Part IX — Monte Carlo Methods

Part VIII built models of time and asked what they predict, and every answer it produced was a closed form — a hitting probability, an arcsine law, a drag term, a ruin exponent. This part is for the questions where no closed form exists, which in strategy evaluation is nearly all of them: the distribution of a maximum drawdown, the odds of a five-year losing stretch, the null distribution of the best of fifty correlated variants, the tail of a book at a probability no history contains. The method is always the same — draw, evaluate, average — and what it produces is never a number but a number and an interval. The interval is the subject. It is estimated from the same draws that produced the number, which makes a simulation a procedure that grades its own homework, and every failure in this part is a version of that one structural weakness.

The dependencies run in file order with two things worth knowing. Random Number Generation and Sampling Methods supply the raw material and Monte Carlo Simulation supplies the estimator and its error bar, so those three read first and everything after assumes them; and Importance Sampling and Rejection Sampling are a pair, since they are the only two things to do with a draw from the wrong distribution — reweight it or throw it away — and they share the hypothesis \(\sup p/q<\infty\) that one degrades quietly under and the other cannot start without. Variance Reduction is independent of both, and Bootstrap Methods and Jackknife Methods are a second pair, resampling data where everything earlier resampled a law that was written down. Where this part stops is worth stating too: no chain is built whose stationary distribution is the target, so Markov chain Monte Carlo is Part XVII; the taxonomy of interval constructions is Part XI and permutation tests as tests are Part XII, with the corrections for correlated families in Part XV; no estimator is decomposed into bias and variance, which is Part XIV; nothing is priced for its own sake, which is Options Pricing, and no risk measure is derived, which is Part XVIII; and no low-discrepancy sequence is constructed anywhere.

One failure runs through the part and it has a single shape: the estimate and its stated precision come out of the same draws, so a defect large enough to corrupt one corrupts the other in the same direction, and the diagnostic that would catch it is never the one a single run computes. A generator passes a Kolmogorov–Smirnov test at \(p=0.5438\) and puts a million of its three-tuples on \(15\) planes. Arithmetic worker seeding estimates \(\mathbb{E}[U]\) without bias and prints a standard error four times too narrow. A Pareto integrand with mean exactly \(3\) is estimated as \(2.997\) while its reported precision improves fourfold and its actual precision stops improving at all. A normal proposal for a Student-\(t\) target returns half the correct tail probability with an effective sample size eighty times healthier than the unbiased estimator's. Antithetic pairing, the standard cure, raises a butterfly's variance by a factor of \(1.9\). An independent bootstrap covers a nominal \(95\%\) interval \(75.5\%\) of the time on an autocorrelated series and is exactly right on a GARCH one. A published \(p\)-value of \(0.06\) computed from \(500\) resamples lands on the far side of \(0.05\) from the truth \(12.7\%\) of the time. And a jackknife standard error for a median is correct on average and wrong by seventy percent in every individual sample, at every sample size. In each case the arithmetic is right, the estimator is consistent, and the second opinion that would have caught it costs less than the consequence of not having it.

Topics

Topic	Focus
Random Number Generation	A generator as a finite state machine whose only freedom is the seed, Hull–Dobell buying a full period and saying nothing about order, RANDU's three-tuples confined to \(15\) planes by the identity \(a^{2}\equiv6a-9\) while it passes a KS test at \(p=0.5438\), MINSTD seeded by worker id correlating \(1.00000\) with that id, and arithmetic seeding printing a standard error \(4.02\) times too narrow on an estimate that is not biased at all
Sampling Methods	The generalized inverse working from right-continuity alone with no continuity assumption, a \(500\)-atom law costing \(73.68\) comparisons by scan and \(1\) by alias table against the \(3{,}079.5\) an iid bootstrap would scan, four routes to a normal agreeing on the law and inheriting hard ceilings of \(8.5717\) and \(6.2303\) from their uniform's mantissa, and a normal matched to a \(t(2.6)\) at the \(99\%\) quantile understating the \(99.99\%\) one by a factor of four
Monte Carlo Simulation	The \(N^{-1/2}\) rate holding in every dimension against a grid rule's \(N^{-2/d}\), \(s\sqrt N\) flat at \(12.53\) across five orders of magnitude while \(10^{6}\) to \(10^{8}\) paths buy one digit, a path-wise stopping rule licensed by a first moment where the error bar needs a second, an infinite-variance integrand whose reported precision improves while its real precision does not, and a correlated variant family whose expected best Sharpe falls to \(0.238\) as its \(95\)th percentile rises to \(0.545\)
Importance Sampling	The zero-variance proposal \(q^{\ast}\propto\lvert f\rvert p\) whose normalizing constant is the answer being sought, crude Monte Carlo returning exactly zero for \(\mathbf{P}(Z>5)\) where tilting returns \(2.868747\times10^{-7}\) at a speed-up of \(617{,}364\), tilted relative error growing like \(\sqrt t\) rather than \(e^{t^{2}/2}\), a light-tailed proposal halving a tail probability while its effective sample size ranks it eighty times healthier than the unbiased estimator, and an expected shortfall that \(20{,}000\) tilted draws resolve and \(1.7\times10^{8}\) crude ones would need
Rejection Sampling	Accepted draws exact by construction with a geometric runtime of mean \(M\), a naive normal-tail envelope needing \(3.5\times10^{6}\) proposals per draw at \(t=5\) where a shaped one needs \(1.0175\) and improves as the tail deepens, an envelope too wide by twenty percent per axis costing \(1.2\) proposals in one dimension and \(8.3\times10^{7}\) in a hundred, and two million base draws buying either \(1.9\) million exact conditional draws or a normalizing constant to three digits and never both
Variance Reduction	The quadratic exchange rate turning a \(4.9\)-fold error reduction into twenty-four times the work, antithetic pairing giving a variance ratio of exactly \(1+\rho\) and so buying \(1.48\) on a call and costing \(1.9\) on a butterfly, a control variate's factor of \(1-\rho^{2}\) converting a correlation of \(0.999628\) into a saving of \(1{,}345\) paths while the naive \(\beta=1\) recovers less than half of it, and common random numbers cutting a comparison's error \(2.93\)-fold and spending independence everywhere else
Bootstrap Methods	The plug-in principle licensed by uniform convergence and by smoothness, four intervals agreeing on five years of daily data and covering between \(0.908\) and \(0.945\) on thirty-six monthly ones where the interval is \(2.31\) wide around a Sharpe of \(0.30\), an iid resample holding \(0.946\) under volatility clustering and collapsing to \(0.755\) under level autocorrelation, a resampled maximum carrying a point mass of \(0.632\) forever, and a \(p\)-value of \(0.0615\) that \(B=500\) misplaces across the threshold \(12.7\%\) of the time
Jackknife Methods	The \((n-1)/n\) inflation reverse-engineered from the mean and the method revealed as the delta method differentiated numerically, an \(O(1/n)\) bias cancelled exactly so that a thirty-six-month Sharpe's \(4.3\%\) overstatement loses \(96.2\%\) of itself, a median whose \(n\) leave-one-out values take three distinct numbers and whose standard error is right on average and off by seventy percent in every sample, and the acceleration constant worth the last eight-tenths of a point of BCa coverage by moving the interval rather than widening it