## **Learning Outcomes**

This article explains how variance reduction methods improve Monte Carlo accuracy without proportionally increasing computational cost. After reading, you will be able to:

- Explain why Monte Carlo sampling error occurs and how it relates to variance and standard error.
- Apply antithetic variates to reduce variance through negative covariance between paired simulation outputs.
- Apply control variates by using a correlated quantity with a known expected value to reduce sampling error.

## **GARP FRM Part 1 Syllabus**

For the GARP FRM Part 1 exam, you are expected to understand Monte Carlo simulation accuracy with a focus on the following syllabus points:

- Describe ways to reduce Monte Carlo sampling error.
- Explain the use of antithetic variates in reducing Monte Carlo sampling error.
- Explain the use of control variates in reducing Monte Carlo sampling error.

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. Which statement best describes _why_ antithetic variates can reduce simulation error?  
   a) They increase the number of scenarios without extra computation  
   b) They create negative covariance between paired simulation outputs  
   c) They remove the need to specify a data generating process (DGP)  
   d) They replace pseudo-random numbers with truly random numbers

2. Two Monte Carlo runs produce estimates $x_1$ and $x_2$ of the same quantity with $\text{var}(x_1)=\text{var}(x_2)=9$. Compute $\text{var}(\bar{x})$ where $\bar{x}=(x_1+x_2)/2$:
   - (i) if the runs are independent
   - (ii) if $\text{cov}(x_1,x_2)=-4$

3. Give two properties of a good control variate in practical risk analytics.

4. True/False (brief explanation required): A control variate estimator remains unbiased even if the analyst uses an incorrect “known” value for the control’s expected value.

## **What Do You Know Already?**

- Can you explain why the Monte Carlo standard error typically falls at a rate proportional to $1/\sqrt{N}$?
- Do you remember how covariance enters the variance of an average such as $(x_1+x_2)/2$?
- If you were pricing an Asian option by simulation, what related instrument might you use as a control variate, and why?

Use these prompts to gauge your starting point; then focus your revision where your answers were least confident.

## **Introduction**

Monte Carlo simulation is widely used in risk management when a valuation or risk metric is hard to compute analytically. Examples include pricing path-dependent derivatives (e.g., Asian options), measuring portfolio tail risk under complex non-linear payoffs, and running multi-factor stress tests where the distribution of outcomes is driven by many correlated risk factors.

A practical limitation is _sampling error_: even if the model and inputs are correct, a finite number of simulation replications produces a noisy estimate. A standard way to reduce this noise is to increase the number of replications $N$, but that can be expensive in multi-asset, multi-period simulations (full revaluation VaR, counterparty credit exposure, XVA, nested simulations for capital, and so on). Variance reduction techniques aim to improve accuracy at a given computational budget by changing _how_ scenarios are generated and combined.

> **Key Term: Monte Carlo sampling error**  
> The random error that arises because a simulation uses a finite number of replications. It is typically summarized by the standard error of the simulated estimator.

## **Monte Carlo sampling error and estimator variance**

A basic Monte Carlo workflow is:

1. Simulate random inputs $x_i$ from an assumed data generating process (DGP).
2. Compute the statistic of interest $g_i=g(x_i)$ (e.g., option payoff, portfolio P\&L).
3. Repeat for $N$ replications to obtain $\{g_1,\dots,g_N\}$.
4. Estimate the desired quantity (often an expectation) with the sample average:
   $$\hat{\mu}_g = \frac{1}{N}\sum_{i=1}^{N} g_i$$
5. Evaluate accuracy via the standard error.

If $g_i$ are independent and identically distributed with variance $\sigma_g^2$, then:
$$\text{var}(\hat{\mu}_g)=\frac{\sigma_g^2}{N} \quad \Rightarrow \quad \text{SE}(\hat{\mu}_g)=\frac{\sigma_g}{\sqrt{N}}$$

This rate is central for the exam: to cut the standard error in half, you need about four times as many simulations.

