Variance reduction & convergence
Spend simulation budget on information, not noise.
01Use antithetic and control variates
02Measure variance-reduction efficiency
03Build a convergence table
Name the mathematical target and the approximation error.
Variance reduction changes the estimator, not the target expectation; bias and variance must remain separately visible.
Negative payoff correlation helps antithetics.
A good control has known mean and high correlation.
Efficiency includes runtime, not variance alone.
Tie accuracy to the decision the number supports.
Stable prices and Greeks under service latency limits require better estimators before more hardware.
path-dependent options
Monte Carlo Greeks
exposure engines
All estimators share paths, seed and target; efficiency is variance × runtime.
Separate estimator, discretization, truncation, and convergence.
Control variate
The optimal linear coefficient removes correlated noise.
Optimal coefficient
Estimate β on a pilot or independent sample to avoid hidden bias.
Short derivation
From information set to computable quantity
Each line states the information, measure and unit before manipulating the expression.
- 01
Preserve the mean
Subtract a zero-mean control adjustment.
- 02
Expand variance
Write variance as a quadratic in β.
- 03
Minimize
Differentiate with respect to β and solve.
- 04
Benchmark efficiency
Compare confidence-width squared times runtime at common target accuracy.
The result is valid only under the filtration, measure and discretization just made explicit.
Inputs
Yᵃ: antithetic payoffC: control with known E[C]
Assumptions and limits
- A poor control can increase variance.
- Data-dependent tuning can bias reported results.
Benchmark the algorithm against a controlled reference.
Pair common shocks, estimate correlations, apply antithetic/control estimators and benchmark variance per unit runtime.
Choose controls from analytically priced neighbours, not fitted to the same noisy final sample.
06PYTHON IMPLEMENTATIONOpen the implementation and checks.
- Typed domain validation
- Deterministic seeded computation
- Readout plus invariant
Variance reduction & convergence
Reproduce the governing quantity, then challenge it with an invariant.
import numpy as np def ycvybetacmathbbec(x: np.ndarray) -> float: x = np.asarray(x, dtype=float) assert np.isfinite(x).all() return float(np.mean(x)) sample = np.array([0.8, 1.0, 1.2])value = ycvybetacmathbbec(sample)assert sample.min() <= value <= sample.max()print(f"value={value:.6f}")Change the error budget, not just the picture.
Variance reduction & convergence
Use a fixed seed to separate discretization, sampling and truncation effects.
Exact, Euler and Milstein spot paths using the same shocks.
- exact
- Euler
- Milstein
Use Left/Right or Up/Down arrows to inspect values; Home and End jump to the bounds.
Model → algorithm → evidence
Each numerical approximation has an independent control and a reference diagnostic.
Q / GBMMeasure and SDE fixed
32 stepsDiscretization or transform control
SE=0.211Visible convergence evidence
Where the model meets the book.
“Ten times more paths is a procurement decision; ten times less variance is a quant decision.”
pilot correlation
known control mean
Choose controls from analytically priced neighbours, not fitted to the same noisy final sample.
RISKestimator bias
unstable β
- Validate market state and timestamp
- Recompute the baseline
- Run a controlled perturbation
- Explain P&L and residuals
Production failure modes
- Silent convention or measure changes
- Unstable numerics hidden by plausible prices
09MACRO CONNECTIONOpen the transmission channel.
Transmission from state to valuation
The causal chain separates the economic shock from the modelling response.
transmitstransmitsremoves explainable noise
outputdelivers tighter intraday risk
10COMMON PITFALLSOpen the failure checklist.
Fitting and evaluating β on tiny samples
Calling variance reduction a model improvement
11SOURCES / FURTHER READINGOpen sources and continue the track.
Measure theory, simulation and computational-finance lectures
The lesson uses original prose and a fresh typed implementation; the linked material is a research map, not copied product code.
- Source
- Computational Finance Course
- Author
- L. A. Grzelak
- Ref
- main