Variance Reduction — Antithetic Variates and Control Variates

Companion notebook to the Variance Reduction article (Monte Carlo Methods, Module 3). All claims are verified numerically using NumPy and SciPy.

• §0 — Imports and shared pricing functions
• §1 — Antithetic correlation: measure ρ and verify VRF = 1/(1+ρ)
• §2 — AV variance reduction: empirical VRF vs theory across moneyness
• §3 — Control variate: optimal β*, VRF = 1/(1−ρ²), empirical β̂ convergence
• §4 — Combined AV + CV: total VRF and comparison table
• §5 — VRF vs moneyness: both techniques across the smile
• §6 — Validation summary

§0 — Imports and shared functions

Defines the Black-Scholes formula, the one-step GBM sampler, and the three estimators: naive MC, antithetic variates (AV), and control variates (CV) with underlying as control. All functions are reused in §1–§5.

§1 — Antithetic correlation ρ and VRF = 1/(1+ρ)

Compute the empirical correlation ρ̂ = Corr[f(Z), f(−Z)] for the ATM call using N = 200 000 pairs. Verify it is negative (as claimed for monotone payoffs) and that the antithetic variance equals σ²_naive × (1+ρ̂)/2.

§2 — AV variance reduction across moneyness

Run M = 500 independent replications of naive MC and AV MC (both with N = 10 000 path evaluations) for each strike K in a grid. Compute empirical VRF at each strike. Confirm AV is most effective deep OTM (ρ → −1) and least effective ATM.

§3 — Control variate: optimal β* and VRF = 1/(1−ρ²)

Use C = e^(−rT)·S_T with known mean μ_C = S_0 as the control. For a range of β values, compute the resulting estimator variance and confirm the parabolic shape — minimum at β*. Measure the empirical VRF and compare to the theoretical 1/(1−ρ²_{f,C}).

§4 — Combined AV + CV: total VRF and comparison

Compare naive MC, AV only, CV only, and AV+CV on equal footing: each uses N = 100 000 path evaluations (so AV uses 50 000 pairs, AV+CV uses 50 000 pairs). Run M = 200 replications to measure empirical RMSE. Report VRF of each technique relative to naive MC.

§5 — VRF vs moneyness for AV and CV

Sweep K from 70 to 130. At each strike, compute theoretical VRF for AV (from ρ̂) and CV (from 1/(1−ρ²_{f,C})). This shows that:

• AV is most effective for deep OTM options (ρ → −1 as payoff becomes very asymmetric)
• CV is most effective for ITM options (payoff ≈ linear in S_T → ρ_{f,C} → 1)
• Combined AV+CV dominates both individually across all moneyness levels

§6 — Validation summary

All claims from the article verified.