Knowing where a random walk ends up (Proposition 1 of the previous note) says nothing about whether it crossed a given level on the way. A barrier option pays off (or is extinguished) based on whether the underlying ever touches a level, not just where it ends up — so "did the path cross zero?" is exactly the kind of question path-counting techniques were built to answer directly, without solving a difference equation. This note develops the reflection principle, a purely combinatorial device for counting random-walk paths that touch a barrier, and uses it to solve the classical ballot problem and the first-return probability for the symmetric walk.
Learning objectives. By the end of this note you should be able to:
- count the number of -step paths from one point to another with a specified number of rightward steps;
- state and apply the reflection principle to count paths that touch a barrier;
- derive and apply the ballot theorem;
- compute the probability that a symmetric random walk does not revisit its starting point within steps.
In this lesson
- Counting sample paths
- The reflection principle
- The ballot theorem
- First returns to the origin
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