Brooks, Introductory Econometrics for Finance, 3rd Edition

Size: px
Start display at page:

Download "Brooks, Introductory Econometrics for Finance, 3rd Edition"

Transcription

1 P1.T2. Quantitative Analysis Brooks, Introductory Econometrics for Finance, 3rd Edition Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM CIPM and Deepa Raju

2 Chris Brooks, Chapter 13. Simulation Methods DESCRIBE THE BASIC STEPS TO CONDUCT A MONTE CARLO SIMULATION... 4 DESCRIBE WAYS TO REDUCE MONTE CARLO SAMPLING ERROR... 5 EXPLAIN HOW TO USE ANTITHETIC VARIATE TECHNIQUE TO REDUCE MONTE CARLO SAMPLING ERROR Please note: The previous syllabus assignment for (P1.T2.) Simulations was Jorion s Chapter 12 called Monte Carlos Methods. Strictly speaking, because the Jorion chapter has been replaced by the Brooks chapter, it is optional for you to review. However, we have included this reading because it remains conceptually relevant. In particular, we do recommend that you review the following learning objectives: Describe how to simulate various distributions using the inverse transform method. Explain how to simulate correlated (normal) random variables 2

3 Chris Brooks, Chapter 13. Simulation Methods Describe the basic steps to conduct a Monte Carlo simulation. Describe ways to reduce Monte Carlo sampling error. Explain how to use antithetic variate technique to reduce Monte Carlo sampling error. Explain how to use control variates to reduce Monte Carlo sampling error and when it is effective. Describe the benefits of reusing sets of random number draws across Monte Carlo experiments and how to reuse them. Describe the bootstrapping method and its advantage over Monte Carlo simulation. Describe the pseudo-random number generation method and how a good simulation design alleviates the effects the choice of the seed has on the properties of the generated series. Describe situations where the bootstrapping method is ineffective. Describe disadvantages of the simulation approach to financial problem solving. In many situations, Monte Carlo simulation is the best approach. Often, analytical solutions are tractable but too simplistic to capture realistic and complex dynamics. Historical simulation, on the other hand, does offer the advantage of plentiful data, but may not give us the means to effectively imagine possible future outcomes. Consider interest rate paths. With relatively few assumptions, simulation allows us to randomize possible future interest rate paths. The two charts below can be found in the Learning Spreadsheet associated with this reading. Each chart simulates ten (10) different interest rate paths over a ten year (120 month) horizon. The models employed are Cox-Ingersoll-Ross (CIR) and the Vasicek model. These interest rate paths become inputs into a credit exposure (in the XLS) or can feed into a value at risk (VaR) model. Such is the flexibility of simulations, which allow us to manufacture data! 3

4 Describe the basic steps to conduct a Monte Carlo simulation Simulations studies are used to investigate the properties and behavior of various statistics of interest. The technique is used in econometrics when the properties of a particular estimation method are not known. The following are the basic steps to conduct a Monte Carlo simulation (Brooks Box 13.1): 1. Generate the data according to the desired data generating process (DGP), with the errors being drawn from some given distribution 2. Do the regression and calculate the test statistic 3. Save the test statistic or whatever parameter is of interest 4. Go back to stage 1 and repeat N times. These four steps contain two key stages : The first stage is the specification of the (data generating) model. This model may be either a pure time series model or a structural model. o o Pure time series models are simpler to implement. A full structural model is harder because it requires (in addition) the specification of the DGP for the explanatory variables also. Once the time series model is selected, the next choice refers to the probability distribution specified for the errors. Usually, standard normal draws are used, although any other empirically plausible distribution (e.g., Student s t) could also be used. The second stage involves estimation of the parameter of interest in the study. The parameter of interest might be, for example, the value of a coefficient in a regression, or the value of an option at its expiry date. It could instead be the value of a portfolio under a particular set of scenarios governing the way that the prices of the component assets move over time. The quantity N is known as the number of replications, and this should be as large as is feasible. The central idea behind Monte Carlo is that of random sampling from a given distribution. If the number of replications is set too small, the results will be sensitive to odd combinations of random number draws. It is also worth noting that asymptotic arguments apply in Monte Carlo studies as well as in other areas of econometrics. The results of a simulation study will be equal to their analytical counterparts (assuming that the latter exist) asymptotically. 4

5 Describe ways to reduce Monte Carlo sampling error There are two basic ways to reduce sampling error a) Increase the sample size, b) Employ a variance reduction technique; aka, acceleration method Sampling variation scales with the square root of the sample size Suppose that the value of the parameter of interest for replication i is denoted xi. If the average value of this parameter is calculated for a set of, say, N=1,000 replications, and another researcher conducts an otherwise identical study with different sets of random draws, a different average value of x is almost guaranteed. This situation is akin to the problem of selecting only a sample of observations from a given population in standard regression analysis. The sampling variation in a Monte Carlo study is measured by the standard error estimate, Sx: = var( ) where var(x) is the variance of the estimates of the quantity of interest over the N replications. It can be seen from this equation that to reduce the Monte Carlo standard error by a factor of 10, the number of replications must be increased by a factor of 100. As a result, to achieve acceptable accuracy, the number of replications may have to be set at an infeasibly high level. An alternative way to reduce Monte Carlo sampling error is to use a variance reduction technique. Two of the intuitively simplest and most widely used variance reduction methods are: Antithetic variate technique: This involves taking the complement of a set of random numbers and running a parallel simulation on those so that the covariance is negative, and therefore the Monte Carlo sampling error is reduced. Control variate technique: This involves using employing a variable similar to that used in the simulation, but whose properties are known prior to the simulation. The effects of sampling error for the problem under study and the known problem will be similar, and hence can be reduced by calibrating the Monte Carlo results using the analytic ones. Explain how to use antithetic variate technique to reduce Monte Carlo sampling error. The antithetic variate technique involves taking the complement of a set of random numbers and running a parallel simulation on those. For example, if the driving stochastic force is a set of T N (0,1) draws, denoted ut, for each replication, an additional replication with errors given by ut is also used. It can be shown that the Monte Carlo standard error is reduced when antithetic variates are used. This is illustrated here: 5

