American Call Option Early Exercise

This question focuses on the time value of an American call option written on a non-dividend-paying stock, viewed as a function of the underlying stock price at a fixed time before expiry. The candidate is asked to reason about how the option's price compares to its immediate exercise value across different moneyness regimes, and to translate that intuition into the qualitative shape of a curve. Because the underlying does not pay dividends, the setting is the canonical textbook case used in interviews for derivatives trading, structuring, and options quant roles.

To answer well, a candidate must draw on no-arbitrage arguments, the decomposition of option value into intrinsic and time value, and the economic trade-off between exercising now versus preserving convexity and leverage. The interviewer is looking for understanding of why early exercise is suboptimal here, recognition of where time value is largest, and a correct qualitative description of how it decays as the option moves deep in- or out-of-the-money. Clear, coherent reasoning and a consistent story tying together payoff geometry, probability of finishing in the money, and discounting are more important than artistic precision in the sketch.

What it tests

The core structure in option pricing is the decomposition of an option's value into its intrinsic value and its time value. The intrinsic value, max[S - X, 0], reflects the immediate exercise value, while the time value captures the premium for potential future favorable movements and the benefit of deferring exercise. For American calls on non-dividend stocks, early exercise is never optimal because holding the option preserves upside without sacrificing downside, so the option's value always exceeds or equals its intrinsic value. The time value is maximized when the probability of a significant move in either direction is highest—this occurs when the option is at-the-money, where uncertainty about exercise is greatest. As the option becomes deep in or out of the money, the likelihood of large gains from waiting diminishes, so the time value shrinks toward zero.