The pricing formula for continuously monitored barriers under GBM rests on the reflection principle: for a Brownian motion (or drifted Brownian motion), the probability that the maximum exceeds a level is computed by "reflecting" the path at and mapping it to a pair of paths with known joint distribution. This principle gives exact analytic formulas for barrier options and — through the Carr lookback hedging argument — a dynamic hedging strategy for lookback options.
Learning objectives. By the end of this note you should be able to:
- state and apply the reflection principle for standard Brownian motion and for drifted Brownian motion;
- derive the down-and-out call formula from the reflection principle;
- define the floating-strike lookback call and write its hedging decomposition;
- price a one-touch option as a special case.
In this lesson
- 2.1 The reflection principle for Brownian motion
- 2.2 Drifted Brownian motion and the barrier formula
- 2.3 One-touch options
- 2.4 The lookback hedging argument
Before you start
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