Monte Carlo: Antithetic Variates, Control Variates, Quasi-MC

Setup

What Monte Carlo Computes

Monte Carlo (MC) methods estimate expectations of the form

$\mu = \mathbb{E}[f(X)],$

where $X$ is a random variable (or vector) with a known distribution and $f$ is a payoff or functional. In derivatives pricing, $X$ is typically a path of the risk-neutral process and $f$ is the discounted payoff.

The standard MC estimator with $N$ i.i.d. samples $X_1, \ldots, X_N$ is

$\hat{\mu}_N = \frac{1}{N}\sum_{i=1}^N f(X_i).$

By the law of large numbers, $\hat{\mu}_N \to \mu$ almost surely. By the central limit theorem,

$\sqrt{N}\left(\hat{\mu}_N - \mu\right) \xrightarrow{d} \mathcal{N}(0, \sigma_f^2), \qquad \sigma_f^2 = \mathrm{Var}(f(X)).$

The standard error is $\mathrm{SE} = \sigma_f / \sqrt{N}$, converging at rate $O(N^{-1/2})$ regardless of the dimension of $X$. This is the central virtue of MC: convergence rate is dimension-independent.

Notation

• $\sigma_f^2 = \mathrm{Var}(f(X))$: variance of the estimator (per sample).
• $N$: number of independent sample paths.
• $\hat{\sigma}_f^2$: sample variance estimator, used to construct confidence intervals.
• The $95\%$ confidence interval for $\mu$ is $\hat{\mu}_N \pm 1.96 \cdot \hat{\sigma}_f / \sqrt{N}$.

A key principle: reducing $\sigma_f$ by a factor $k$ is equivalent to increasing $N$ by $k^2$. Variance reduction techniques achieve this without generating more samples.

Antithetic Variates

Principle

For each sample $X_i$, generate a paired sample $X_i'$ from the antithetic distribution — typically by negating the underlying normal draw. The antithetic estimator is:

$\hat{\mu}_N^{\mathrm{AV}} = \frac{1}{N}\sum_{i=1}^N \frac{f(X_i) + f(X_i')}{2}.$

This is unbiased: $\mathbb{E}\!\left[\frac{f(X) + f(X')}{2}\right] = \mu$ since $X'$ has the same marginal distribution as $X$.

Variance Analysis

$\mathrm{Var}\!\left(\frac{f(X) + f(X')}{2}\right) = \frac{\mathrm{Var}(f(X)) + \mathrm{Var}(f(X')) + 2\,\mathrm{Cov}(f(X), f(X'))}{4}.$

Since $X$ and $X'$ are identically distributed, $\mathrm{Var}(f(X)) = \mathrm{Var}(f(X')) = \sigma_f^2$. Therefore:

$\mathrm{Var}\!\left(\frac{f(X) + f(X')}{2}\right) = \frac{\sigma_f^2 + \mathrm{Cov}(f(X), f(X'))}{2}.$

Variance is reduced whenever $\mathrm{Cov}(f(X), f(X')) < 0$. Compared to the standard estimator variance $\sigma_f^2 / N$ using $2N$ samples (matching the computational cost — two paths per antithetic pair), the antithetic estimator variance is:

$\frac{1}{N} \cdot \frac{\sigma_f^2 + \mathrm{Cov}(f(X), f(X'))}{2} \quad \text{vs.} \quad \frac{\sigma_f^2}{2N}.$

The efficiency gain is:

$\text{Efficiency gain} = \frac{\sigma_f^2 / (2N)}{(\sigma_f^2 + \mathrm{Cov})/2N} = \frac{\sigma_f^2}{\sigma_f^2 + \mathrm{Cov}(f(X), f(X'))} = \frac{1}{1 + \rho_{f,f'}},$

where $\rho_{f,f'}$ is the correlation between $f(X)$ and $f(X')$. For this to be a gain, we need $\rho_{f,f'} < 0$.

Application to GBM Call Option

Under GBM, one path realises $W_T = \sqrt{T} Z$ for $Z \sim \mathcal{N}(0,1)$. The antithetic path uses $-Z$:

$S_T = S_0 \exp\!\left((r - \tfrac{1}{2}\sigma^2)T + \sigma\sqrt{T}\, Z\right), \qquad S_T' = S_0 \exp\!\left((r - \tfrac{1}{2}\sigma^2)T - \sigma\sqrt{T}\, Z\right).$

The payoff $f(Z) = e^{-rT}(S_T - K)^+$ is an increasing function of $Z$. When $Z$ is large (call in the money), $-Z$ is large and negative (call out of the money). So $\mathrm{Cov}(f(Z), f(-Z)) < 0$ for a convex, monotone payoff — antithetic variates are effective.

For path-dependent options, antithetic variates require generating the full antithetic path $\{-Z_1, \ldots, -Z_n\}_{n=1}^{N_\text{steps}}$.

Control Variates

Principle

Let $Y$ be a random variable correlated with $f(X)$ such that $\mathbb{E}[Y] = \mu_Y$ is known analytically. The control variate estimator is:

$\hat{\mu}_N^{\mathrm{CV}}(c) = \frac{1}{N}\sum_{i=1}^N \left[f(X_i) - c(Y_i - \mu_Y)\right].$

This is unbiased for any constant $c$: the correction term $c(Y_i - \mu_Y)$ has mean zero.

Optimal Control Coefficient

The variance of $f(X) - c(Y - \mu_Y)$ is:

$\mathrm{Var}(f(X) - c Y) = \sigma_f^2 - 2c\,\mathrm{Cov}(f,Y) + c^2 \sigma_Y^2.$

Minimising over $c$:

$c^* = \frac{\mathrm{Cov}(f(X), Y)}{\mathrm{Var}(Y)}.$