Brownian Motion & Hitting Times

Hard5–6 hrsBrownian MotionHitting TimesReflection PrincipleIto

Overview

Brownian motion is the mathematical model for random noise — it's continuous, nowhere differentiable, and has Gaussian increments. The key practical fact: over time horizon T, Brownian motion moves approximately ±σ√T. Stock prices in Black-Scholes are modeled as geometric Brownian motion — the log is normal. You don't need to derive Itô's lemma; you need to apply it and understand why the σ²/2 drift correction exists.

How to Recognize

→Continuous-time random process, Gaussian increments
→First passage time, maximum, or probability of hitting a level
→Options pricing context — Black-Scholes uses GBM
→'Geometric Brownian Motion' or 'log-normal' stock price model

Step-by-Step Approach

1.BM: B(0)=0, independent increments, B(t)−B(s) ~ N(0, t−s)
2.Reflection principle: P(max B(s) ≥ a for s≤t) = 2·P(B(t) ≥ a)
3.GBM: S(t) = S(0)·exp((μ−σ²/2)t + σB(t)) — log returns are normal
4.Itô's lemma: df(B,t) = f'dB + ½f''dt + f_t dt

Key Formulas

B(t) \sim N(0,\,t)

;

B(t)-B(s) \sim N(0,\,t-s)

, independent

P(\max_{s\leq t} B(s) \geq a) = 2\,\Phi\!\left(-\frac{a}{\sqrt{t}}\right) = 2P(B(t)\geq a)

03GBM:

\log S(t)/S(0) \sim N\!\left((\mu - \tfrac{1}{2}\sigma^2)t,\;\sigma^2 t\right)

04Itô:

dS = \mu S\,dt + \sigma S\,dB_t

Worked Examples

0Quick Example

P(BM hits level 2 by time 1) = 2·Φ(−2) ≈ 2·0.0228 = 4.56%.

Problem

Standard Brownian motion starts at 0. What is P(it ever reaches level a = 2 by time t = 1)?

Solution

P(max B(s) ≥ a for s ≤ t) — probability of hitting level a at or before time t.

Reflection principle: any path that reaches a before t and ends above a corresponds to a path that ends above a. Any path that reaches a and ends below a corresponds (by reflection) to a path ending above a.

Therefore: P(max ≥ a) = 2·P(B(t) ≥ a).

P(B(1) ≥ 2) = 1 − Φ(2) ≈ 1 − 0.9772 = 0.0228.

P(max B ≥ 2 by t=1) = 2·0.0228 = 4.56%.

Answer

$P(\text{hit level 2 by time 1}) = 2\Phi(-2) \approx 4.56\%$ . The reflection principle doubles the terminal probability.

Common Mistakes

!
Using B(t) ~ N(0,1) instead of N(0,t). Standard BM has variance = t, not 1.
!
Forgetting the σ²/2 term in GBM. The log-return has drift μ−σ²/2, not μ.
!
Applying the reflection principle to GBM directly — it only works for standard BM (zero drift). For BM with drift, use Girsanov's theorem.

Practice Problems

Click "Show Answer" to reveal

What is P(BM > 3 at time t=4)?

Hint: B(4) ~ N(0,4), so B(4)/2 ~ N(0,1).

Stock with μ=15%, σ=25%, S(0)=50. What is E[S(2)]?

Hint: E[S(T)] = S(0)·e^(μT).

A stock has S(0)=100, σ=30%, T=1yr, μ=5%. What is P(S < 80)?

Hint: Log return ~ N((0.05−0.045)·1, 0.09). Find P(log(S/100) < log(0.8)).

Back to Study Plan

Gambler's Ruin Martingales & Optional Stopping