Gamma Blows Up at Expiry

This question focuses on the behavior of option Greeks near expiry, specifically how the gamma of an at-the-money European option evolves as time to maturity shrinks. The setup is a standard equity option with continuous dividends in a Black–Scholes-style framework, and the candidate must reason qualitatively about the limiting behavior rather than compute a precise number. It is a typical type of interview question for derivatives quant and trading roles, where understanding the extremal behavior of risk metrics close to expiry is critical for hedging and risk control.

The problem leans heavily on understanding the link between price, delta, and gamma, and how the option payoff's increasing "all or nothing" character near expiry affects these sensitivities. It calls for comfort with asymptotic reasoning in the Black–Scholes formula, including how normal densities and time scaling interact. An interviewer is looking for intuition about why second-order sensitivities can become very large, how this affects hedging strategies, and whether the candidate can connect the formal expression for gamma with a clear picture of the delta profile around the strike.

What it tests

The core structure governing this class of problems is the relationship between an option's gamma and the sharpness of the transition in its delta as the underlying price crosses the strike, especially as time to maturity shrinks. Gamma measures the curvature of the option price with respect to the underlying, and as expiry nears, the option's payoff becomes increasingly binary: either fully exercised or not at all. This means the delta, which is the slope, transitions from near 0 to near 1 (for calls) over an ever-narrower range of underlying prices, making the slope itself extremely steep at the strike. The mathematical reason is that the normal density in the Black-Scholes formula remains bounded, but the denominator (proportional to the square root of time) vanishes, causing gamma to diverge. This is a universal feature of European options: as time to expiry approaches zero, the at-the-money region becomes where all the action (change in delta) is concentrated, and gamma reflects this by blowing up.