6 Suppose that the average value of the parameter of interest across two sets of Monte Carlo replications is given by = ( + )/2 where x 1 and x 2 are the average parameter values for replications sets 1 and 2, respectively. The variance of will be given by: ( ) = 1 4 ( ( ) + ( ) + 2 (, )) If no antithetic variates are used, the two sets of Monte Carlo replications will be independent, so that their covariance will be zero, i.e. var( ) = 1 4 (var( ) + var( )) However, the use of antithetic variates would lead the covariance to be negative, and therefore the Monte Carlo sampling error to be reduced. It may appear that the reduction in Monte Carlo sampling variation from using antithetic variates will be huge since, by definition, (, ) = (, ) = 1. It is however important to remember that the relevant covariance is between the simulated quantity of interest for the standard replications and those using the antithetic variates. The perfect negative covariance is between the random draws (i.e. the error terms) and their antithetic variates. For eg. in the context of option pricing, the production of a price for the underlying security constitutes a non-linear transformation of ut. Therefore, the covariances between the terminal prices of the underlying assets based on the draws and based on the antithetic variates will be negative, but not 1. Several other variance reduction techniques that operate using similar principles are available, including stratified sampling, moment-matching and low discrepancy sequencing. Low discrepancy sequencing is also known as quasi-random sequences of draws. These involve the selection of a specific sequence of representative samples from a given probability distribution. Successive samples are selected so that the unselected gaps left in the probability distribution are filled by subsequent replications. The result is a set of random draws that are appropriately distributed across all of the outcomes of interest. The use of low-discrepancy sequences leads the Monte Carlo standard errors to be reduced in direct proportion to the number of replications rather than in proportion to the square root of the number of replications. o For example, to reduce the Monte Carlo standard error by a factor of 10, the number of replications would have to be increased by a factor of 100 for standard Monte Carlo random sampling, but only 10 for low-discrepancy sequencing. 6

Week 7 Quantitative Analysis of Financial Markets Simulation Methods

Week 7 Quantitative Analysis of Financial Markets Simulation Methods Week 7 Quantitative Analysis of Financial Markets Simulation Methods Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 November

More information

Introduction Dickey-Fuller Test Option Pricing Bootstrapping. Simulation Methods. Chapter 13 of Chris Brook s Book.

Introduction Dickey-Fuller Test Option Pricing Bootstrapping. Simulation Methods. Chapter 13 of Chris Brook s Book. Simulation Methods Chapter 13 of Chris Brook s Book Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 April 26, 2017 Christopher

More information

P2.T8. Risk Management & Investment Management. Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition.

P2.T8. Risk Management & Investment Management. Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition. P2.T8. Risk Management & Investment Management Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition. Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Raju

More information

Value at Risk Ch.12. PAK Study Manual

Value at Risk Ch.12. PAK Study Manual Value at Risk Ch.12 Related Learning Objectives 3a) Apply and construct risk metrics to quantify major types of risk exposure such as market risk, credit risk, liquidity risk, regulatory risk etc., and

More information

Monte Carlo Methods in Financial Engineering

Monte Carlo Methods in Financial Engineering Paul Glassennan Monte Carlo Methods in Financial Engineering With 99 Figures

More information

P2.T5. Tuckman Chapter 9. Bionic Turtle FRM Video Tutorials. By: David Harper CFA, FRM, CIPM

P2.T5. Tuckman Chapter 9. Bionic Turtle FRM Video Tutorials. By: David Harper CFA, FRM, CIPM P2.T5. Tuckman Chapter 9 Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal copy and

More information

P2.T5. Market Risk Measurement & Management. Bruce Tuckman, Fixed Income Securities, 3rd Edition

P2.T5. Market Risk Measurement & Management. Bruce Tuckman, Fixed Income Securities, 3rd Edition P2.T5. Market Risk Measurement & Management Bruce Tuckman, Fixed Income Securities, 3rd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Tuckman, Chapter 6: Empirical

More information

P2.T5. Market Risk Measurement & Management. Bruce Tuckman, Fixed Income Securities, 3rd Edition

P2.T5. Market Risk Measurement & Management. Bruce Tuckman, Fixed Income Securities, 3rd Edition P2.T5. Market Risk Measurement & Management Bruce Tuckman, Fixed Income Securities, 3rd Edition Bionic Turtle FRM Study Notes Reading 40 By David Harper, CFA FRM CIPM www.bionicturtle.com TUCKMAN, CHAPTER

