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Stochastic Calculus / Brownian Motion: Construction and Properties

The Strong Markov Property and the Reflection Principle

A down-and-out call option pays (ST−K)+(S_T - K)^+ if the stock price never falls below a barrier HH during [0,T][0, T]. The price depends on the running minimum min⁡s≤TSs\min_{s \le T} S_s — a path-dependent quantity involving uncountably many times. Yet it can be priced analytically. The key is the reflection principle: for Brownian motion, the distribution of the running maximum max⁡s≤tWs\max_{s \le t} W_s is directly computable from the distribution of the endpoint WtW_t.
But why does the reflection argument work? Because at the first-passage time τb\tau_b (the first time WW hits level bb), Brownian motion "starts afresh" — the future evolution after τb\tau_b is an independent copy of BM. This is the strong Markov property: the restart property holds not just at deterministic times but at stopping times. The ordinary Markov property (Lesson 3) is not enough — stopping times are random, and a Markov process can fail to restart at them.
Learning objectives. By the end of this note you should be able to:
  • State the distinction between the Markov property (restart at fixed times) and the strong Markov property (restart at stopping times).
  • Prove the reflection principle: P(τb≤t)=2P(Wt≥b)\mathbb{P}(\tau_b \le t) = 2\mathbb{P}(W_t \ge b) for b>0b > 0.
  • Derive the CDF and density of the first-passage time τb\tau_b and the running maximum Mt=max⁡s≤tWsM_t = \max_{s \le t} W_s.
  • Apply the reflection principle to price barrier options in the Black-Scholes model.
  • State the strong Markov property for Brownian motion as a theorem and give its financial interpretation.

In this lesson

  1. The Ordinary vs. Strong Markov Property
  2. The First-Passage Time
  3. The Reflection Principle (Désiré André)
  4. Distributions of the Running Maximum and First-Passage Time
  5. Application: Barrier Options in the Black-Scholes Model

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