A barrier option is the simplest "path-dependent" option: it pays off like a vanilla only if the stock has (or has not) crossed a barrier level during the option's life. This single additional condition makes barrier options far more sensitive to the dynamics of the volatility surface — to the forward smile and to the SSR (skew stickiness ratio) — than vanilla options, which depend only on the terminal stock price. Two models that produce identical vanilla prices can price a barrier option very differently.
Barrier options are ubiquitous in structured products (capital-guaranteed notes, autocallables, trigger notes) and FX derivatives. Understanding their pricing and hedging is essential preparation for the exotic products topic that follows.
By the end of this topic you should be able to:
- state the in-out parity and use it to price one barrier from another;
- derive the reflection-principle pricing formula for continuous barriers under GBM;
- construct a quasi-static hedge using put-call symmetry;
- apply the Broadie-Glasserman-Kou correction for discrete monitoring;
- price barriers in a jump-diffusion model via the PIDE approach.
In this lesson
- Contents
- The big picture
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