Monte Carlo Methods for Strategy Validation: Beyond the Bootstrap

1. Why Resampling Matters

A backtest produces a single equity curve from a single historical path. The fundamental question is: how much of this outcome is due to the strategy’s edge, and how much is due to the specific sequence of market events that happened to occur? Resampling methods answer this by generating thousands of alternative paths and computing the strategy’s performance distribution across them.

2. Three Resampling Approaches

Block bootstrap (Künsch, 1989): resample fixed-length blocks of returns, concatenate them into synthetic histories, and re-run the strategy. The block length controls the trade-off between preserving serial dependence (longer blocks) and generating diverse paths (shorter blocks). The problem: fixed block length creates artificial discontinuities at block boundaries.

Stationary bootstrap (Politis and Romano, 1994): blocks have random length drawn from a geometric distribution with mean L. This eliminates boundary effects and produces stationary resampled series. The parameter L controls the expected autocorrelation structure.

Parametric Monte Carlo: fit a parametric model (e.g., AR(1)-GARCH(1,1) with Student-t innovations) to the return series, then simulate from the fitted model. This preserves the exact autocorrelation and volatility clustering structure but introduces model risk.

import numpy as np

def stationary_bootstrap(returns, n_paths, mean_block_length):
    """Politis-Romano stationary bootstrap."""
    T = len(returns)
    p = 1.0 / mean_block_length  # geometric probability
    paths = np.zeros((n_paths, T))
    for i in range(n_paths):
        t = 0; idx = np.random.randint(T)
        while t < T:
            paths[i, t] = returns[idx % T]
            t += 1; idx += 1
            if np.random.random() < p:  # start new block
                idx = np.random.randint(T)
    return paths

def parametric_mc(returns, n_paths, model='ar1_garch'):
    """Parametric MC using AR(1)-GARCH(1,1)-t model."""
    from arch import arch_model
    am = arch_model(returns*100, vol='Garch', p=1, q=1,
                    mean='AR', lags=1, dist='StudentsT')
    res = am.fit(disp='off')
    sims = res.forecast(horizon=len(returns), method='simulation',
                       simulations=n_paths)
    return sims.simulations.residuals.values / 100

3. Comparison on 12 Strategies

Table 1: Resampling method comparison. “ACF(1) Preserved” = % of resampled paths where lag-1 autocorrelation is within 20% of the original. “CI Coverage” = empirical coverage of nominal 95% CIs for the Sharpe ratio.

4. Practical Recommendations

The stationary bootstrap provides the best balance of simplicity, reliability, and assumption-freedom. We recommend a mean block length of L = 2/ρ̂ where ρ̂ is the estimated lag-1 autocorrelation of the strategy’s returns, which preserves the serial dependence structure without overfitting. For strategies with no significant autocorrelation, L = 10–20 is a reasonable default. Use at least 10,000 resampled paths for stable confidence intervals.

5. Conclusion

Block bootstrap understates uncertainty by destroying serial dependence. Stationary bootstrap corrects this with minimal additional complexity. Parametric Monte Carlo preserves all statistical structure but introduces model risk. For most practitioners, the stationary bootstrap is the right default choice for strategy validation.