A one-cent edge, repeated a million times

Question

Every trade has an expected profit of \0.01 with a standard deviation of \1.00. Trades are independent. What is the probability you're profitable after 10{,}000 trades? After 1{,}000{,}000? What happens to the relative size of the noise?

Accepted Answer

After N independent trades, total P&L is approximately Gaussian with mean N\mu = 0.01N and standard deviation \sigma\sqrt{N} = \sqrt{N}. The probability of ending profitable is P(	ext{P\&L} 0) = \Phi\!\left(\frac{N\mu}{\sigma\sqrt{N}}ight) = \Phi\!\left(\frac{0.01\,N}{\sqrt{N}}ight) = \Phi\!\left(0.01\sqrt{N}ight). At N = 10{,}000: 0.01\sqrt{10{,}000} = 0.01 	imes 100 = 1.0, so P = \Phi(1.0) \approx 84\%. At N = 1{,}000{,}000: 0.01\sqrt{1{,}000{,}000} = 0.01 	imes 1000 = 10, so P = \Phi(10) \approx 100\% (a near-certainty). The engine is how the two quantities scale differently: expe…

A one-cent edge, repeated a million times

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