Tune Option Expiry to Volatility

This question looks at a single-stock European call option whose volatility briefly spikes before reverting, and asks how to modify the option's time to expiry so that the resulting price change matches what the volatility shock would have done. The setup forces the candidate to think in terms of "total randomness experienced before expiry" rather than just "volatility today," and to compare a short-lived volatility regime change with a permanent change in maturity. It is a style of question that appears in derivatives trading, volatility trading, and exotics or structuring interviews, where understanding the time structure of volatility is central.

Answering it well requires comfort with how Black–Scholes prices scale with volatility and time, and with expressing the effect of a volatility path via integrated variance. The candidate needs to translate a transient change in volatility into an equivalent change in effective maturity and argue why this comparison is valid. Interviewers look for clear reasoning about variance, not just volatility, and for an appreciation of approximations such as vega linearity and when they are acceptable.

What it tests

In option pricing, the key driver of uncertainty is the total variance accumulated over the option's life, which is the product of the variance per unit time ($\sigma^2$) and the time to expiry ($T-t$). This means that the price of an option is sensitive not just to the level of volatility, but to how much total variance is experienced before expiry. Any temporary change in volatility can be understood as a change in the path of accumulated variance, and its effect can be replicated by adjusting the time to expiry so that the total variance remains the same. This equivalence arises because, under the Black-Scholes framework, the option price (especially for at-the-money options) is approximately a function of $\sigma \sqrt{T-t}$, so only the product of variance and time matters for pricing, not how it is distributed.