More information

Bias Reduction Using the Bootstrap

Bias Reduction Using the Bootstrap Bias Reduction Using the Bootstrap Find f t (i.e., t) so that or E(f t (P, P n ) P) = 0 E(T(P n ) θ(p) + t P) = 0. Change the problem to the sample: whose solution is so the bias-reduced estimate is E(T(P

More information

P2.T5. Market Risk Measurement & Management

P2.T5. Market Risk Measurement & Management P2.T5. Market Risk Measurement & Management Kevin Dowd, Measuring Market Risk Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Raju www.bionicturtle.com Dowd Chapter 3: Estimating

More information

2.1 Mathematical Basis: Risk-Neutral Pricing

2.1 Mathematical Basis: Risk-Neutral Pricing Chapter Monte-Carlo Simulation.1 Mathematical Basis: Risk-Neutral Pricing Suppose that F T is the payoff at T for a European-type derivative f. Then the price at times t before T is given by f t = e r(t

More information

P2.T5. Market Risk Measurement & Management. Jorion, Value-at Risk: The New Benchmark for Managing Financial Risk, 3 rd Edition

P2.T5. Market Risk Measurement & Management. Jorion, Value-at Risk: The New Benchmark for Managing Financial Risk, 3 rd Edition P2.T5. Market Risk Measurement & Management Jorion, Value-at Risk: The New Benchmark for Managing Financial Risk, 3 rd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Raju

More information

Alternative VaR Models

Alternative VaR Models Alternative VaR Models Neil Roeth, Senior Risk Developer, TFG Financial Systems. 15 th July 2015 Abstract We describe a variety of VaR models in terms of their key attributes and differences, e.g., parametric

More information

Hull, Options, Futures, and Other Derivatives, 9 th Edition

Hull, Options, Futures, and Other Derivatives, 9 th Edition P1.T4. Valuation & Risk Models Hull, Options, Futures, and Other Derivatives, 9 th Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Sounder www.bionicturtle.com Hull, Chapter

More information

P2.T5. Market Risk Measurement & Management. Kevin Dowd, Measuring Market Risk, 2nd Edition

P2.T5. Market Risk Measurement & Management. Kevin Dowd, Measuring Market Risk, 2nd Edition P2.T5. Market Risk Measurement & Management Kevin Dowd, Measuring Market Risk, 2nd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Dowd Chapter 3: Estimating Market

More information

Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and Operational Risk: The Value at Risk Approach

Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and Operational Risk: The Value at Risk Approach P1.T4. Valuation & Risk Models Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and Operational Risk: The Value at Risk Approach Bionic Turtle FRM Study Notes Reading 26 By

More information

MONTE CARLO EXTENSIONS

MONTE CARLO EXTENSIONS MONTE CARLO EXTENSIONS School of Mathematics 2013 OUTLINE 1 REVIEW OUTLINE 1 REVIEW 2 EXTENSION TO MONTE CARLO OUTLINE 1 REVIEW 2 EXTENSION TO MONTE CARLO 3 SUMMARY MONTE CARLO SO FAR... Simple to program

More information

Publication date: 12-Nov-2001 Reprinted from RatingsDirect

Publication date: 12-Nov-2001 Reprinted from RatingsDirect Publication date: 12-Nov-2001 Reprinted from RatingsDirect Commentary CDO Evaluator Applies Correlation and Monte Carlo Simulation to the Art of Determining Portfolio Quality Analyst: Sten Bergman, New

More information

A Glimpse of Representing Stochastic Processes. Nathaniel Osgood CMPT 858 March 22, 2011

A Glimpse of Representing Stochastic Processes. Nathaniel Osgood CMPT 858 March 22, 2011 A Glimpse of Representing Stochastic Processes Nathaniel Osgood CMPT 858 March 22, 2011 Recall: Project Guidelines Creating one or more simulation models. Placing data into the model to customize it to

More information

Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition

Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition P1.T3. Financial Markets & Products Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Raju

More information

Introduction to Algorithmic Trading Strategies Lecture 8

Introduction to Algorithmic Trading Strategies Lecture 8 Introduction to Algorithmic Trading Strategies Lecture 8 Risk Management Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Outline Value at Risk (VaR) Extreme Value Theory (EVT) References

More information

P2.T6. Credit Risk Measurement & Management. Malz, Financial Risk Management: Models, History & Institutions

P2.T6. Credit Risk Measurement & Management. Malz, Financial Risk Management: Models, History & Institutions P2.T6. Credit Risk Measurement & Management Malz, Financial Risk Management: Models, History & Institutions Portfolio Credit Risk Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1 Portfolio

More information

Computational Finance Improving Monte Carlo

Computational Finance Improving Monte Carlo Computational Finance Improving Monte Carlo School of Mathematics 2018 Monte Carlo so far... Simple to program and to understand Convergence is slow, extrapolation impossible. Forward looking method ideal

More information

Anthony Saunders and Marcia Millon Cornett, Financial Institutions Management: A Risk Management Approach

Anthony Saunders and Marcia Millon Cornett, Financial Institutions Management: A Risk Management Approach P1.T3. Financial Markets & Products Anthony Saunders and Marcia Millon Cornett, Financial Institutions Management: A Risk Management Approach Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM

