3.8.4 Variance Reduction & Monte Carlo Efficiency

Basic Prerequisites: 03 · Monte Carlo Pricing.

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1. Intuition & Practical Objective

Plain Monte Carlo is unbiased and dimension-free, and therefore slow: error falls as $n^{-1/2}$. Variance reduction attacks the constant, not the rate - replace ${\sigma }_f$ by something smaller while keeping the estimator unbiased. Four standard devices plus one budget rule:

Device	One-line mechanism	Gain
Antithetic	pair each draw with its mirror $Z\mapsto -Z$ so the two errors cancel	modest (needs monotone payoff)
Control variate	subtract a correlated quantity whose mean is known	$\times \frac{1}{1-{\rho }^2}$ - the big one when $\rho$ is high
Stratified / LHS	sample each stratum (or each marginal) evenly	never hurts; removes inter-stratum variance
Importance sampling	sample where the payoff is, reweight by the likelihood ratio	enormous for rare events; dangerous if mis-tuned

The practical objective is the work-normalised comparison: reduction factors are only meaningful at equal path counts and equal per-path cost, and the payoff is the same $\sigma /\sqrt{n}$ with a smaller $\sigma$ (or the same accuracy with fewer paths).

Three “aha”s:

1. A control variate is a regression. With optimal coefficient $b^{\ast }=\mathrm{Cov}(X,Y)/\mathrm{Var}(X)$ the variance ratio is exactly $1-{\rho }_{XY}^2$ (Glasserman eq. 4.4) - Corollary: $\rho =0.95$ buys $10\times$, $\rho =0.70$ buys only $2\times$. Correlation must be high to pay.
2. Antithetics and stratification work on opposite ends of the path. Antithetics pair whole paths; stratification partitions a dominant scalar (the terminal value, via the Brownian bridge). Only stratification removes the inter-stratum piece $\mathrm{Var}[\mathbb{E}[Y\mid X]]$ (eq. 4.44) - stronger than a control variate, which removes only the linear part.
3. Importance sampling is the only method that can be worse than useless. It can give infinite variance for a badly chosen proposal, and its weights degenerate on long horizons (Glasserman §4.6: the likelihood ratio tends to $0$ a.s. even though its mean is $1$). It is also the only way to price deep-OTM contracts at all.

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2. Mathematical Ground Truth & Derivations

Control variates (Glasserman eqs. 4.1–4.4). With $Y$ the payoff and $X$ a control with known $\mathbb{E}[X]$:

$$\bar{Y}(b)=\frac{1}{n}\sum _{i=1}^n(Y_i-b(X_i-\mathbb{E}[X])),\mathrm{Var}[Y_i(b)]={\sigma }_Y^2-2b{\sigma }_X{\sigma }_Y{\rho }_{XY}+b^2{\sigma }_X^2,$$ $$b^{\ast }=\frac{\mathrm{Cov}[X,Y]}{\mathrm{Var}[X]}={\rho }_{XY}\frac{{\sigma }_Y}{{\sigma }_X},\frac{\mathrm{Var}[\bar{Y}-b^{\ast }(\bar{X}-\mathbb{E}[X])]}{\mathrm{Var}[\bar{Y}]}=1-{\rho }_{XY}^2.$$

Multiple controls: $b^{\ast }={\Sigma }_X^{-1}{\Sigma }_{XY}$ (eq. 4.13) and the ratio is $1-R^2$. The coefficient is estimated by the OLS slope ${\hat{b}}_n=\frac{\sum (X_i-\bar{X})(Y_i-\bar{Y})}{\sum (X_i-\bar{X}{)}^2}$ (eq. 4.5), which biases the estimator by only $O(1/n)$ while remaining asymptotically as precise as $b^{\ast }$ - quoting a CV price still requires care about the bias, but it is second-order.

Antithetic variates (eqs. 4.27–4.30).

$${\hat{Y}}_{\mathrm{AV}}=\frac{1}{n}\sum _{i=1}^n\frac{Y_i+{\tilde{Y}}_i}{2},\mathrm{Var}\left[\frac{Y+\tilde{Y}}{2}\right]=\frac{{\sigma }_Y^2}{2}(1+{\rho }_{Y\tilde{Y}}),$$

so the gain exists iff ${\rho }_{Y\tilde{Y}}<0$; a monotone simulation map guarantees it, and a linear payoff in $Z$ gives zero variance. Splitting $f=f_0+f_1$ into symmetric/antisymmetric parts shows antithetics kill $\mathrm{Var}[f_1]$ exactly.

Stratified sampling (eqs. 4.31–4.44). With strata of probability $p_i$ and allocation $q_i=n_i/n$:

$$\hat{Y}=\sum _i\frac{p_i}{q_i}\frac{1}{n_i}\sum _j{Y}_{ij},{\sigma }^2(q)=\sum _i\frac{p_i^2}{q_i}{\sigma }_i^2,q_i^{\text{Neyman}}=\frac{p_i{\sigma }_i}{{\sum }_j{p}_j{\sigma }_j}(\text{cost-aware: }\propto p_i{\sigma }_i/\sqrt{{\tau }_i}).$$

Proportional allocation ($q_i=p_i$) gives $\mathrm{Var}={\sum }_i{p}_i{\sigma }_i^2\le {\sigma }^2$ always - “stratified sampling can only help” - because

$$\mathrm{Var}[Y]=\underset{\text{removed by proportional stratification}}{\underbrace{\mathrm{Var}[\mathbb{E}[Y\mid \eta ]]}}+\underset{\text{left}}{\underbrace{\mathbb{E}[\mathrm{Var}[Y\mid \eta ]]}}.$$

Latin hypercube (eqs. 4.55–4.58) stratifies every marginal coordinate with $K$ bins: for any square-integrable $f$ and $K\ge 2$, $\mathrm{Var}[{\hat{\alpha }}_{\text{LHS}}]\le {\sigma }^2/(K-1)$ (Owen Prop. 3), and asymptotically ${\sigma }_{\epsilon }^2/K$ (Stein) - LHS removes the variance of the additive part of $f$. Caveat: it stratifies the increments of a Brownian path, not the terminal value; directing that requires the bridge.