> **Key Term: Standard error (SE) in simulation**  
> The estimated standard deviation of the Monte Carlo estimator (often a simulated mean). For independent replications it scales as:
>
> $$
> \frac{1}{\sqrt{N}}
> $$

### Worked Example 1.1

A Monte Carlo model produces a sample standard deviation of simulated outcomes of $14.80$.

1. Compute the standard error with $N=100$ replications.
2. Compute the standard error with $N=400$ replications.
3. By what factor did accuracy improve?

> **Answer:**
>
> 1. With N=100,
>    $$\text{SE}=\frac{14.80}{\sqrt{100}}=\frac{14.80}{10}=1.48$$
> 2. With N=400,
>    $$\text{SE}=\frac{14.80}{\sqrt{400}}=\frac{14.80}{20}=0.74$$
> 3. The standard error fell from 1.48 to 0.74, so it was cut in half (accuracy improved twofold) when N increased by a factor of 4.

This motivates variance reduction: if increasing $N$ is costly, we try to reduce the variance of the estimator by introducing _negative dependence_ (antithetic variates) or by using _correlated information with known expectation_ (control variates).

## **Antithetic variates and negative covariance**

### Core idea

Antithetic variates reduce variance by pairing each random draw with a matched draw chosen to create negative correlation. A common construction is:

- Generate a random draw $u$ (often uniform on $[0,1]$).
- Create its complement $1-u$ (or, in symmetric cases such as a normal, use $z$ and $-z$).
- Run the model twice using these paired values and average the two outputs.

In many finance applications, the model output $g(\cdot)$ moves monotonically with the driving randomness (for example, a European call payoff increases with the terminal asset price). In such cases, pairing “high” draws with “low” draws makes the two outputs move in opposite directions, and averaging them reduces dispersion.

> **Key Term: Antithetic variates**  
> A variance reduction method that reruns the simulation using specially paired random draws (e.g., u and 1-u, or z and -z) to induce negative correlation between paired outputs.

### Variance of an average with covariance

Suppose two simulation runs produce estimates $x_1$ and $x_2$ of the same quantity, and the combined estimate is:
$$\bar{x}=\frac{x_1+x_2}{2}$$

The variance of $\bar{x}$ is:
$$\text{var}(\bar{x})=\frac{\text{var}(x_1)+\text{var}(x_2)+2\text{cov}(x_1,x_2)}{4}$$

- If the two runs are independent, then $\text{cov}(x_1,x_2)=0$ and:
  $$\text{var}(\bar{x})=\frac{\text{var}(x_1)+\text{var}(x_2)}{4}$$
- With antithetic variates, the goal is $\text{cov}(x_1,x_2)<0$, which further reduces $\text{var}(\bar{x})$.

This is the exam-friendly intuition: negative covariance reduces the variance of the average.

### Implementation mechanics

A standard approach is to generate uniforms in pairs:
$$(u_1,1-u_1),(u_2,1-u_2),\ldots,(u_{N/2},1-u_{N/2})$$

Then transform to the required distribution using the inverse CDF $F^{-1}$:
$$x=F^{-1}(u)$$

For a standard normal, symmetry implies that the antithetic draw is simply the negative:
$$z \sim N(0,1) \quad \Rightarrow \quad -z \sim N(0,1)$$

### When does it work well

Antithetic variates are most reliable when the function $g(\cdot)$ is monotonic in the random driver. Monotonicity makes the negative correlation in inputs carry through to negative correlation in outputs.

Finance examples where monotonicity is often reasonable:

- European call payoff $g(S_T)=\max(S_T-K,0)$ is increasing in $S_T$.
- Present value of a fixed set of cash flows is decreasing in interest rates (in many standard rate models, higher shocks push rates up, reducing PV).

Cases where monotonicity may fail:

- A straddle payoff is U-shaped in the asset price and not monotonic.
- Portfolios with strong convexity or path dependence can have regions where the payoff is increasing and other regions where it is decreasing in the same factor.

### Worked Example 2.1

Two simulation runs estimate the same parameter. Each has variance $9$.