More information

Monte Carlo Methods in Structuring and Derivatives Pricing

Monte Carlo Methods in Structuring and Derivatives Pricing Monte Carlo Methods in Structuring and Derivatives Pricing Prof. Manuela Pedio (guest) 20263 Advanced Tools for Risk Management and Pricing Spring 2017 Outline and objectives The basic Monte Carlo algorithm

More information

Practical example of an Economic Scenario Generator

Practical example of an Economic Scenario Generator Practical example of an Economic Scenario Generator Martin Schenk Actuarial & Insurance Solutions SAV 7 March 2014 Agenda Introduction Deterministic vs. stochastic approach Mathematical model Application

More information

Market Risk Analysis Volume IV. Value-at-Risk Models

Market Risk Analysis Volume IV. Value-at-Risk Models Market Risk Analysis Volume IV Value-at-Risk Models Carol Alexander John Wiley & Sons, Ltd List of Figures List of Tables List of Examples Foreword Preface to Volume IV xiii xvi xxi xxv xxix IV.l Value

More information

Strategies for Improving the Efficiency of Monte-Carlo Methods

Strategies for Improving the Efficiency of Monte-Carlo Methods Strategies for Improving the Efficiency of Monte-Carlo Methods Paul J. Atzberger General comments or corrections should be sent to: paulatz@cims.nyu.edu Introduction The Monte-Carlo method is a useful

More information

Estimating term structure of interest rates: neural network vs one factor parametric models

Estimating term structure of interest rates: neural network vs one factor parametric models Estimating term structure of interest rates: neural network vs one factor parametric models F. Abid & M. B. Salah Faculty of Economics and Busines, Sfax, Tunisia Abstract The aim of this paper is twofold;

More information

Kevin Dowd, Measuring Market Risk, 2nd Edition

Kevin Dowd, Measuring Market Risk, 2nd Edition P1.T4. Valuation & Risk Models Kevin Dowd, Measuring Market Risk, 2nd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Dowd, Chapter 2: Measures of Financial Risk

More information

Chapter 5 Univariate time-series analysis. () Chapter 5 Univariate time-series analysis 1 / 29

Chapter 5 Univariate time-series analysis. () Chapter 5 Univariate time-series analysis 1 / 29 Chapter 5 Univariate time-series analysis () Chapter 5 Univariate time-series analysis 1 / 29 Time-Series Time-series is a sequence fx 1, x 2,..., x T g or fx t g, t = 1,..., T, where t is an index denoting

More information

ELEMENTS OF MONTE CARLO SIMULATION

ELEMENTS OF MONTE CARLO SIMULATION APPENDIX B ELEMENTS OF MONTE CARLO SIMULATION B. GENERAL CONCEPT The basic idea of Monte Carlo simulation is to create a series of experimental samples using a random number sequence. According to the

More information

Implementing Models in Quantitative Finance: Methods and Cases

Implementing Models in Quantitative Finance: Methods and Cases Gianluca Fusai Andrea Roncoroni Implementing Models in Quantitative Finance: Methods and Cases vl Springer Contents Introduction xv Parti Methods 1 Static Monte Carlo 3 1.1 Motivation and Issues 3 1.1.1

More information

Bloomberg. Portfolio Value-at-Risk. Sridhar Gollamudi & Bryan Weber. September 22, Version 1.0

Bloomberg. Portfolio Value-at-Risk. Sridhar Gollamudi & Bryan Weber. September 22, Version 1.0 Portfolio Value-at-Risk Sridhar Gollamudi & Bryan Weber September 22, 2011 Version 1.0 Table of Contents 1 Portfolio Value-at-Risk 2 2 Fundamental Factor Models 3 3 Valuation methodology 5 3.1 Linear factor

More information

B. Maddah INDE 504 Discrete-Event Simulation. Output Analysis (3)

B. Maddah INDE 504 Discrete-Event Simulation. Output Analysis (3) B. Maddah INDE 504 Discrete-Event Simulation Output Analysis (3) Variance Reduction Variance reduction techniques (VRT) are methods to reduce the variance (i.e. increase precision) of simulation output

More information

Estimation of dynamic term structure models

Estimation of dynamic term structure models Estimation of dynamic term structure models Greg Duffee Haas School of Business, UC-Berkeley Joint with Richard Stanton, Haas School Presentation at IMA Workshop, May 2004 (full paper at http://faculty.haas.berkeley.edu/duffee)

More information

3. Monte Carlo Simulation

3. Monte Carlo Simulation 3. Monte Carlo Simulation 3.7 Variance Reduction Techniques Math443 W08, HM Zhu Variance Reduction Procedures (Chap 4.5., 4.5.3, Brandimarte) Usually, a very large value of M is needed to estimate V with

More information

A Quantitative Metric to Validate Risk Models

A Quantitative Metric to Validate Risk Models 2013 A Quantitative Metric to Validate Risk Models William Rearden 1 M.A., M.Sc. Chih-Kai, Chang 2 Ph.D., CERA, FSA Abstract The paper applies a back-testing validation methodology of economic scenario

More information

P1.T1. Foundations of Risk Management Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition Bionic Turtle FRM Study Notes

P1.T1. Foundations of Risk Management Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition Bionic Turtle FRM Study Notes P1.T1. Foundations of Risk Management Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com BODIE, CHAPTER

