Double-Barrier Option Pricing

This question focuses on the relationship between double-barrier knock-out options and their single-barrier counterparts in a standard diffusion-based option pricing framework. The candidate is asked to reason about how path-dependent features combine when multiple barriers are present, and whether a payoff that extinguishes upon hitting either of two barriers can be decomposed as a simple sum of payoffs that each extinguish on only one side. The setup is typical of structuring and exotics discussions in derivatives desks, where intuition about barrier interactions and survival probabilities is essential.

Answering it well relies on a clear grasp of events defined by barrier hits, including unions, intersections, and complements in probability space. It leans heavily on set-theoretic reasoning applied to paths of the underlying, plus a firm understanding of knock-out versus knock-in logic and portfolio replication arguments. Interviewers watch for candidates who can translate informal statements like "hit either barrier first" into precise event relationships, carefully distinguish payoff profiles from survival probabilities, and avoid linearity assumptions that ignore overlapping path events. Strong answers articulate these relationships cleanly and may touch on how such structures appear in practical exotics pricing and hedging.

What it tests

In barrier option pricing, the key structure is that payoffs depend not just on whether a barrier is breached, but on the first time and which barrier is breached. For single-barrier options, the price reflects the risk of breaching one specific barrier. For double-barrier options, the contract is extinguished as soon as either barrier is hit, so the probability of survival is lower than for either single-barrier case. The principle is that the intersection (both barriers not breached) is a stricter condition than the union (at least one barrier not breached), so the price of a double-barrier knock-out is less than the sum of the two single-barrier knock-outs. This is because the double-barrier option only survives if neither barrier is hit, while a portfolio of up-and-out and down-and-out options pays off unless both barriers are breached, which is a less restrictive event.