Understanding Theta for Call Options

This question focuses on the behavior of a European call option's time decay and on the mechanics of a simple option-hedging strategy. The first part asks when a call option's theta, usually negative, can actually become positive as the underlying spot price and time to maturity vary. Candidates must reason qualitatively about how remaining time, moneyness, and the discounting of the strike interact to determine whether the passage of time helps or hurts the option's value. The second part then moves to a practical setting: building a delta-neutral position by combining the call with the underlying stock and interpreting how that portfolio reacts to an immediate move in the stock price.

To answer well, a candidate must be comfortable with the Black–Scholes Greeks, especially theta and delta, and how they behave across different regions of the option's payoff diagram. The problem leans on no-arbitrage arguments, risk-neutral pricing, and the interpretation of delta-hedged positions as being primarily exposed to higher-order Greeks. Interviewers look for an understanding of why "delta-neutral" does not mean risk-free, a clear explanation of the role of gamma and theta in the hedge's profit and loss, and logically consistent reasoning about time decay and arbitrage.

What it tests

Options pricing is fundamentally governed by sensitivities (the 'Greeks') to underlying variables, which reflect how the value of an option responds to changes in time, price, volatility, and other parameters. The sign and magnitude of theta, in particular, arise from the interplay between the time value of money, the probability of exercise, and any cash flows (like dividends) associated with the underlying asset. When the benefit from holding the option (such as potential for favorable price movement or receiving dividends) outweighs the cost of waiting (time decay), theta can become positive. This is not an arbitrary occurrence, but a consequence of how the option's payoff structure and external cash flows interact over time. The principle generalizes: whenever an external factor (like dividends or deep moneyness) shifts the balance of value toward holding the option as expiration approaches, the usual negative time decay can reverse.