More information

STOCHASTIC DIFFERENTIAL EQUATION APPROACH FOR DAILY GOLD PRICES IN SRI LANKA

STOCHASTIC DIFFERENTIAL EQUATION APPROACH FOR DAILY GOLD PRICES IN SRI LANKA STOCHASTIC DIFFERENTIAL EQUATION APPROACH FOR DAILY GOLD PRICES IN SRI LANKA Weerasinghe Mohottige Hasitha Nilakshi Weerasinghe (148914G) Degree of Master of Science Department of Mathematics University

More information

P1.T1. Foundations of Risk. Bionic Turtle FRM Practice Questions. Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition

P1.T1. Foundations of Risk. Bionic Turtle FRM Practice Questions. Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition P1.T1. Foundations of Risk Bionic Turtle FRM Practice Questions Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition By David Harper, CFA FRM CIPM www.bionicturtle.com Bodie, Chapter 10:

More information

Math Computational Finance Double barrier option pricing using Quasi Monte Carlo and Brownian Bridge methods

Math Computational Finance Double barrier option pricing using Quasi Monte Carlo and Brownian Bridge methods . Math 623 - Computational Finance Double barrier option pricing using Quasi Monte Carlo and Brownian Bridge methods Pratik Mehta pbmehta@eden.rutgers.edu Masters of Science in Mathematical Finance Department

More information

F19: Introduction to Monte Carlo simulations. Ebrahim Shayesteh

F19: Introduction to Monte Carlo simulations. Ebrahim Shayesteh F19: Introduction to Monte Carlo simulations Ebrahim Shayesteh Introduction and repetition Agenda Monte Carlo methods: Background, Introduction, Motivation Example 1: Buffon s needle Simple Sampling Example

More information

Gamma. The finite-difference formula for gamma is

Gamma. The finite-difference formula for gamma is Gamma The finite-difference formula for gamma is [ P (S + ɛ) 2 P (S) + P (S ɛ) e rτ E ɛ 2 ]. For a correlation option with multiple underlying assets, the finite-difference formula for the cross gammas

More information

Math Computational Finance Option pricing using Brownian bridge and Stratified samlping

Math Computational Finance Option pricing using Brownian bridge and Stratified samlping . Math 623 - Computational Finance Option pricing using Brownian bridge and Stratified samlping Pratik Mehta pbmehta@eden.rutgers.edu Masters of Science in Mathematical Finance Department of Mathematics,

More information

Contents Critique 26. portfolio optimization 32

Contents Critique 26. portfolio optimization 32 Contents Preface vii 1 Financial problems and numerical methods 3 1.1 MATLAB environment 4 1.1.1 Why MATLAB? 5 1.2 Fixed-income securities: analysis and portfolio immunization 6 1.2.1 Basic valuation of

More information

Fast Convergence of Regress-later Series Estimators

Fast Convergence of Regress-later Series Estimators Fast Convergence of Regress-later Series Estimators New Thinking in Finance, London Eric Beutner, Antoon Pelsser, Janina Schweizer Maastricht University & Kleynen Consultants 12 February 2014 Beutner Pelsser

More information

Preprint: Will be published in Perm Winter School Financial Econometrics and Empirical Market Microstructure, Springer

Preprint: Will be published in Perm Winter School Financial Econometrics and Empirical Market Microstructure, Springer STRESS-TESTING MODEL FOR CORPORATE BORROWER PORTFOLIOS. Preprint: Will be published in Perm Winter School Financial Econometrics and Empirical Market Microstructure, Springer Seleznev Vladimir Denis Surzhko,

More information

Subject CS1 Actuarial Statistics 1 Core Principles. Syllabus. for the 2019 exams. 1 June 2018

Subject CS1 Actuarial Statistics 1 Core Principles. Syllabus. for the 2019 exams. 1 June 2018 ` Subject CS1 Actuarial Statistics 1 Core Principles Syllabus for the 2019 exams 1 June 2018 Copyright in this Core Reading is the property of the Institute and Faculty of Actuaries who are the sole distributors.

More information

Modelling the Sharpe ratio for investment strategies

Modelling the Sharpe ratio for investment strategies Modelling the Sharpe ratio for investment strategies Group 6 Sako Arts 0776148 Rik Coenders 0777004 Stefan Luijten 0783116 Ivo van Heck 0775551 Rik Hagelaars 0789883 Stephan van Driel 0858182 Ellen Cardinaels

More information

Credit Risk Modeling Using Excel and VBA with DVD O. Gunter Loffler Peter N. Posch. WILEY A John Wiley and Sons, Ltd., Publication

Credit Risk Modeling Using Excel and VBA with DVD O. Gunter Loffler Peter N. Posch. WILEY A John Wiley and Sons, Ltd., Publication Credit Risk Modeling Using Excel and VBA with DVD O Gunter Loffler Peter N. Posch WILEY A John Wiley and Sons, Ltd., Publication Preface to the 2nd edition Preface to the 1st edition Some Hints for Troubleshooting

More information

Statistical Models and Methods for Financial Markets

Statistical Models and Methods for Financial Markets Tze Leung Lai/ Haipeng Xing Statistical Models and Methods for Financial Markets B 374756 4Q Springer Preface \ vii Part I Basic Statistical Methods and Financial Applications 1 Linear Regression Models

More information

John Hull, Risk Management and Financial Institutions, 4th Edition

John Hull, Risk Management and Financial Institutions, 4th Edition P1.T2. Quantitative Analysis John Hull, Risk Management and Financial Institutions, 4th Edition Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1 Chapter 10: Volatility (Learning objectives)

