Reconciling Theta vs Call Price Spike

This question focuses on the apparent contradiction between time decay and a rising expected option value. The setup is a standard European call on an underlying asset evolving over a short horizon, and the candidate is asked to reconcile why the "fair" or model-implied price tomorrow could be higher, even though the Greek associated with the passage of time is negative. It probes whether the candidate can separate conditional expectations under different measures, and distinguish between "all else equal" sensitivities and the actual joint evolution of time and the underlying.

Answering it well requires comfort with the Greek decomposition of an option's short-term price dynamics, especially the roles of theta, delta, and gamma. The interviewer is looking for someone who understands how convexity and the underlying's drift and volatility interact with time decay, and who can clearly articulate why a local, partial-derivative notion of decay need not match the unconditional expected price change. Strong candidates will reason cleanly under the risk-neutral measure, avoid intuitive but incorrect arbitrage arguments, and explain the economics of why options can still be valuable despite negative carry.

What it tests

In option pricing, the expected change in the value of a derivative over a small time interval can be decomposed into contributions from time decay (`theta`), sensitivity to the underlying (`delta`), and convexity with respect to the underlying (`gamma`). The key insight is that while `theta` measures the instantaneous loss in value from the passage of time (holding all else constant), the expected price movement also incorporates the drift and variance of the underlying asset. The gamma term, in particular, reflects the convex payoff structure of options: even if the average move of the underlying is zero, the option benefits from volatility because the payoff grows faster in favorable moves than it shrinks in unfavorable ones. Thus, the total expected change can be positive even if time decay is negative, due to the positive contribution from convexity and drift.