Barrier Cost for IBM

This question introduces a simple barrier-style contingent claim on a single stock, framed in the cleanest possible market assumptions: zero interest rates, no dividends, and continuous trading. The payoff is triggered the first time the stock price hits a specified upper level, and the candidate is asked to determine the fair cost of that claim today. It sits squarely in the option pricing family, but with a focus on hitting times rather than standard European payoffs, and is typical of interview questions for derivatives, exotics, and quantitative research roles where intuition about barrier behavior matters.

To tackle it, a candidate must recognize the relevance of martingale methods under the risk-neutral measure and the replication of path-dependent claims using simple building blocks. The core idea is to relate the expected payoff at the hitting time to the current stock price and the barrier, exploiting the fact that with zero drift and no discounting, the stock price is a martingale. Interviewers look for fluency with risk-neutral reasoning, comfort using stopping-time arguments, and the ability to distill a seemingly path-dependent payoff into a simple, static pricing expression.

What it tests

For problems involving the first time a process (like a stock price) hits a barrier, the key is that the expected present value of a payout at the hitting time can often be replicated by a linear combination of the underlying asset and risk-free assets. When the risk-free rate is zero and there are no dividends, the expected discounted payout is proportional to the current price divided by the barrier. This arises because the process is a martingale under the risk-neutral measure, so the expected value of the stock at the hitting time equals its current value. The absence of drift and discounting means that the probability-weighted payoff is just a fraction of the current price, regardless of volatility.