1. Compute $\text{var}(\bar{x})$ when the runs are independent.
2. Compute $\text{var}(\bar{x})$ when antithetic construction makes $\text{cov}(x_1,x_2)=-4$.
3. Compute the percentage reduction in variance from (1) to (2).

> **Answer:**
>
> Given:
>
> $$
> \text{var}(x_1)=\text{var}(x_2)=9 \quad \text{and} \quad \bar{x}=(x_1+x_2)/2
> $$
>
> 1. Independent runs: cov(x1,x2)=0,
>    $$\text{var}(\bar{x})=\frac{9+9}{4}=\frac{18}{4}=4.5$$
> 2. Antithetic runs: cov(x1,x2)=-4,
>    $$\text{var}(\bar{x})=\frac{9+9+2(-4)}{4}=\frac{18-8}{4}=\frac{10}{4}=2.5$$
> 3. Variance reduction:
>    $$\frac{4.5-2.5}{4.5}=\frac{2}{4.5}\approx 44.44\%$$

## **Control variates using known expectations**

### Core idea

Control variates reduce sampling error by using a second quantity that:

1. Is correlated with the simulation output of interest, and
2. Has a known expected value (or can be priced accurately by an analytic model).

Intuition: if two quantities tend to be high and low together, then the simulation error in one can be partially offset using the other.

> **Key Term: Control variate**  
> A correlated variable with known expectation used to adjust a Monte Carlo estimate, reducing sampling error without (when set up correctly) introducing bias.

### A simple control variate estimator

Let $x$ be the unknown quantity we want, and let $y$ be a related quantity with known value (typically its expectation). Simulate both using the same random numbers to get $\hat{x}$ and $\hat{y}$. A common adjusted estimator is:
$$x^* = y + (\hat{x}-\hat{y})$$

Interpretation:

- $\hat{x}$ is the raw Monte Carlo estimate of the target.
- $\hat{y}$ is the Monte Carlo estimate of the control, based on the same scenarios.
- $y$ is the known value (for example, a Black-Scholes-Merton price, or a known expected value).
- The adjustment $(y-\hat{y})$ corrects for the sampling error in the control, and because $\hat{x}$ and $\hat{y}$ move together, it tends to correct $\hat{x}$ as well.

Unbiasedness (key exam point): if $E[\hat{y}]=y$ and $E[\hat{x}]=x$, then:
$$E[x^*]=E[y+\hat{x}-\hat{y}]=y+x-y=x$$

### Why variance can fall

Because $y$ is known, it has no sampling error. The variance of the adjusted estimator is:
$$\text{var}(x^*)=\text{var}(\hat{x}-\hat{y})=\text{var}(\hat{x})+\text{var}(\hat{y})-2\text{cov}(\hat{x},\hat{y})$$

Variance is reduced relative to the raw estimate $\hat{x}$ when:
$$\text{var}(\hat{x})+\text{var}(\hat{y})-2\text{cov}(\hat{x},\hat{y}) < \text{var}(\hat{x})$$
which simplifies to:
$$\text{var}(\hat{y})-2\text{cov}(\hat{x},\hat{y})<0 \quad \Rightarrow \quad \text{cov}(\hat{x},\hat{y})>\frac{\text{var}(\hat{y})}{2}$$

This highlights a practical rule: the control must be sufficiently correlated with the target (and not excessively noisy relative to the target) for the adjustment to be helpful.

### The general (optimal) control variate form

Many texts use a more general estimator:
$$x^{cv}=\hat{x}+\beta (y-\hat{y})$$

- If $\beta=1$, you recover $x^*=y+(\hat{x}-\hat{y})$.
- The variance-minimizing coefficient is:
  $$\beta^*=\frac{\text{cov}(\hat{x},\hat{y})}{\text{var}(\hat{y})}$$

In practice, $\beta^*$ is estimated by regressing $\hat{x}$ on $\hat{y}$ using the simulated sample. This is closely related to the idea that control variates “subtract out” the component of $\hat{x}$ that can be explained by $\hat{y}$.

### Practical example in derivatives

A standard application is pricing an Asian option (payoff depends on an average price). A closely related European option has a price available from the Black-Scholes-Merton model, which provides a natural control.

