Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Variance Reduction Techniques in Simulation › Convergence Analysis

Michael BrenndoerferDecember 3, 20253 min read

Quantitative FinanceAll chapters

Part 6 of 15Numerical Methods for Derivative Pricing

Chapter 35Variance Reduction Techniques in Simulation › Convergence Analysis

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Section 7 of 12 · 3 min read

Variance Reduction Techniques in Simulation › Convergence AnalysisOne section · Sections in this chapter7 / 12 ↓

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Understanding how variance reduction affects convergence helps in practical deployment. Let's visualize the convergence of different methods.

In[30]:

Code

# Convergence analysis
n_values = [1000, 2000, 5000, 10000, 20000, 50000, 100000]
methods = {
    "Standard MC": [],
    "Antithetic": [],
    "Control Variate": [],
    "Combined": [],
}
errors = {k: [] for k in methods.keys()}

for n in n_values:
    # Standard MC
    np.random.seed(42)
    Z = np.random.standard_normal(n)
    ST = S0 * np.exp((r - 0.5 * sigma**2) * T + sigma * np.sqrt(T) * Z)
    payoffs = np.exp(-r * T) * np.maximum(ST - K, 0)
    methods["Standard MC"].append(np.mean(payoffs))
    errors["Standard MC"].append(np.std(payoffs) / np.sqrt(n))

    # Antithetic
    p, se = monte_carlo_european_call(S0, K, T, r, sigma, n, antithetic=True)
    methods["Antithetic"].append(p)
    errors["Antithetic"].append(se)

    # Control Variate
    res = monte_carlo_call_control_variate(S0, K, T, r, sigma, n)
    methods["Control Variate"].append(res["price_cv"])
    errors["Control Variate"].append(res["se_cv"])

    # Combined
    p, se = monte_carlo_call_combined(S0, K, T, r, sigma, n)
    methods["Combined"].append(p)
    errors["Combined"].append(se)

Out[31]:

Visualization

Line plot showing convergence of four Monte Carlo methods toward the true option price. — Convergence of option price estimates to the Black-Scholes true value ($10.4506, dashed reference line) as simulations increase from 1,000 to 100,000. Shaded regions represent approximate 95% confidence intervals (plus or minus two standard errors). The combined antithetic and control-variate method shows the tightest error bands throughout, demonstrating how variance reduction achieves useful precision with fewer simulations.

The shaded regions show 95% confidence intervals (two standard errors). Methods with stronger variance reduction have tighter confidence intervals, reaching the same precision with fewer simulations.

Out[32]:

Visualization

Log-log plot of standard error versus number of simulations for standard Monte Carlo, antithetic variates, control variates, and a combined method, with all lines showing parallel 1/sqrt(n) slopes at different intercepts. — Standard error trajectories on a log-log scale for all methods from 1,000 to 100,000 simulations. All methods display the characteristic parallel slopes corresponding to the 1/sqrt(n) convergence rate, confirming that variance reduction does not change the fundamental convergence order. However, variance reduction techniques shift the intercepts downward, effectively lowering the error constant. This shift means that for any fixed sample size, these techniques achieve significantly lower standard errors, making the same target precision achievable with fewer simulations.