200-Day Call Option Pricing Drill

This question presents a simple pair of European call options that differ only in their time to maturity and asks how their prices should relate. The candidate is told the price of the shorter-dated option and is asked to infer the approximate price of the longer-dated one under a standard continuous-time pricing framework. The setup is stripped of market complications such as interest rates or dividends so that the focus is entirely on how time to expiry affects the value. This style of question is common in interviews for quantitative trading and derivatives roles, where intuition for option behavior under Black-Scholes-style dynamics is essential.

To answer it, the candidate must recognize how volatility feeds into option prices through the time dimension, and recall that the relevant uncertainty scales differently from calendar time. The interviewer is looking for comfort with square-root-of-time scaling, the link between variance and standard deviation, and an ability to translate that into price intuition without detailed calculation. Clear reasoning about how additional time alters the distribution of outcomes, and thus the option's value, is more important than producing an exact numerical figure.

What it tests

The value of an option, especially an at-the-money European option with zero interest rates, is fundamentally governed by the way uncertainty (volatility) accumulates over time. In the Black-Scholes framework, this uncertainty enters through the term $\sigma \sqrt{T-t}$, reflecting that the standard deviation of log-returns grows with the square root of time. This means that, all else equal, the option's price does not scale linearly with time to maturity, but rather with its square root. The reason is that the probability distribution of the underlying asset's future price spreads out more slowly than time itself, so the incremental value of extra time diminishes as maturity increases. This scaling is a direct consequence of the properties of Brownian motion, which underlies the Black-Scholes model.