Combined tolerance
|actual - expected| <= atol + rtol |expected|Absolute tolerance protects behavior near zero; relative tolerance scales with result magnitude.
Architecture & Systems
Numerical Tolerances for Regression Tests
Tolerances must reflect trade scale, numerical method, and noise sources, balancing stability against necessary change.
Mental model
A tolerance must separate harmless numerical noise from meaningful pricing change.
Absolute tolerances are suitable for small-magnitude quantities, while relative tolerances scale with the size of the value.
A common rule is |new − old| ≤ abs_tol + rel_tol·|old|, with parameters tailored to product type and currency.
Explore the mechanism
How large must a numerical difference be before it signals changed behavior rather than expected computation noise?
Numerical tolerances for regression tests
Choose tolerances to balance stability against necessary change.
Tolerance rule
Pass if |new − old| ≤ abs_tol + rel_tol·|old|
abs_tol
0.020
rel_tol
0.0010
Engine and change
Suite failures
0 / 5
Balanced: tests are sensitive but not fragile.
Per-case deviation versus tolerance
Bar shows |new − old|. Band shows allowed tolerance.
FX option (small premium)
old: 0.3800 · new: 0.3800
|diff|: 0.00000 · tol: 0.02038
PASS
0|diff| / tol0.00×
Equity call (normal)
old: 2.5400 · new: 2.5400
|diff|: 0.00000 · tol: 0.02254
PASS
0|diff| / tol0.00×
Rates swap PV (large)
old: 145.2000 · new: 145.2000
|diff|: 0.00000 · tol: 0.16520
PASS
0|diff| / tol0.00×
Commodity spread (mid)
old: 5.8000 · new: 5.8000
|diff|: 0.00000 · tol: 0.02580
PASS
0|diff| / tol0.00×
Credit index PV (large)
old: 980.0000 · new: 980.0000
|diff|: 0.00000 · tol: 1.00000
PASS
0|diff| / tol0.00×
Interpretation
- Absolute tolerance protects small values. Relative tolerance scales for large PVs.
- Deterministic methods justify tight tolerances. Monte Carlo needs SE-aware tolerances.
- Too tight → false fails. Too loose → behaviour drifts without being noticed.
UNDERSTAND
What to notice
Deterministic methods such as PDE, lattice, or closed-form pricers justify tight tolerances dominated by floating-point limits.
Monte Carlo tolerances should be related to estimated standard errors so tests do not fail on expected sampling variation alone.
The quantitative structure
Notation
actual- new numerical result
expected- reviewed baseline
atol- absolute tolerance
rtol- relative tolerance
A small example
A scale-aware comparison
Expected value is 1,000, actual value is 1,000.08, atol is 0.01, and rtol is 0.0001.
- Observed difference is 0.08.
- Allowed difference is 0.01 + 0.0001 * 1,000 = 0.11.
- Because 0.08 <= 0.11, the comparison passes.
ResultThe result passes the combined tolerance while retaining a clear numerical bound.
Where it appears
- comparing Monte Carlo outputs
- testing cross-platform pricing results
- controlling solver and calibration regressions
Where the intuition breaks
What this view leaves out
- A tolerance can hide real defects when it is too loose and create unstable tests when it ignores scale, conditioning, or stochastic error.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Numerical Tolerances for Regression Tests
Numerical Tolerances for Regression TestsYOU ARE HERE
Used byGolden Regression TestsRegression tests need tolerances that distinguish acceptable numerical drift from real defects.
Used byValidation Toolbox for PricersA validation toolbox needs explicit tolerance policy so numerical differences are interpreted consistently.
SupportsFinite-Difference Checks for GreeksFinite-difference checks need explicit tolerances because discretisation and floating-point effects prevent exact equality.
Used by · Golden Regression Tests
Regression tests need tolerances that distinguish acceptable numerical drift from real defects.
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