Book 2. Quantitative Analysis

FRM Part 1

QA 13. Simulation and Bootstrapping

Presented by: Sudhanshu

Module 1. Monte-Carlo Simulation and Sampling Error Reduction

Module 2. Bootstrapping and Random Number Generation

Module 1. Monte-Carlo Simulation and Sampling Error Reduction

Topic 1. Introduction to Simulation and Bootstrapping

Topic 2. Steps in Monte Carlo Simulation

Topic 3. Monte Carlo Sampling Error & Its Reduction

Topic 4. Antithetic Variates Method

Topic 5. Control Variates Method

Topic 1.  Introduction to Simulation and Bootstrapping

  • Simulation is a statistical technique to model uncertainty in complex systems.
  • Monte Carlo Simulation (MCS): Uses random numbers from theoretical distributions to simulate possible outcomes.
  • Bootstrapping: Uses historical data instead of assumptions about distributions.
  • Applications in Finance:
    • Valuation of exotic derivatives
    • Risk estimation under macroeconomic stress
    • Regulatory capital projections

Practice Questions: Q1

Q1. Which of the following statements regarding Monte Carlo simulation is least accurate? When using Monte Carlo simulation:
A. simulated data is used to numerically approximate the expected value of a function.
B. the user specifies a complete data generating process (DGP) that is used to produce simulated data.
C. the observed data are used directly to generate a simulated data set.
D. a full statistical model is used that includes an assumption about the distribution of the shocks.

Practice Questions: Q1 Answer

Explanation: C is correct.

In both Monte Carlo simulation and bootstrapping, the goal is to numerically
approximate the expected value of a complex function through the use of
computer-generated values (i.e., simulated data). The main difference between
Monte Carlo simulation and bootstrapping is the source of the simulated data: in
Monte Carlo simulation, the user specifies a complete DGP that is used to produce
the simulated data, while in bootstrapping, the observed data are used directly to
generate the simulated data set—without specifying a complete DGP.

Topic 2. Steps in Monte Carlo Simulation

Five Key Steps:

  • Generate random inputs                                   from an assumed data-generating process (DGP).
  • Compute statistic of interest                .
  • Repeat the above steps N times to form                        
  • Estimate quantity of interest using the replicated outputs.
  • Estimate quantity of interest using the replicated outputs.

 

  • Common distributions used: Normal, t-distribution
  • Output: Distribution of the estimate instead of a single value

  • Example: Estimating ending capital of a portfolio:

 

x_i=\left[x_{1 i}, x_{2 i}, \ldots, x_{n i}\right]
g_i=g\left(x_i\right)
\left\{g_1, g_2, \ldots, g N\right\} .
\mathrm{SE}=\frac{s}{\sqrt{N}}
C_1=C_0(1+r)

Topic 3.  Monte Carlo Sampling Error & Its Reduction

  • Sampling Error: Occurs due to limited number of simulations (N)
  • Formula:

 

  • Increasing N improves accuracy:
    • 4×N → SE reduced by 2×
    • 100×N → SE reduced by 10×

Illustration:

  •  

 

  •  
N=100: S E=\frac{14.80}{10}=1.48
\mathrm{SE}=\frac{s}{\sqrt{N}}
N=400: S E=\frac{14.80}{20}=0.74

Topic 4.  Antithetic Variates Method

  • ​Goal: Reduce variance by pairing each simulation with its negative counterpart.
  • Random input set:
  • Antithetic pair:
  • Perfect negative correlation: Corr(ut,−ut)=−1\text{Corr}(u_t, -u_t) = -1−ut-u_t

​Mechanism:

  • Generate x1,x2x_1, x_2x1​,x2​ → compute average:

 

  • Reduces variance due to negative covariance.
  • Used when full sampling range is expensive to generate.
u_t
-u_t
\operatorname{Corr}\left(u_t,-u_t\right)=-1
\bar{x}=\frac{x_1+x_2}{2}

Topic 5.  Control Variates Method

  • ​Use of a known variable yyy (control) to reduce error in estimating unknown xxx.
  • New estimator:

 

  • Effective when:

 

Example:

  • Pricing Asian options using European option (with known Black-Scholes price) as control:

 

x^*=x+\left(y_{\text {true }}-y_{\text {sim }}\right)
\operatorname{Corr}(x, y) \approx 1
P_A^*=P_A+\left(P_{B S}-P_{B S}^*\right)

Practice Questions: Q2

Q2. Suppose an analyst is concerned about Monte Carlo sampling error. Based on an initial Monte Carlo simulation with 100 replications, the results indicated a standard deviation of 12.64. The simulation was rerun with 900 replications and the standard
deviation remained at 12.64. What are the standard error estimates for the simulations with 100 replications and 900 replications, respectively?