More information

Introductory Econometrics for Finance

Introductory Econometrics for Finance Introductory Econometrics for Finance SECOND EDITION Chris Brooks The ICMA Centre, University of Reading CAMBRIDGE UNIVERSITY PRESS List of figures List of tables List of boxes List of screenshots Preface

More information

Faster solutions for Black zero lower bound term structure models

Faster solutions for Black zero lower bound term structure models Crawford School of Public Policy CAMA Centre for Applied Macroeconomic Analysis Faster solutions for Black zero lower bound term structure models CAMA Working Paper 66/2013 September 2013 Leo Krippner

More information

P2.T5. Market Risk Measurement & Management. Jorion, Value-at Risk: The New Benchmark for Managing Financial Risk, 3 rd Edition

P2.T5. Market Risk Measurement & Management. Jorion, Value-at Risk: The New Benchmark for Managing Financial Risk, 3 rd Edition P2.T5. Market Risk Measurement & Management Jorion, Value-at Risk: The New Benchmark for Managing Financial Risk, 3 rd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com

More information

Hull, Options, Futures & Other Derivatives

Hull, Options, Futures & Other Derivatives P1.T3. Financial Markets & Products Hull, Options, Futures & Other Derivatives Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM CIPM and Deepa Raju www.bionicturtle.com Hull, Chapter 1: Introduction

More information

Hull, Options, Futures & Other Derivatives Exotic Options

Hull, Options, Futures & Other Derivatives Exotic Options P1.T3. Financial Markets & Products Hull, Options, Futures & Other Derivatives Exotic Options Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1 Exotic Options Define and contrast exotic derivatives

More information

International Finance. Estimation Error. Campbell R. Harvey Duke University, NBER and Investment Strategy Advisor, Man Group, plc.

International Finance. Estimation Error. Campbell R. Harvey Duke University, NBER and Investment Strategy Advisor, Man Group, plc. International Finance Estimation Error Campbell R. Harvey Duke University, NBER and Investment Strategy Advisor, Man Group, plc February 17, 2017 Motivation The Markowitz Mean Variance Efficiency is the

More information

by Kian Guan Lim Professor of Finance Head, Quantitative Finance Unit Singapore Management University

by Kian Guan Lim Professor of Finance Head, Quantitative Finance Unit Singapore Management University by Kian Guan Lim Professor of Finance Head, Quantitative Finance Unit Singapore Management University Presentation at Hitotsubashi University, August 8, 2009 There are 14 compulsory semester courses out

More information

CPSC 540: Machine Learning

CPSC 540: Machine Learning CPSC 540: Machine Learning Monte Carlo Methods Mark Schmidt University of British Columbia Winter 2019 Last Time: Markov Chains We can use Markov chains for density estimation, d p(x) = p(x 1 ) p(x }{{}

More information

Chapter 2 Uncertainty Analysis and Sampling Techniques

Chapter 2 Uncertainty Analysis and Sampling Techniques Chapter 2 Uncertainty Analysis and Sampling Techniques The probabilistic or stochastic modeling (Fig. 2.) iterative loop in the stochastic optimization procedure (Fig..4 in Chap. ) involves:. Specifying

More information

UPDATED IAA EDUCATION SYLLABUS

UPDATED IAA EDUCATION SYLLABUS II. UPDATED IAA EDUCATION SYLLABUS A. Supporting Learning Areas 1. STATISTICS Aim: To enable students to apply core statistical techniques to actuarial applications in insurance, pensions and emerging

More information

P2.T8. Risk Management & Investment Management. Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition

P2.T8. Risk Management & Investment Management. Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition P2.T8. Risk Management & Investment Management Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Bodie,

More information

SIMULATION OF ELECTRICITY MARKETS

SIMULATION OF ELECTRICITY MARKETS SIMULATION OF ELECTRICITY MARKETS MONTE CARLO METHODS Lectures 15-18 in EG2050 System Planning Mikael Amelin 1 COURSE OBJECTIVES To pass the course, the students should show that they are able to - apply

More information

MFE Course Details. Financial Mathematics & Statistics

MFE Course Details. Financial Mathematics & Statistics MFE Course Details Financial Mathematics & Statistics Calculus & Linear Algebra This course covers mathematical tools and concepts for solving problems in financial engineering. It will also help to satisfy

More information

CPSC 540: Machine Learning

CPSC 540: Machine Learning CPSC 540: Machine Learning Monte Carlo Methods Mark Schmidt University of British Columbia Winter 2018 Last Time: Markov Chains We can use Markov chains for density estimation, p(x) = p(x 1 ) }{{} d p(x

More information

Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition

Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition P1.T3. Financial Markets & Products Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com

More information

P2.T8. Risk Management & Investment Management. Grinold, Chapter 14: Portfolio Construction

P2.T8. Risk Management & Investment Management. Grinold, Chapter 14: Portfolio Construction P2.T8. Risk Management & Investment Management Grinold, Chapter 14: Portfolio Construction Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Grinold, Chapter 14: Portfolio

More information

P1.T4.Valuation Tuckman, Chapter 5. Bionic Turtle FRM Video Tutorials

P1.T4.Valuation Tuckman, Chapter 5. Bionic Turtle FRM Video Tutorials P1.T4.Valuation Tuckman, Chapter 5 Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal

