Exercise: Hitting Times and the Reflection Principle for Random Walks

Prerequisites: Stopping Times, Random Walk, Brownian Motion

Problem

1. Sketch the proof. The idea is: any path that hits level $a$ and ends at $b < a$ can be "reflected" after its first hit of $a$ to produce a new path of the same probability ending at $2a - b$.

2. Use the reflection principle to derive $\mathbb{P}(M_n \ge a) = \mathbb{P}(S_n \ge a) + \mathbb{P}(S_n > a)$. (This is the discrete-time analogue of $\mathbb{P}(M_t \ge a) = 2\mathbb{P}(W_t \ge a)$ for Brownian motion.)

3. For a barrier option with knock-out level $a > 0$ and maturity $n$, the option survives iff the underlying path never crosses $a$. Using the reflection principle, write an expression for the probability that the walk never hits $a$ by time $n$.

4. Monte Carlo check. Simulate $N = 100{,}000$ paths of a 100-step simple random walk and empirically estimate $\mathbb{P}(M_{100} \ge 10)$. Compare to the reflection-principle formula.

Hint

For the reflection principle, the first hitting time $\tau_a$ is the stopping time at which the mirroring happens: reflect the post-$\tau_a$ piece of the path about the level $a$.