Computing the Cost of an At-the-Money Straddle

This quant interview question is about understanding how to price a basic volatility strategy using a continuous-time option pricing model. It focuses on an at-the-money payoff structure commonly used in trading desks and risk management, and it forces you to connect model inputs like volatility, maturity, and spot level to a concrete premium. In quant prep, this is a canonical way to see if you can move from theory to an actual number.

It trains your command of option pricing intuition, symmetry between calls and puts, and how volatility and time combine to drive option values. It checks whether you can translate model assumptions about price dynamics into the value of a volatility trade, and whether you are comfortable with scaling properties in diffusion models.

This matters in quant interviews because straddles are fundamental building blocks of volatility trading and hedging. Interviewers use this kind of problem to test whether you understand what drives option premia, can manipulate model inputs quickly, and can reason about risk and payoff under uncertainty. Strong performance here signals you are ready for more advanced derivatives modeling, calibration, and trading strategy design in real-world quant roles.

What it tests

For at-the-money options, especially when pricing straddles, the dominant factors are the volatility of the underlying asset, the time to expiration, and the current price. The Black-Scholes formula simplifies in this regime because the call and put prices are nearly symmetric, and their sum (the straddle price) can be approximated by a formula that highlights the square root dependence on both volatility and time. This approximation arises because, at-the-money, the expected movement of the asset (driven by volatility over time) is the main contributor to option value, while the effects of drift and discounting are secondary. The formula reflects the fact that, over short time horizons and moderate volatilities, the distribution of returns is nearly normal, and the expected payoff of a straddle is proportional to the expected absolute movement of the underlying. The $\sqrt{T}$ scaling comes from the properties of Brownian motion, which governs asset price evolution in these models.