Volatility Across Maturities

This question is about implied volatility for European options when the strike is fixed but maturities differ. The setup contrasts a shorter-dated option with a longer-dated one and then changes the time to expiration at which volatilities are quoted, asking how the long-dated implied volatility should look under a standard diffusion model. It forces the candidate to reconcile option maturities with the horizon over which volatility is being measured, a common sanity-check in sell-side derivatives and market-making interviews when discussing term structures of volatility and time scaling of risk.

The problem leans on basic properties of geometric Brownian motion and the relationship between volatility, variance, and time. It tests whether the candidate understands that implied volatility is an annualized standard deviation, not a variance, and how it behaves when the time horizon changes. The interviewer is watching for comfort with scaling arguments, clarity about what "implied vol" actually represents, and the ability to keep track of time units consistently without being tripped up by the presence of different maturities.

What it tests

For any asset following geometric Brownian motion, the variance of log returns over a time interval grows linearly with the length of that interval. This means that if you know the volatility (standard deviation) for a certain time period, the variance for a longer period is simply the shorter-period variance multiplied by the ratio of the times. Volatility, being the square root of variance, therefore scales with the square root of time. This relationship is fundamental to option pricing and risk management, as it allows us to translate risk and price sensitivities across different time horizons. The reason this pattern holds is that independent increments in Brownian motion accumulate variance additively, but the standard deviation (volatility) grows more slowly, reflecting the diffusive nature of random walks.