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The Role of Drift vs Volatility in Markets

Drift against noise
dS=μS dt⏟drift+σS dW⏟noisedS = \underbrace{\mu S\,dt}_{\text{drift}} + \underbrace{\sigma S\,dW}_{\text{noise}}

Drift grows with t and noise with the square root of t, so which term dominates is entirely a question of horizon.

Two terms, and which one matters depends entirely on the horizon.

Hold the drift and volatility still and stretch the horizon. At one year the paths are a spray with no visible trend; at thirty the drift has separated from the noise. The readout names the crossover, and it is later than most people expect.

The scaling mismatch

Over a period TT, drift contributes μT\mu T while volatility contributes σT\sigma\sqrt{T}. So

signalnoise=μTσT=μσT\frac{\text{signal}}{\text{noise}} = \frac{\mu T}{\sigma\sqrt{T}} = \frac{\mu}{\sigma}\sqrt{T}

Drift grows linearly, noise only as the square root. For small TT the square root dominates and the drift is invisible; for large TT the reverse.

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