 

 

 

 

Practice Questions: Q2 Answer

Explanation: C is correct.

The standard error is determined by dividing the standard deviation by the square root of the number of replications            . The standard error estimate for the first simulation of 100 replications is 1.264 (i.e., 12.64 / 10). With 900 replications, the standard error estimate is reduced to 0.4213 (i.e., 12.64 / 30).

s / \sqrt{\mathrm{N}} .

Practice Questions: Q3

Q3. A concern for Monte Carlo simulations is the size of the sampling error. One way to
reduce the sampling error is to use the antithetic variate technique. Which of the
following statements best describes this technique?
A. The simulation is rerun using a complement set of the original set of random
variables.

Practice Questions: Q3 Answer

Explanation: A is correct.

The antithetic variate technique reduces Monte Carlo sampling error by rerunning the simulation using a complement set of the original set of random variables.

Module 2. Bootstrapping and Random Number Generation

Topic 1. Bootstrapping Method & Comparison with MCS

Topic 2. When Bootstrapping Fails

Topic 3. Pseudo-Random Number Generation

Topic 4. Disadvantages of Simulation

Topic 1.  Bootstrapping Method & Comparison with MCS

  • Draws samples with replacement from historical data
  • Doesn’t assume any parametric distribution
  • Two types:
    • i.i.d. Bootstrapping: Assumes data independence; randomly draws from full data
    • CBB (Circular Block Bootstrapping): Maintains time dependency by sampling blocks

Example of i.i.d.:

  • From 10 observations → Sample 3 with replacement: {x2, x7, x9}, etc.

Example of CBB:

  • 10 blocks of 3 consecutive data points:
\{\mathrm{x} 1, \mathrm{x} 2, \mathrm{x} 3\},\{\mathrm{x} 2, \mathrm{x} 3, \mathrm{x} 4\}, \ldots,\{\mathrm{x} 10, \mathrm{x} 1, \mathrm{x} 2\}

Topic 2.  When Bootstrapping Fails

Situations Where Bootstrapping Is Ineffective

  • Non-Representative Historical Data
    • Bootstrapping assumes that past data reflects the future.
    • If market conditions today are abnormal, bootstrapped results may mislead.
    • Example:
      • During the 2007–2009 financial crisis, using bootstrapping on low-volatility pre-crisis data would understate the risk (e.g., Value-at-Risk or VaR estimates too low).
  • Structural Market Changes
    • When permanent shifts in market conditions occur, past data becomes irrelevant.
    • Example:
      • U.S. T-bill rates remained near zero for a decade post-2008, a condition never seen in historical data.
      • Bootstrapping pre-2008 data would fail to replicate post-2008 reality.
  • Inability to Handle Tail Events
    • Bootstrapping may miss rare but important events (e.g., crashes) if not present in historical data.
  • Bootstrapping is useful only if:
    • The distribution of data is stable
    • No fundamental shifts in the economy or financial structure

Topic 3.  Pseudo-Random Number Generation

 What Are Pseudo-Random Numbers?

  • Pseudo-Random Number Generators (PRNGs):
    Algorithms that simulate random sequences using mathematical formulas.
  • Not truly random, but appear random for practical purposes.
    They produce numbers in the uniform (0,1) interval.

Key Term:

  • Seed Value – Starting input to the PRNG.
    • The same seed always produces the same sequence.

Benefits of PRNG in Finance:

  • Repeatability
    • Ideal for testing multiple models under identical simulated conditions.
    • Regulators may require simulations to be reproduced for audit or validation.
  • Use in Distributed Computing
    • Large simulations (e.g., thousands of financial instruments) can be run on computing clusters.
    • Using the same seed across clusters ensures synchronized scenarios.
  • Control Over Simulation Inputs
    • Enables comparison of strategies under identical “random” paths.
  • Limitations:
    • Deterministic → Not suitable for cryptographic or high-security needs.
    • Patterns may emerge if not enough entropy is introduced.

Topic 3.  Disadvantages of Simulation Approach

Limitations of Using Simulations in Financial Problem Solving

  • Model Misspecification Risk
    • Simulations rely on assumptions for the data-generating process (DGP).
      • Example: Using a normal distribution for options data that actually follows a fat-tailed distribution.

    • Incorrect assumptions → Biased or misleading results regardless of simulation size.

