# Monte Carlo Simulation - Methodologies and Applications for Pricing and Risk Management

Duration: 3 days
• Monte Carlo Simulation in Finance
• Random Number Generation
• Cholesky Decomposition
• Binomial Lattice Models
• Stochastic Differential Equations
• Variance Reduction Techniques
• Pricing Exotic Options
• Measuring Value at Risk
The objective of this advanced-level course is to give the participants a good understanding of the Monte Carlo Method and its applications in finance. We shall start by motivating the use of Monte Carlo methods and we shall give an overview of the widespread use of Monte Carlo methods in securities and derivatives pricing and in risk management. We then give an in-depth explanation of the Monte Carlo method, enumerating its fundamental building blocks. We shall work our way through generation of pseudo-random numbers including numbers drawn from arbitrary probability distributions, discrete as well as continuous. We explain how the “Cholesky decomposition” technique can be used when sampling from multivariate distributions, when assets are correlated. We will use lattice-pricing models to price exotic options using various stochastic processes, including Black-Scholes as a benchmark. Further, we will show how to construct discrete versions of widely used Stochastic Differential Equations. These versions will be used to simulate trajectories of assets and to measure the Value at Risk of a portfolio of securities. Finally, we present quite a few variance reduction techniques for use with Monte Carlo Simulation, including the use of antithetic variables, control variates and importance sampling. The effect of these techniques on computational accuracy and/or performance will be evaluated. Throughout the course the participants will be given the opportunity to work on exercises, gaining hands-on experience with some of the Monte Carlo methods. (Excel and Visual Basic). The exercises/workshops will be based upon Microsoft Excel 2000 and Visual Basic.

## Day One

### 09.15 - 12.00 Introduction to Monte Carlo Simulation

• What is "Monte Carlo Simulation"?
• Applications of Monte Carlo Simulation in Finance
• A Couple of Examples of What You Can Do
• Introductory Exercise

### The Monte Carlo Toolkit

• Generating Random Numbers
• Random number generators - how they work
• Testing the Excel/VB random number generator

### 13.00 - 16.30 The Monte Carlo Toolkit (cont' d)

• Statistical Distributions
• Uniform, normal and log-normal distributions
• Binomial distribution
• Other distributions
• Sampling Techniques
• Generating normally distributed random numbers
• Drawing form multivariate distributions
• Stochastic Differential Equations
• Exercises

## Day  Two

### 09.15 - 12.00 Pricing Options Using Monte Carlo Simulation

• Overview of Option Pricing Models
• Pricing Standard European Options
• Pricing "Path Dependent" Options
• Barrier options
• Lookback
• Asian
• Range Floaters/EARNs
• Pricing American Options
• Greeks in Monte Carlo
• Exercises/Workshop

### 13.00 - 16.30 Calculating "Value-at-Risk"

• What is "Value-at-Risk"?
• VaR due to market risk
• VaR due to credit risk
• Approaches to Calculating VaR
• Calculating VaR Using Monte Carlo Simulation
• VaR for Single Asset Portfolios
• Formulating the price process
• Discretising the price process
• Constructing the P&L Histogram
• Inferring the VaR
• Exercises

## Day  Three

### 09.15 - 12.00 Calculating Value-at-Risk (continued)

• VaR for Multiple Asset Portfolios
• When prices are independent
• When prices are perfectly correlated
• When prices are imperfectly correlated
• Choleksky decomposition
• Constructing the P&L Histogram
• Inferring the VaR
• Stress Testing
• Exercises/Workshop

### 13.00 - 16.00 Making Monte Carlo Simulation More Efficient

• Problems with Conventional MCS
• "Clustering" and other problems
• Quasi-Monte Carlo Approaches
• Scrambled Nets Approach
• Scenario Simulation - an Alternative Approach
• Examples and Exercises