European Digital Option Black-Scholes

This question focuses on pricing a European digital option in the Black-Scholes framework, where the payoff is a fixed amount if the underlying finishes above a strike and zero otherwise. The candidate is asked to translate this simple, all-or-nothing payoff into a present value under the risk-neutral measure, and then to relate the resulting expression to the familiar Black-Scholes call formula. It probes whether the candidate can see digital options as closely related to vanilla options, rather than as an entirely different object, and how the binary payoff structure connects to standard models of asset dynamics.

On the technical side, the problem leans on risk-neutral valuation, lognormal asset price dynamics, and the cumulative normal distribution that appears in Black-Scholes. It also touches on the link between digital payoffs and derivatives of option prices with respect to the strike, a common trick in derivatives pricing. An interviewer is looking for clean identification of the relevant risk-neutral probability, correct discounting, and an articulate explanation of the structural connection between digital and vanilla calls.

What it tests

Digital options are priced by recognizing that their payoffs are contingent on a binary event: whether the underlying asset exceeds a threshold at expiration. Under risk-neutral valuation, the present value of the fixed payout is weighted by the risk-neutral probability that the event occurs. This probability is given by $N(d_2)$ in the Black-Scholes framework, reflecting the likelihood that the asset price will exceed the strike under the risk-neutral measure. The core structure is that pricing reduces to multiplying the discounted payout by this probability, regardless of the specific numbers involved. This principle generalizes to any contingent claim where the payout is a fixed amount conditional on a simple event.