Let:

- $P_A$ = simulated Asian option price (raw estimate).
- $P_{BS}$ = known European option price from Black-Scholes-Merton.
- $P_{BS}^*$ = simulated European option price, using the same scenarios as the Asian simulation.

Then an adjusted Asian price is:
$$P_A^*=(P_A-P_{BS}^*)+P_{BS}$$

This is the same structure as $x^*=y+(\hat{x}-\hat{y})$.

### Worked Example 3.1

An analyst simulates an Asian call and, using the same scenarios, also simulates a European call.

- Raw Asian estimate: $P_A=5.20$
- Simulated European estimate: $P_{BS}^*=6.10$
- Known European price from Black-Scholes-Merton: $P_{BS}=6.00$

Compute the control variate adjusted estimate $P_A^*$.

> **Answer:**
>
> Use:
> $$P_A^*=(P_A-P_{BS}^*)+P_{BS}$$
>
> Substitute values:
> $$P_A^*=(5.20-6.10)+6.00=-0.90+6.00=5.10$$
>
> The adjusted estimate is 5.10. The adjustment reduced the estimate because the simulated European price was above its known value, suggesting the scenarios (by chance) were somewhat “favorable” to call payoffs.

## **Choosing between methods and using both**

Antithetic variates and control variates are compatible and can be used together:

- Antithetic variates change _how scenarios are generated_ (pairing complements) to create negative covariance across paired outputs.
- Control variates change _how outputs are combined_ by subtracting simulation noise using a correlated quantity with known expectation.

A common workflow in derivatives and VaR engines is:

1. Simulate factors with antithetic pairing (e.g., Gaussian shocks $z$ and $-z$).
2. For each path, compute both the target payoff/P\&L and one or more controls (cheap approximations or analytically priced instruments).
3. Combine results using a control variate adjustment (possibly with an estimated $\beta$).
4. Report the standard error reduction relative to a baseline run.

### **Exam Warning**

> Do not claim that antithetic variates “always” reduce variance for any payoff. The negative correlation is guaranteed for the paired _random draws_, but variance reduction in the _payoff_ is most dependable when the payoff is monotonic in the driving randomness. For control variates, remember that using the wrong “known” value biases the result.

## **Summary**

Monte Carlo simulation error is a sampling problem: with finite replications, the estimated mean (or other statistic) is random, and its standard error typically falls as $1/\sqrt{N}$. Increasing $N$ improves accuracy but can be expensive in realistic risk systems.

Variance reduction methods improve accuracy per unit of computation. Antithetic variates create paired scenarios designed to produce negative covariance, reducing the variance of an average estimator. Control variates adjust a simulated estimate using a correlated quantity whose expected value is known, with variance reduction driven by covariance (or correlation) between the target and the control.

For the FRM Part 1 exam, you should be comfortable with the variance-of-an-average formula including covariance, the logic of negative covariance under antithetic pairing, and the structure of a control variate estimator such as $x^*=y+(\hat{x}-\hat{y})$.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Monte Carlo standard error scales approximately as $\sigma/\sqrt{N}$ for independent replications.
- Variance of an average includes a covariance term: negative covariance reduces estimator variance.
- Antithetic variates rerun simulations with paired random draws (e.g., $u$ and $1-u$, or $z$ and $-z$).
- Control variates adjust an estimate using a correlated quantity with known expectation: $x^*=y+(\hat{x}-\hat{y})$.
- Control variates reduce sampling error when correlation is sufficiently high and the control is not too noisy.
- Antithetic variates and control variates can be applied together in one simulation engine.

## **Key Terms and Concepts (Summary of Variance Reduction)**

- Monte Carlo sampling error
- Standard error (SE)
- Data generating process (DGP)
- Antithetic variates
- Covariance and correlation (in variance reduction)
- Control variate
- Control variate estimator $x^*=y+(\hat{x}-\hat{y})$
- Variance of an average with covariance
- Monotonic payoff (as a condition supporting antithetic effectiveness)


Simulation and numerical methods - Variance reduction (antithetic variates, control variates)