Black-Scholes Price of a Binary Cash-or-Nothing Call (Part 3)

Black-Scholes pricing of binary digital options

This is a medium-difficulty option-theory question that tests whether you can apply the Black-Scholes framework to exotic derivatives. Binary (digital) options are simpler payoff structures than vanilla calls, but pricing them correctly requires understanding how the model's assumptions translate to the risk-neutral measure and probability of exercise.

The key insight is recognizing that a cash-or-nothing binary call's value depends on the probability that the option expires in-the-money, discounted at the risk-free rate. Candidates must recall the correct form of the boundary-crossing probability under geometric Brownian motion and match it to the appropriate d-term from the standard Black-Scholes derivation. The distinction between d₁ and d₂—and the role of the volatility term in each—is crucial.

• Risk-neutral pricing and the martingale property
• Digital payoff functions and their probability representation
• The d₁ and d₂ terms and their probabilistic meaning
• Discounting under the risk-free measure