European Digital Option Payoffs

This question focuses on pricing European digital options in a continuous-time, no-arbitrage setting, specifically the asset-or-nothing and cash-or-nothing variants. The candidate is asked to reason about contingent payoffs that depend only on whether the underlying finishes above or below a strike at maturity, not on how far it moves. The setup is the standard Black–Scholes world with a tradable underlying and a money-market account, and is common in derivatives quant and structuring interviews at banks and trading firms. The aim is to see whether the candidate can move comfortably from vanilla European calls and puts to more path-independent exotics with discontinuous payoffs.

The solution leans on risk-neutral pricing, change of measure, and familiarity with the lognormal distribution of the underlying. It requires expressing the option value as a discounted expectation under an appropriate numeraire, and then reducing that expectation to a probability involving the normal cumulative distribution function. Interviewers look for clear identification of the correct measure, correct conditioning on the terminal underlying price, and a clean linkage between digital prices and vanilla option Greeks and payoffs, plus awareness of any differentiability or hedging implications.

What it tests

Pricing digital options under the Black-Scholes framework relies on risk-neutral valuation, where the expected payoff is discounted using a probability measure that makes the discounted asset price a martingale. The key insight is that the probability of the option finishing in-the-money under this measure can be computed explicitly using the cumulative distribution function of the standard normal, due to the lognormal distribution of the underlying asset. For asset-or-nothing options, changing the numeraire to the asset itself simplifies the expectation, while for cash-or-nothing options, the risk-free asset is the natural numeraire. The structure of these payoffs means their prices are closely tied to the likelihood of crossing the strike, not the magnitude of the move, which is why the normal CDF appears directly in the pricing formulas. This approach generalizes to any option whose payoff is a simple function of whether a barrier is crossed, rather than how far past it the asset ends up.