Theta vs Gamma in Black-Scholes

This question focuses on the interplay between two core Greeks for options priced under the Black-Scholes model: the time sensitivity of the option's value and its curvature with respect to the underlying price. Candidates must reason about how these quantities are linked by the Black-Scholes partial differential equation and by no-arbitrage arguments, rather than by memorizing formulas. The discussion often centers on standard European options on non-dividend-paying underlyings, but the question also pushes candidates to think about how this relationship might change when the usual assumptions are relaxed or when payoffs are less vanilla.

From a technical perspective, the problem leans on understanding the Black-Scholes PDE, risk-neutral valuation, and the decomposition of an option's value into intrinsic and time value. It rewards familiarity with the static and dynamic replication of options, and how gamma risk is financed through theta. Interviewers watch for clarity in connecting the PDE terms to economic intuition, awareness of model assumptions and boundary conditions, and an ability to identify and justify scenarios in which the usual sign relationship between these Greeks may break down.

What it tests

In the Black-Scholes framework, the relationship between the time decay of an option's value (`theta`) and its convexity with respect to the underlying price (`gamma`) is governed by the structure of the Black-Scholes PDE. The key insight is that, for non-dividend-paying assets and under typical market conditions, the dominant contributors to the PDE are the `theta` and `gamma` terms, especially when the effects of `delta` and the risk-free rate approximately offset each other. This leads to an approximate inverse relationship: increasing convexity (positive `gamma`) means the option's value grows faster with volatility, but this benefit must be paid for by a faster time decay (negative `theta`). The underlying reason is that the option's ability to profit from large price moves (high `gamma`) comes at the cost of losing value over time if the price does not move.