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Random Walks and Brownian Motion

Random walk

Random walk
Xt+1=Xt+εtX_{t+1} = X_t + \varepsilon_t

Each step is the last value plus an independent shock, so the best forecast of tomorrow is today.

Independent steps with no drift. The discrete prototype of an unpredictable price, and the formal content of weak-form market efficiency: the current level contains everything the history had to say.

Set the drift to zero and the paths still wander a long way from where they started. A random walk has no tendency to come back, and mistaking that wandering for a trend is the most common error in reading a price chart.

Brownian motion

The continuous-time limit, defined by four properties:

  • W0=0W_0 = 0
  • Wt−Ws∼N(0,t−s)W_t - W_s \sim \mathcal{N}(0, t-s)
  • Independent increments
  • Continuous paths

The second property is the one to internalise: variance grows linearly with time, so standard deviation grows with t\sqrt{t}. Every σT\sigma\sqrt{T} in finance traces back here.

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