Vega of a Digital Option

This question focuses on the vega of a European digital (binary) option, where the payoff depends purely on whether the underlying finishes above a strike, not by how much. The candidate is asked to reason about how this all-or-nothing payoff responds to changes in volatility, given a standard lognormal model for the underlying and risk-neutral pricing. It highlights the contrast with plain vanilla calls and puts, whose value responds more straightforwardly to volatility. Variants of this style of question appear in derivatives and volatility trading interviews, particularly for roles involving exotics pricing or volatility products.

To answer well, a candidate needs familiarity with Black–Scholes digital pricing, the link between option values and the distribution of terminal prices, and the probabilistic meaning of the cumulative distribution function. It leans on understanding how volatility reshapes that distribution and how the strike's location relative to the current forward level changes the sensitivity. Interviewers watch for clear reasoning about probability mass moving across the strike, correct qualitative sign and shape of vega, and an ability to contrast digital vega with vanilla vega without relying on memorized formulas.

What it tests

The core structure in digital (binary) option pricing is that the value is determined by the probability that a threshold is crossed, not by how far past the threshold the underlying asset moves. This means the price is a direct function of the cumulative distribution function (CDF) of the underlying asset's terminal distribution. Volatility, in this context, spreads out the distribution: it makes extreme outcomes more likely, but also makes the central (most probable) outcome less concentrated. For options that pay only if a barrier is crossed, this means volatility can either increase or decrease the price, depending on whether the current probability of crossing is close to 0, 1, or in between. This is fundamentally different from vanilla options, where the payoff grows with the distance past the strike, so increased volatility always increases expected payout.