the same expiry. The payoff of the put is bounded by the strike whereas the payoff of the call is unbounded. But the put-call parity dictates that they have the same price. How do you explain this "contradiction"?
The explanations I have found are all about how stock prices are lognormally distributed etc. This explanation cannot possibly be true since the put-call parity is model-independent. To make it a bit formal I consider the first theorem of asset pricing. Then for the put and the call we have
$$C = E^Q[\max(S_T - S_0,0)]$$ $$P = E^Q[\max(S_0 - S_T,0)]$$
Here $Q$ is the risk-neutral measure (or the $T$-terminal measure). Subtracting $P$ from $C$ we get $$C - P = E^Q[\max(S_T - S_0,0)] - E^Q[\max(S_0-S_T,0)]$$ Since expectation is linear and $S_T - S_ 0 = \max(S_T - S_0,0) - \max(S_0-S_T,0)$ we have $$C - P = E^Q[S_T - S_0] = E^Q[S_T] - S_0$$ According to the first theorem of asset pricing absence of arbitrage implies that discounted asset prices are martingales. The discount rate is $0$ in this example. Hence, $E^Q[S_T] = S_0$. Finally, $$C = P$$ So I arrived at this conclusion based on two things: there is no arbitrage and interest rates are $0$. Nowhere I directly or indirectly imposed an assumption on the distribution of $S_T$ under $Q$. So what is the explanation of this phenomenon? The only explanation I can think of is that it is not necessarily an arbitrage opportunity if two contracts, one with bounded payoff and the other unbounded payoff, have the same price. Another example is a bond and a stock that coincidentally have the same price. Bond has a bounded (fixed) payoff whereas the stock has unbounded payoff. Nothing weird about that. Likewise, nothing fishy about the put and the call in the example. Is there more to this?