One-Month Call Pricing with Trending Returns

This question focuses on pricing a short-dated European call when the underlying index clearly violates the simple random walk assumption. You are given return data at multiple frequencies and told that longer-horizon variance grows faster than linearly with time. The task is to think through what this implies about the return-generating process and then decide how to adjust a standard option pricing framework to accommodate trending or serially correlated returns over a one-month horizon.

The problem leans on ideas from time series analysis, scaling of variance, and the link between autocorrelation and effective volatility. It asks you to recognize when the usual Black–Scholes assumptions break down, and to rebuild the pricing input (in particular, the relevant monthly volatility and drift structure) from the data rather than from theory alone. Interviewers watch for the ability to translate statistical features of historical returns into a risk-neutral pricing setup, to handle non-iid dynamics without overfitting, and to articulate clearly how and why the option price will differ from a naïve random-walk-based valuation.

What it tests

In time series analysis for asset returns, the random walk hypothesis implies that returns are serially uncorrelated and their variances scale linearly with time. That is, for independent increments, the variance over $n$ periods should be $n$ times the single-period variance. Deviations from this scaling law signal the presence of autocorrelation—returns are either positively or negatively related across time, altering the predictability of future prices. This pattern is foundational because it determines whether models like Black-Scholes, which assume log returns are i.i.d. (independent and identically distributed), are appropriate. When autocorrelation is present, the effective volatility over longer horizons is not a simple multiple of the short-term variance, and option pricing must adjust for this to avoid systematic mispricing.