Pricing Digital Call Options

This interview question focuses on pricing a European cash-or-nothing digital call option on an underlying equity modeled under the standard continuous-time diffusion framework. The candidate is asked to express the value of a payoff that depends only on whether the underlying crosses a level at expiry, and not on the magnitude of that move, within the usual arbitrage-free pricing setting. In many banks' equity derivatives or exotics interviews, this style of question probes whether the candidate can adapt vanilla option intuition to a payoff with a discontinuity and understand how that affects valuation and trading behavior.

To answer well, you need a solid grasp of risk-neutral valuation, the link between option prices and probabilities, and how the familiar closed-form machinery changes when the payoff becomes digital. The discussion of hedging pushes you toward differentiating prices with respect to the underlying, interpreting the resulting Greeks, and recognizing the pathologies near the strike and near expiry. Interviewers listen for an understanding of replication and dynamic hedging, awareness of model assumptions, and a realistic discussion of why continuous-time hedging breaks down in practice for such highly discontinuous payoffs.

What it tests

Digital options, like the cash-or-nothing call, are priced using risk-neutral probabilities because their payoffs depend solely on whether a certain event occurs (e.g., the underlying exceeds the strike at expiry), not on how far in-the-money the option finishes. The risk-neutral framework transforms the real-world probability measure into one where all assets earn the risk-free rate, allowing us to discount the expected payoff at this rate. For digital options, the price is the present value of the risk-neutral probability of the event, which is why the cumulative normal distribution appears in the formula. The sharp discontinuity in the payoff at the strike means the option's price is highly sensitive to the probability mass near the strike, making its price and hedge ratios behave very differently from vanilla options. This structure means that the Greeks, especially delta, can become extremely large and unstable as expiry approaches or as the underlying nears the strike, reflecting the binary nature of the payoff.