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Monte Carlo That Converges

Monte Carlo problems are a fixture of quant developer assessments, usually as a pricing or probability question with a simulation attached. The code is short. What is being tested is whether you know how accurate the answer is, because a simulation without an error estimate is a number with no claim attached to it.

The square-root law

Average NN independent draws of a quantity with standard deviation σ\sigma. The standard error of the mean is

SE=σN\text{SE} = \frac{\sigma}{\sqrt{N}}

Everything about Monte Carlo follows from the square root in that denominator. One more digit of accuracy costs a hundred times the paths. Doubling the paths reduces the error by 29 per cent, not by half. A million paths on a payoff with a standard deviation of 10 gives a standard error of 0.01, which is fine for a price quoted in cents and useless for one quoted in basis points.

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