Confidence intervals for empirical convergence rates of randomised quasi-Monte Carlo, with applications to option pricing

Giacomo Case

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Abstract: Empirical comparisons of quasi-Monte Carlo rules often report fitted convergence exponents without intervals. We estimate the root mean square error from independent randomisations, fit its log slope over a declared sample-size window, and resample whole randomisations to construct an interval. For a fixed window, we establish asymptotic validity under finite fourth moments and a positive limiting variance. We measure coverage of four interval constructions on integrands with known exponents. With Gaussian errors, all four are compatible with nominal coverage at 128 and 512 randomisations, and jackknife-$t$ already at 32. All four undercover for the two higher-kurtosis families even at 512 randomisations. For differences between slopes, including null and small effects, preserving the dependence between paired randomisations can greatly narrow intervals, but it does not ensure nominal coverage. In option pricing, a steeper slope and a smaller error at a fixed budget can rank rules differently. Preintegration lowers a digital Asian option's exponent by about 0.5 in every tested window. For a barrier option it reduces the error six- to ninefold while changing the exponent by less than four hundredths. An arithmetic basket's gain grows with sample size. A geometric basket remains a ridge function as assets are added, limiting its use as a dimension benchmark. For multi-asset Asian options, different bases within a degenerate PCA eigenspace can change the measured exponent, in a direction that depends on the payoff.

Date: 2025-02, Revised 2026-09
New Economics Papers: this item is included in nep-cmp
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