More information

VOLATILITY EFFECTS AND VIRTUAL ASSETS: HOW TO PRICE AND HEDGE AN ENERGY PORTFOLIO

VOLATILITY EFFECTS AND VIRTUAL ASSETS: HOW TO PRICE AND HEDGE AN ENERGY PORTFOLIO VOLATILITY EFFECTS AND VIRTUAL ASSETS: HOW TO PRICE AND HEDGE AN ENERGY PORTFOLIO GME Workshop on FINANCIAL MARKETS IMPACT ON ENERGY PRICES Responsabile Pricing and Structuring Edison Trading Rome, 4 December

More information

Appendix A (Pornprasertmanit & Little, in press) Mathematical Proof

Appendix A (Pornprasertmanit & Little, in press) Mathematical Proof Appendix A (Pornprasertmanit & Little, in press) Mathematical Proof Definition We begin by defining notations that are needed for later sections. First, we define moment as the mean of a random variable

More information

P2.T5. Market Risk Measurement & Management. BIS # 19, Messages from the Academic Literature on Risk Measuring for the Trading Books

P2.T5. Market Risk Measurement & Management. BIS # 19, Messages from the Academic Literature on Risk Measuring for the Trading Books P2.T5. Market Risk Measurement & Management BIS # 19, Messages from the Academic Literature on Risk Measuring for the Trading Books Bionic Turtle FRM Study Notes Reading 38 By David Harper, CFA FRM CIPM

More information

Computational Finance Least Squares Monte Carlo

Computational Finance Least Squares Monte Carlo Computational Finance Least Squares Monte Carlo School of Mathematics 2019 Monte Carlo and Binomial Methods In the last two lectures we discussed the binomial tree method and convergence problems. One

More information

Math Option pricing using Quasi Monte Carlo simulation

Math Option pricing using Quasi Monte Carlo simulation . Math 623 - Option pricing using Quasi Monte Carlo simulation Pratik Mehta pbmehta@eden.rutgers.edu Masters of Science in Mathematical Finance Department of Mathematics, Rutgers University This paper

More information

Market interest-rate models

Market interest-rate models Market interest-rate models Marco Marchioro www.marchioro.org November 24 th, 2012 Market interest-rate models 1 Lecture Summary No-arbitrage models Detailed example: Hull-White Monte Carlo simulations

More information

SYLLABUS. IEOR E4728 Topics in Quantitative Finance: Inflation Derivatives

SYLLABUS. IEOR E4728 Topics in Quantitative Finance: Inflation Derivatives SYLLABUS IEOR E4728 Topics in Quantitative Finance: Inflation Derivatives Term: Summer 2007 Department: Industrial Engineering and Operations Research (IEOR) Instructor: Iraj Kani TA: Wayne Lu References:

More information

MFE Course Details. Financial Mathematics & Statistics

MFE Course Details. Financial Mathematics & Statistics MFE Course Details Financial Mathematics & Statistics FE8506 Calculus & Linear Algebra This course covers mathematical tools and concepts for solving problems in financial engineering. It will also help

More information

Management Services Reviewer by Ma. Elenita Balatbat-Cabrera

Management Services Reviewer by Ma. Elenita Balatbat-Cabrera Course Name: Course Title: Instructors: Required Text: Course Description: XMASREV Management Services Review David, Dimalanta and Morales Management Services Reviewer by Ma. Elenita Balatbat-Cabrera This

More information

Financial Risk Management and Governance Other VaR methods. Prof. Hugues Pirotte

Financial Risk Management and Governance Other VaR methods. Prof. Hugues Pirotte Financial Risk Management and Governance Other VaR methods Prof. ugues Pirotte Idea of historical simulations Why rely on statistics and hypothetical distribution?» Use the effective past distribution

More information

Jaime Frade Dr. Niu Interest rate modeling

Jaime Frade Dr. Niu Interest rate modeling Interest rate modeling Abstract In this paper, three models were used to forecast short term interest rates for the 3 month LIBOR. Each of the models, regression time series, GARCH, and Cox, Ingersoll,

More information

Fixed Income and Risk Management

Fixed Income and Risk Management Fixed Income and Risk Management Fall 2003, Term 2 Michael W. Brandt, 2003 All rights reserved without exception Agenda and key issues Pricing with binomial trees Replication Risk-neutral pricing Interest

More information

Volume 37, Issue 2. Handling Endogeneity in Stochastic Frontier Analysis

Volume 37, Issue 2. Handling Endogeneity in Stochastic Frontier Analysis Volume 37, Issue 2 Handling Endogeneity in Stochastic Frontier Analysis Mustafa U. Karakaplan Georgetown University Levent Kutlu Georgia Institute of Technology Abstract We present a general maximum likelihood

More information

Monte Carlo Methods in Finance

Monte Carlo Methods in Finance Monte Carlo Methods in Finance Peter Jackel JOHN WILEY & SONS, LTD Preface Acknowledgements Mathematical Notation xi xiii xv 1 Introduction 1 2 The Mathematics Behind Monte Carlo Methods 5 2.1 A Few Basic

More information

Accelerated Option Pricing Multiple Scenarios

Accelerated Option Pricing Multiple Scenarios Accelerated Option Pricing in Multiple Scenarios 04.07.2008 Stefan Dirnstorfer (stefan@thetaris.com) Andreas J. Grau (grau@thetaris.com) 1 Abstract This paper covers a massive acceleration of Monte-Carlo

