ATM Option Pricing in Black-Scholes

This question asks for a fast mental estimate of the price of an at-the-money European call option with a short maturity, given the stock price, implied volatility, and a zero interest-rate environment. The setting is a very standard Black-Scholes one, with parameters chosen so the arithmetic can be done in your head within a few seconds. The final part asks you to compare the call's price with that of the corresponding put, probing your grasp of call-put symmetry at the money when discounting is absent. This style of back-of-the-envelope option-pricing question is common in interviews for derivatives trading, volatility trading, and some quantitative research roles.

To answer well, a candidate has to use the key Black-Scholes scaling relationships, especially how ATM option values scale with volatility and the square root of time. It leans on recognizing that, at the money with zero rates, drift and discounting largely drop out, leaving volatility as the main driver. An interviewer is watching for comfort with mental approximation, understanding of ATM behavior, and qualitative use of put-call parity rather than mechanical formula-chasing.

What it tests

The price of an at-the-money European option under the Black-Scholes model, especially with zero interest rates, is governed by the scaling property of Brownian motion and the linearity of option value in volatility and the square root of time. The intuition is that for ATM options, the probability of finishing in or out of the money is symmetric, and the expected payoff is primarily driven by the typical magnitude of price fluctuations, which is proportional to $S \sigma \sqrt{T-t}$. This is why the price grows linearly with both the underlying's volatility and the square root of the time to maturity. The absence of interest rates removes discounting and drift, making the relationship particularly transparent and symmetric between calls and puts at the money.