Black-Scholes Call Price Variables

This question is about visualizing how a European call option's value depends on different underlying price arguments in a Black-Scholes style setting. The candidate must interpret and sketch three distinct but related functions: the simple payoff profile at maturity against the terminal underlying price, and two pre-maturity pricing profiles, one as a function of the futures (or forward) price and one as a function of the current spot. It probes whether the candidate can translate the usual verbal and formula-based intuition about options into clean geometric pictures, and explain how those pictures change as time moves away from maturity, as well as what remains structurally the same.

The solution leans on understanding of payoff diagrams, risk-neutral valuation, and the relationship between spot, forward or futures prices, and discounting. It tests whether the candidate can distinguish piecewise-linear payoff from smooth pricing functions, and can articulate why convexity arises before maturity. Interviewers look for clear reasoning about asymptotic behavior, slope, and curvature of the price curves, recognition of the role of the strike and risk-free rate, and an ability to explain the equivalence between pricing off spot and pricing off forwards in a no-arbitrage framework.

What it tests

The core structure underlying these problems is the mapping between the option's value and the underlying asset's price, shaped by the option's payoff function and the time value of money. For European options, the payoff at maturity is a piecewise-linear function, but before maturity, the price becomes a smooth, convex function due to the probabilistic nature of future payoffs and discounting. The relationship to the spot price incorporates both the expected payoff and the discounting from risk-free rates, while the relationship to the futures price leverages the risk-neutral valuation principle, making the option price a function of the forward or futures price discounted at the risk-free rate. This structure holds because the option's value is always the present value of its expected payoff under the risk-neutral measure, which is fundamentally determined by the distribution of the underlying asset's price at maturity and the strike price.