More information

Quantitative Finance Investment Advanced Exam

Quantitative Finance Investment Advanced Exam Quantitative Finance Investment Advanced Exam Important Exam Information: Exam Registration Order Study Notes Introductory Study Note Case Study Past Exams Updates Formula Package Table Candidates may

More information

The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay. Solutions to Final Exam

The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay. Solutions to Final Exam The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (40 points) Answer briefly the following questions. 1. Consider

More information

A Poor Man s Guide. Quantitative Finance

A Poor Man s Guide. Quantitative Finance Sachs A Poor Man s Guide To Quantitative Finance Emanuel Derman October 2002 Email: emanuel@ederman.com Web: www.ederman.com PoorMansGuideToQF.fm September 30, 2002 Page 1 of 17 Sachs Summary Quantitative

More information

MFE8825 Quantitative Management of Bond Portfolios

MFE8825 Quantitative Management of Bond Portfolios MFE8825 Quantitative Management of Bond Portfolios William C. H. Leon Nanyang Business School March 18, 2018 1 / 150 William C. H. Leon MFE8825 Quantitative Management of Bond Portfolios 1 Overview 2 /

More information

Market Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk

Market Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk Market Risk: FROM VALUE AT RISK TO STRESS TESTING Agenda The Notional Amount Approach Price Sensitivity Measure for Derivatives Weakness of the Greek Measure Define Value at Risk 1 Day to VaR to 10 Day

More information

Simulation Wrap-up, Statistics COS 323

Simulation Wrap-up, Statistics COS 323 Simulation Wrap-up, Statistics COS 323 Today Simulation Re-cap Statistics Variance and confidence intervals for simulations Simulation wrap-up FYI: No class or office hours Thursday Simulation wrap-up

More information

The Fixed Income Valuation Course. Sanjay K. Nawalkha Natalia A. Beliaeva Gloria M. Soto

The Fixed Income Valuation Course. Sanjay K. Nawalkha Natalia A. Beliaeva Gloria M. Soto Dynamic Term Structure Modeling The Fixed Income Valuation Course Sanjay K. Nawalkha Natalia A. Beliaeva Gloria M. Soto Dynamic Term Structure Modeling. The Fixed Income Valuation Course. Sanjay K. Nawalkha,

More information

Lecture 3: Factor models in modern portfolio choice

Lecture 3: Factor models in modern portfolio choice Lecture 3: Factor models in modern portfolio choice Prof. Massimo Guidolin Portfolio Management Spring 2016 Overview The inputs of portfolio problems Using the single index model Multi-index models Portfolio

More information

Are Your Risk Tolerance and LDI Glide Path in Sync?

Are Your Risk Tolerance and LDI Glide Path in Sync? Are Your Risk Tolerance and LDI Glide Path in Sync? Wesley Phoa, LDI Portfolio Manager, Capital Group Luke Farrell, LDI Investment Specialist, Capital Group The Plan Sponsor s Mission Dual accountability

More information

Chapter 3. Dynamic discrete games and auctions: an introduction

Chapter 3. Dynamic discrete games and auctions: an introduction Chapter 3. Dynamic discrete games and auctions: an introduction Joan Llull Structural Micro. IDEA PhD Program I. Dynamic Discrete Games with Imperfect Information A. Motivating example: firm entry and

More information

Choice Probabilities. Logit Choice Probabilities Derivation. Choice Probabilities. Basic Econometrics in Transportation.

Choice Probabilities. Logit Choice Probabilities Derivation. Choice Probabilities. Basic Econometrics in Transportation. 1/31 Choice Probabilities Basic Econometrics in Transportation Logit Models Amir Samimi Civil Engineering Department Sharif University of Technology Primary Source: Discrete Choice Methods with Simulation

More information

P2.T7. Operational & Integrated Risk Management. Michael Crouhy, Dan Galai and Robert Mark, The Essentials of Risk Management, 2nd Edition

P2.T7. Operational & Integrated Risk Management. Michael Crouhy, Dan Galai and Robert Mark, The Essentials of Risk Management, 2nd Edition P2.T7. Operational & Integrated Risk Management Bionic Turtle FRM Practice Questions Michael Crouhy, Dan Galai and Robert Mark, The Essentials of Risk Management, 2nd Edition By David Harper, CFA FRM CIPM

More information

Handbook of Financial Risk Management

Handbook of Financial Risk Management Handbook of Financial Risk Management Simulations and Case Studies N.H. Chan H.Y. Wong The Chinese University of Hong Kong WILEY Contents Preface xi 1 An Introduction to Excel VBA 1 1.1 How to Start Excel

More information

Three Components of a Premium

Three Components of a Premium Three Components of a Premium The simple pricing approach outlined in this module is the Return-on-Risk methodology. The sections in the first part of the module describe the three components of a premium

More information

List of tables List of boxes List of screenshots Preface to the third edition Acknowledgements

List of tables List of boxes List of screenshots Preface to the third edition Acknowledgements Table of List of figures List of tables List of boxes List of screenshots Preface to the third edition Acknowledgements page xii xv xvii xix xxi xxv 1 Introduction 1 1.1 What is econometrics? 2 1.2 Is

More information