# Why are put and call options worth the same despite that put has no upside whereas call has unlimited upsides?

The following is an interview question.

All Black-Scholes assumptions hold. Assume no dividends. Consider a standard European call and a standard European put on the same stock. Assume that each option has the same maturity, and is struck-at-the-money (i.e. strike equals current spot). For the sake of simplicity, assume that the interest rate is zero, Draw the payoff diagrams for each option (i.e. terminal payoff to option versus level of underlying).

This part of question is easy. Just the usual kinked payoff diagram.

However, the second part of the question throws me off.

The put has limited downside potential and no upside; the call has unlimited upside and no downside. Given the random direction of the stock price movements between now and expiration, the disparity in potential payoffs seems to suggest that the call should be worth more than the put. However, put-call parity says that this is not so. Verify the put-call parity implications and reconcile them with the seemingly disparate potential payoffs.

I have a feeling that it is due to nature of lognormal distribution as stock price follows a lognormal distribution. But I can't pinpoint this concretely.

• Yes it is the lognormality. In logmoneyness terms, strike 0 is equidistant from ATM as strike infinity is. – ilovevolatility Dec 5 '19 at 14:14

• For $1),$ is there a mathematical proof? – Idonknow Dec 5 '19 at 13:37