Importance sampling (eqs. 4.73–4.91). If $g$ dominates $f$,

$$\alpha =\mathbb{E}[h(X)]=\tilde{\mathbb{E}}\left[h(X)\frac{f(X)}{g(X)}\right],{\hat{\alpha }}_g=\frac{1}{n}\sum _{i=1}^{n}h(X_i)\frac{f(X_i)}{g(X_i)},X_i\sim g,$$

with the zero-variance (unusable) choice $g\propto hf$. Two practical tilts:

• exponential tilting: $f_{\theta }(x)=e^{\theta x-\psi (\theta )}f(x)$, with likelihood ratio $e^{-\theta \sum X_i+n\psi (\theta )}$ (eq. 4.84);
• drift tilt for Gaussian paths ($Z\mapsto Z+\mu$): likelihood ratio $e^{-{\mu }^{\prime }Z+\frac{1}{2}{\mu }^{\prime }\mu }$, where $\mu$ solves the fixed-point $\nabla F(\mu )={\mu }^{\prime }$ - the optimal path (eqs. 4.89–4.90).

The efficiency rule (Glasserman §1.1.3, §6.3.3): with bias $b{\delta }^{\beta }$, cost per path $\propto {\delta }^{-\eta }$ and work budget $s$,

$$\mathrm{RMSE}=O\left(s^{-\beta /(2\beta +\eta )}\right),\text{discretisation-aware: }{\delta }^{\ast }\propto s^{-1/(2\beta +1)},\sqrt{\mathrm{MSE}}\propto s^{-\beta /(2\beta +1)}.$$

For $\beta =1$ (Euler) the budget rate is $s^{-1/3}$ versus $s^{-1/2}$ for an unbiased estimator: discretisation costs you half the convergence exponent, which is why exact sampling (page 03) and better schemes are worth more than more paths.

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3. Computational Implementation - four reductions, measured

Each device is measured at a fixed path budget, against the closed form where one exists.

The numbers confirm the theory each time. Antithetic: $4.41\times$. Control variate for the European call: measured $6.80\times$ against the theoretical $1/(1-{\rho }^2)=1/(1-0.9235^2)=6.80$ - an exact match. Control variate for the arithmetic Asian: $1256.7\times$, driven entirely by $\rho =0.99960$ (a correlation of $0.9996$ is what “same payoff, closed form” buys). Importance sampling on a $0.0435\%$ event: RMSE falls from $1.21\times 10^{-4}$ to $4\times 10^{-6}$, a variance ratio of $\approx 915\times$ - and the plain-MC mean is $9.4\%$ high while the IS mean is $0.2\%$ high.

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4. Failure Modes & First-Principles Breakdowns

1. A control variate that is only mildly correlated is a waste of code. $\rho =0.70$ gives $1.96\times$; $\rho =0.30$ gives $1.10\times$. Corollary: use controls that are almost the payoff (geometric Asian, the underlying itself) and reject the rest.
2. Antithetics can hurt. If the payoff is not monotone in $Z$ (a straddle, a butterfly, a barrier), ${\rho }_{Y\tilde{Y}}>0$ and the paired estimator is worse than plain MC. Check $\rho <0$ before using it.
3. Importance sampling with the wrong tilt has infinite variance. Tilting too far concentrates on paths the payoff never reaches; the sampled variance reflects a heavy-tailed weight distribution, and the reported standard error is then a lie. The safe diagnostic is the estimated second moment of the weight, which must be finite and tame (Glasserman §4.6, Table 4.5: stratification on top of IS is what delivers the $10^3$–$10^4$ factors).
4. Long-horizon likelihood-ratio degeneracy. Under the twisted measure, $\frac{1}{m}\sum \log (f/g)\to c<0$ (Jensen/Glynn–Iglehart), so the likelihood ratio of a long path tends to $0$ a.s. even though $\mathbb{E}[\text{LR}]=1$ at every $m$. Any IS scheme whose weights are spread over many steps inherits this; the cure is a drift change (much narrower spread) rather than a per-step density change.
5. Conflating variance with accuracy. A $1256\times$ variance reduction on the Asian does not mean $1256\times$ accuracy: it means the same accuracy as $1256\times$ more paths, at the cost of one extra closed-form evaluation per path. Report the work-normalised efficiency, or the comparison is meaningless.
6. The score/likelihood-ratio Greek explodes with the number of dates. The LR score is a mean-zero martingale, so its variance grows with the monitoring count and blows up as the first interval $t_1\to 0$. For path-dependent Greeks, use pathwise (page 06) with a smoothing correction, not raw LR.

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5. References

• Glasserman, Monte Carlo Methods in Financial Engineering
• Glasserman
• Hull, Options, Futures, and Other Derivatives
• Haug, Complete Guide to Option Pricing Formulas

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6. Connected Graph Bridges

• Back: 03 · Monte Carlo Pricing · Index Hub
• Forward: 05 · Failure Modes → 06 · Advanced Extensions (QMC, American MC, pathwise Greeks)
• Sibling: BSM · Advanced Extensions (jump-diffusion MC and the same variance toolbox)