  • High Computational Cost
    • Realistic simulations may require:
      • 10,000+ replications
      • Multi-step time paths
      • High-dimensional asset correlations
  • This increases:
    • Time complexity
    • Hardware requirements
    • Energy consumption
  • No Analytical Guarantees
    • Simulation results can be sensitive to:
      • Initial inputs
      • Random seed
      • Numerical rounding errors
    • Unlike closed-form solutions, simulations do not provide mathematical guarantees of accuracy or convergence.

Practice Questions: Q1

Q1. Which of the following statements regarding the bootstrapping method is least accurate? Bootstrapping simulations:

A. draw data from historical data sets.
B. replace drawn data so it can be redrawn.
C. require assumptions with respect to the true distribution of the parameter
estimates.
D. rely on the key assumption that the present resembles the past.

Practice Questions: Q1 Answer

Explanation: C is correct.

The bootstrapping technique does not require any assumptions with respect to the true distribution of the parameter estimates. Bootstrapping simulations repeatedly draw data from historical data sets, and then replace the data so it can be redrawn. The bootstrapping method is only as valid as the assumption that the
present resembles the past.

Topic 2. Jarque-Bera Test

  • Purpose: Used to test whether a distribution is normal.
    • A normal distribution implies zero skewness and no excess kurtosis (K−3=0).
  • Hypotheses : Null Hypothesis         : S=0 and K=3 (zero skewness and kurtosis of three)
    • Alternative Hypothesis                                           (skewness not zero and kurtosis not three)
  • Test Statistic ​
  • Where T is the sample size, S is skewness, and K is kurtosis.
  • Interpretation : Smaller JB values suggest the null hypothesis is likely true (distribution is normal).
    • Larger JB values suggest the null hypothesis is likely to be rejected (distribution is not normal).
    • Critical values at 5% and 1% are 5.99 and 9.21, respectively. Reject null if JB is above these levels.
  • Impact of Time Period : Longer measurement periods often result in smaller JB statistics, approximating a normal distribution.
(H_0)
(H_A): S \neq 0 \text{ and }K \neq 3
JB=(T-1)\left(\frac{\hat{S}^2}{6}+\frac{(\hat{K}-3)^2}{24}\right)

Practice Questions: Q2

Q2. Which of the following statements regarding the pseudo-random number generation method is least accurate? Pseudo-random numbers are:
A. not truly random.
B. actually generated from a formula.
C. determined by the choice of the initial seed value.
D. impossible to predict.

Practice Questions: Q2 Answer

Explanation: D is correct.

Pseudo-random numbers appear random because they are difficult to predict. However, they are produced by deterministic functions that are complex rather than truly random. The initial choice of a seed value determines the series of random numbers that is generated.

Topic 3. The Power Law

  • Relevance to Financial Returns 
    • Financial returns tend to follow non-normal distributions.
    • Studying the tails helps explain return distribution.
  • Normal Distribution Tails 
    • Kurtosis of three (excess kurtosis of zero).
    • Thin tails.
  • Power Law Tails 
    • Found in some distributions, including Student's t-distribution.
    • Probability of a return larger than x:                                  (where k and a are constants).
    • "Fatter" tails than normal distributions
    • Tails do not decline as quickly as normal distributions.
    • Observations away from the mean are more common.
P(X>x)=k x^{-\alpha}

Practice Questions: Q3

Q3. The bootstrapping method is most likely to be effective when the:
A. data contains outliers.
B. present is different from the past.
C. data is independent.
D. markets have experienced structural changes.

Practice Questions: Q3 Answer

Explanation: C is correct.

The bootstrapping method is most likely to be effective when the data is independent and there are no outliers in the data. Bootstrapping uses the entire data set to generate a simulated sample, so the bootstrapping method should be reliable if the current state of the financial market is the same as its normal state, meaning that no structural changes have taken place.

Practice Questions: Q4

Q4. Monte Carlo simulation is a widely used technique in solving economic and financial
problems. Which of the following statements is least likely to represent a limitation
of the Monte Carlo technique when solving problems of this nature?
A. High computational costs arise with complex problems.
B. Simulation results are experiment-specific because financial problems are
analyzed based on a specific data generating process (DGP) and set of equations.
C. Results of most Monte Carlo experiments are difficult to replicate.
D. If the input variables have fat tails, Monte Carlo simulation is not relevant
because it always draws random variables from a normally distributed population.

Practice Questions: Q4 Answer

Explanation: D is correct.

A disadvantage of Monte Carlo simulations is that imprecise results may occur when the assumptions of model inputs or DGP are unrealistic. The distribution of  input variables does not need to be the normal distribution. Problems will arise if a real-world variable is fat-tailed, but the model erroneously draws option prices from a normal distribution.