Vega of a double no-touch

Question

An option pays $1$ at expiry provided the spot stays inside a fixed band for the entire life of the option, and nothing otherwise. What sign does its vega have?

Short answer

Negative. The option only pays if the path never reaches either barrier; higher volatility makes the path wander further and hit a barrier sooner, so the survival probability, and hence the price, falls as volatility rises.

Adapted from Joshi, Denson & Downes, Quant Job Interview Questions and Answers, Q2.45, p.20; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Greeks and Numerical Sensitivities
Derivation — From Binomial Trees to Black-Scholes in the Continuous Limit
The Black-Scholes Formula Derived via Change of Measure

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