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Risk Management and Simulation

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内容提示: Risk Management and SimulationK11 622_FM.indd 15/13/1 3 3:08 PM K11 622_FM.indd 25/13/1 3 3:08 PM Risk Management and SimulationAparna GuptaK11 622_FM.indd 35/13/1 3 3:08 PM MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® soft-ware or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedag...

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Risk Management and SimulationK11 622_FM.indd 15/13/1 3 3:08 PM K11 622_FM.indd 25/13/1 3 3:08 PM Risk Management and SimulationAparna GuptaK11 622_FM.indd 35/13/1 3 3:08 PM MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® soft-ware or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742© 2014 by Taylor & Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group, an Informa businessNo claim to original U.S. Government worksPrinted on acid-free paperVersion Date: 20130509International Standard Book Number-13: 978-1-4398-3594-4 (Hardback)This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit-ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.Library of Congress Cataloging‑in‑Publication DataGupta, Aparna.Risk management and simulation / Aparna Gupta.pages cmIncludes bibliographical references and index.ISBN 978-1-4398-3594-4 (alk. paper)1. Risk management. 2. Risk management--Simulation methods. I. Title. HD61.G86 2013338.5--dc23 2013007014Visit the Taylor & Francis Web site athttp://www.taylorandfrancis.comand the CRC Press Web site athttp://www.crcpress.comK11 622_FM.indd 45/13/1 3 3:08 PM To my parents, Amar-Sneh ContentsIRisk and Regulation1Defining Risk1.1Types of Risk355678. . . . . . . . . . . . . . . . . . . . . . . . . .Pure Risk . . . . . . . . . . . . . . . . . . . .Speculative Risk . . . . . . . . . . . . . . . .Classification of Pure Risk . . . . . . . . . . . . . . . .Classification of Speculative Risk . . . . . . . . . . . .Getting Started with Modeling Risk . . . . . . . . . . . . . .1.2.1Random Variable and Probability1.2.1.1Summarizing Random Variables . . . . . . .1.2.1.2Several Random Variables and Correlation .1.2.1.3Conditional Probability . . . . . . . . . . . .1.2.2Specific Models of Risk1.2.2.1Normal Distribution . . . . . . . . . . . . . .1.2.2.2Uniform Distribution1.2.2.3Central Limit Theorem . . . . . . . . . . . .1.2.2.4Binomial Distribution . . . . . . . . . . . . .1.2.2.5Poisson Distribution . . . . . . . . . . . . . .1.2.2.6Exponential Distribution . . . . . . . . . . .1.2.2.7Weibull Distribution . . . . . . . . . . . . . .1.2.2.8Lognormal Distribution . . . . . . . . . . . .1.2.2.9Chi-Square Distribution . . . . . . . . . . . .1.2.2.10 Gamma Distribution . . . . . . . . . . . . . .MATLABTools for DistributionsSummary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Questions and Exercises . . . . . . . . . . . . . . . . . . . . .1.1.0.11.1.0.21.1.11.1.21.212121416171919202123232426272729303031. . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . .1.31.41.5. . . . . . . . . . . . . .2Framework for Risk Management2.1How to Handle Risk . . . . . . . . . . . . . . . . . . . . . . .2.1.1The Risk Management Framework . . . . . . . . . . .2.1.2Risk Preference vs. Risk Aversion . . . . . . . . . . . .2.1.2.1Normative vs. Behavioral Choice . . . . . . .2.1.3Risk Measures. . . . . . . . . . . . . . . . . . . . . .2.1.4Risk Management. . . . . . . . . . . . . . . . . . . .2.1.5Elements of the Framework . . . . . . . . . . . . . . .3536374043454850vii viiiContents2.1.5.12.1.5.22.1.5.32.1.5.4Avoid . . . . . . . . . . . . . . . . . . . . . .Mitigate. . . . . . . . . . . . . . . . . . . .Transfer . . . . . . . . . . . . . . . . . . . . .Keep. . . . . . . . . . . . . . . . . . . . . .Example Contexts to Apply the Framework . . . . . . . . . .2.2.1Analysis Using Central Measures . . . . . . . . . . . .2.2.2Tail Analysis . . . . . . . . . . . . . . . . . . . . . . .2.2.3Scenario Analysis . . . . . . . . . . . . . . . . . . . . .2.2.4Stress Testing . . . . . . . . . . . . . . . . . . . . . . .MATLAB Tools for Risk Measures . . . . . . . . . . . . . . .Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Questions and Exercises . . . . . . . . . . . . . . . . . . . . .5151535454555658596061612.22.32.42.53Regulations and Risk Management3.1Regulations Overview3.1.1Regulatory Evolution for Banking3.1.2Regulatory Evolution for Investment Banking . . . . .3.1.3Regulatory Evolution for Insurance . . . . . . . . . . .3.2Regulations and Banking. . . . . . . . . . . . . . . . . . . .3.3Regulations and Investment Banking3.4Regulations and Insurance3.5Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.6Questions and Exercises . . . . . . . . . . . . . . . . . . . . .65666771737480848686. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .IIModeling and Simulation of Risk4Principles of Simulation and Generating Random Variates4.1Principles of Simulation . . . . . . . . . . . . . . . . . . . . .4.1.1What Is Simulation? . . . . . . . . . . . . . . . . . . .4.2Random Number Generation . . . . . . . . . . . . . . . . . .4.2.1Linear Congruential Generator . . . . . . . . . . . . .4.2.2Lagged Fibonacci Generator . . . . . . . . . . . . . . .4.3Generation of Discrete Random Variates4.3.1n-Outcome Random Variate . . . . . . . . . . . . . . .4.3.2Poisson Random Variate . . . . . . . . . . . . . . . . .4.4Generation of Continuous Random Variates . . . . . . . . . .4.4.1Inverse Transform Method . . . . . . . . . . . . . . . .4.4.2Acceptance-Rejection Method . . . . . . . . . . . . . .4.4.3Normal Random Variate . . . . . . . . . . . . . . . . .4.4.3.1Box-Muller Method . . . . . . . . . . . . . .4.4.3.2Polar-Marsaglia Method . . . . . . . . . . . .4.4.3.3Generation of Multi-Variate Normal . . . . .4.4.4Chi-Square and Other Random Variates . . . . . . . .4.5Testing Random Variates. . . . . . . . . . . . . . . . . . . .4.5.1Testing for Independence of Random Numbers . . . .939394969797989899100100101103104104106107107108. . . . . . . . . . . Contentsix4.5.1.1Testing for Correctness of Distribution . . . . . . . . .4.5.2.1The χ2Goodness of Fit Test . . . . . . . . .4.5.2.2Kolmogorov-Smirnov Test . . . . . . . . . . .Validation of Model. . . . . . . . . . . . . . . . . . . . . . .4.6.1Techniques for Model Verification . . . . . . . . . . . .4.6.2Techniques for Model Validation . . . . . . . . . . . .Output Analysis. . . . . . . . . . . . . . . . . . . . . . . . .4.7.1Descriptive Output Analysis4.7.1.1Designing Simulation Run by Properties ofEs-timators . . . . . . . . . . . . . . . . . . . . .4.7.2Inferential Output Analysis . . . . . . . . . . . . . . .MATLAB Tools for Simulation . . . . . . . . . . . . . . . . .Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.10 Questions and Exercises . . . . . . . . . . . . . . . . . . . . .Shuffling Procedure . . . . . . . . . . . . . .1091101101121131141151171184.5.24.64.7. . . . . . . . . . . . . .1191201211221224.84.95Modeling Risk Evolving over Time5.1Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . .5.2Discrete-Time Evolution of Risk5.2.1Discrete-Time Markov Chains . . . . . . . . . . . . . .5.2.2Simple Random Walk . . . . . . . . . . . . . . . . . .5.2.3Geometric Random Walk . . . . . . . . . . . . . . . .5.3Continuous-Time Evolution of Risk5.3.1Continuous-Time Markov Chains . . . . . . . . . . . .5.3.2Poisson Process . . . . . . . . . . . . . . . . . . . . . .5.3.3Birth-Death Process . . . . . . . . . . . . . . . . . . .5.3.4Markov Process . . . . . . . . . . . . . . . . . . . . . .5.3.5Gaussian Process . . . . . . . . . . . . . . . . . . . . .5.3.6Brownian Motion . . . . . . . . . . . . . . . . . . . . .5.3.6.1Approximating Brownian Motion by a Ran-dom Walk5.3.6.2Convergence of Random Variables . . . . . .5.3.6.3Properties of the Wiener Process . . . . . . .5.3.7Brownian Motion with Drift and Geometric BrownianMotion. . . . . . . . . . . . . . . . . . . . . . . . . .5.3.8Additional Concepts for Stochastic Processes . . . . .5.4Modeling Correlation. . . . . . . . . . . . . . . . . . . . . .5.4.1Correlated Brownian Motion . . . . . . . . . . . . . .5.4.2Copulas for Correlation . . . . . . . . . . . . . . . . .5.5MATLAB Tools for Modeling Risk Evolving over Time5.6Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.7Questions and Exercises . . . . . . . . . . . . . . . . . . . . .127127128129133135136136138140141142144. . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .145146147149150152152153156156157. . . xContents6Building and Solving Models of Risk6.1Deterministic Financial Modeling6.2Introducing Stochasticity in the Modeling . . . . . . . . . . .6.3Defining New Integrals. . . . . . . . . . . . . . . . . . . . .6.3.1Ito Integral . . . . . . . . . . . . . . . . . . . . . . . .6.3.2Properties of the Ito Integral . . . . . . . . . . . . . .6.3.3Chain Rule of Ito Calculus - The Ito Formula . . . . .6.4Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . .6.4.1Solving the Model Exactly . . . . . . . . . . . . . . . .6.5Solving Models Using Simulation . . . . . . . . . . . . . . . .6.5.1The Euler Method for Solving Differential Equations .6.5.2Evaluating Simulation Solutions6.5.2.1Convergence Properties of Solutions . . . . .6.5.2.2Error Analysis - Absolute Error Criterion . .6.5.2.3Error Analysis - Mean Error Criterion . . . .6.5.3Higher Order Methods . . . . . . . . . . . . . . . . . .6.5.3.1Trapezoidal Method . . . . . . . . . . . . . .6.6Estimating Parameters. . . . . . . . . . . . . . . . . . . . .6.6.1Geometric Brownian Motion6.6.2Method of Maximum Likelihood . . . . . . . . . . . .6.6.3Method of Quasi-Maximum Likelihood . . . . . . . . .6.6.4Method of Moments . . . . . . . . . . . . . . . . . . .6.6.4.1Ornstein-Uhlenbeck Process6.7MATLAB Tools for Building and Solving Models of Risk6.8Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.9Questions and Exercises . . . . . . . . . . . . . . . . . . . . .161161164166166168170171172175175180181181183186186188188189191192192193194194. . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . .. .IIIRisk Management7Managing Equity Market Risk7.1Mitigating Equity Risk7.1.1Portfolio Diversification . . . . . . . . . . . . . . . . .7.1.1.1Classical Mean-Variance Reward-Risk Mea-sures. . . . . . . . . . . . . . . . . . . . . .7.1.1.2Dynamic Investment Strategy7.1.2Portfolio Optimization . . . . . . . . . . . . . . . . . .7.1.2.1Optimum Risk-Return Trade-Off . . . . . . .7.1.2.2Simulation Analysis for Portfolio Decisions .7.2Transferring Equity Risk. . . . . . . . . . . . . . . . . . . .7.2.1Option Pricing - Black-Scholes-Merton Approach . . .7.2.1.1Solving Black-Scholes Partial Differential Equa-tion . . . . . . . . . . . . . . . . . . . . . . .7.2.1.2Estimating Option Price by Simulation . . .7.2.1.3Making Model Simpler - Binomial Tree Ap-proach. . . . . . . . . . . . . . . . . . . . .199200200. . . . . . . . . . . . . . . . . . . . .201203205205208210211. . . . . . . .216219220 Contentsxi7.2.2Implied Volatility and Calibration for Risk-NeutralPricing. . . . . . . . . . . . . . . . . . . . . . . . . .Sensitivity to the Parameters . . . . . . . . . . . . . .Exotic Options . . . . . . . . . . . . . . . . . . . . . .American Options . . . . . . . . . . . . . . . . . . . .Generalizing the Models in Black-Scholes-Merton . . .7.2.6.1Constant Elasticity of Variance (CEV) Model7.2.6.2Model for Several Correlated Stocks . . . . .7.2.6.3Extensions in Option Pricing - StochasticVolatility . . . . . . . . . . . . . . . . . . . .7.2.6.4Large Sudden Changes in Prices - Jump Dif-fusion Model . . . . . . . . . . . . . . . . . .Equity Hedging Strategies. . . . . . . . . . . . . . . . . . .7.3.1Static Hedging Strategies . . . . . . . . . . . . . . . .7.3.2Optimal Hedge Problem . . . . . . . . . . . . . . . . .7.3.3Dynamic Hedging Strategies . . . . . . . . . . . . . . .MATLAB Tools for Equity and Portfolios . . . . . . . . . . .Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Questions and Exercises . . . . . . . . . . . . . . . . . . . . .2232252292332352362377.2.37.2.47.2.57.2.62392442462472532542582582597.37.47.57.68Managing Interest Rates and Other Market Risks8.1Pricing Fixed Income Instruments8.1.1Bond Pricing . . . . . . . . . . . . . . . . . . . . . . .8.1.2Stochastic Interest Rate Models . . . . . . . . . . . . .8.1.2.1Short Rate Models . . . . . . . . . . . . . . .8.1.2.2Multi-Factor Interest Rate Models . . . . . .8.1.2.3Other Fixed-Income Instruments . . . . . . .8.1.3Simulation of Interest Rate Models . . . . . . . . . . .8.2Interest-Rate Risk Management8.2.1Interest-Rate Sensitivity in Fixed-Income Instruments8.2.1.1Bond Portfolio Immunization . . . . . . . . .8.2.2Interest-Rate Derivatives8.2.3Interest-Rate Hedging Strategies . . . . . . . . . . . .8.3Managing Commodities Risk8.3.1Modeling Commodity Spot Prices8.3.1.1Energy, Electricity, and Weather Risk . . . .8.3.2Management of Commodity Risk . . . . . . . . . . . .8.3.2.1Commodity Futures and Other Derivatives .8.4Managing Foreign Exchange Risk8.4.1Models for Spot and Forward Exchange Rates . . . . .8.4.2Currency Derivatives . . . . . . . . . . . . . . . . . . .8.5Value-at-Risk and Stress Testing for Market Risk Management 3128.6MATLAB Tools for Fixed Income, Commodities, and ExchangeRates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.7Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265266266270270276277279280281285287291294297299301304306309310. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . .317318 xiiContents8.8Questions and Exercises . . . . . . . . . . . . . . . . . . . . .3189Credit Risk Management9.1Retail Credit Risk9.1.1Measuring Retail Credit Risk . . . . . . . . . . . . . .9.1.1.1Credit Scoring Methods . . . . . . . . . . . .9.1.2Retail Credit Risk Management . . . . . . . . . . . . .9.2Commercial Credit Risk . . . . . . . . . . . . . . . . . . . . .9.2.1Credit Rating System . . . . . . . . . . . . . . . . . .9.2.1.1Risk Assessment by Credit Rating Migration9.2.2Models for Credit Risk . . . . . . . . . . . . . . . . . .9.2.2.1Structural Model of Credit Risk . . . . . . .9.2.2.2Reduced-Form Model of Credit Risk . . . . .9.3Credit Risk Hedging Instruments9.3.1Single-Name Credit Derivatives . . . . . . . . . . . . .9.3.1.1Credit Default Swaps . . . . . . . . . . . . .9.3.1.2Spread Options . . . . . . . . . . . . . . . . .9.3.2Multi-Name Credit Derivatives . . . . . . . . . . . . .9.3.2.1Collateralized Debt Obligations9.4Portfolio Credit Risk Management . . . . . . . . . . . . . . .9.5MATLAB Tools for Credit Risk9.6Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.7Questions and Exercises . . . . . . . . . . . . . . . . . . . . .325326329332336340341342347348350351354355357357358361364364365. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . .10 Strategic, Business, and Operational Risk Management10.1 Strategic Risk Management . . . . . . . . . . . . . . . . . . .10.1.1 Objective of Strategic Risk Management . . . . . . . .10.1.2 Approaches for Strategic Risk Management . . . . . .10.2 Business Risk Management10.3 Asset-Liability Management10.3.1 Components of Asset-Liability Management . . . . . .10.3.2 Risk Management in ALM10.3.2.1 Gap Analysis . . . . . . . . . . . . . . . . . .10.3.2.2 Cumulative Gap Analysis . . . . . . . . . . .10.3.2.3 Duration Gap Analysis and Gap Convexity .10.3.2.4 Dynamic Gap and Long-Term Value at RiskAnalysis10.3.2.5 Scenario Analysis and Stress Testing . . . . .10.4 Operational Risk Management10.4.1 Assessing Operational Risk . . . . . . . . . . . . . . .10.4.2 Managing Operational Risk . . . . . . . . . . . . . . .10.4.2.1 Risk Measures for Operational Risk . . . . .10.4.2.2 Operational Risk Management Strategy . . .10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.6 Questions and Exercises . . . . . . . . . . . . . . . . . . . . .371371373374378380382385385387387. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .388390391393395396397399399. . . . . . . . . . . . . . . . . Contentsxiii11 Risk Management Using Insurance11.1 Basic Concepts of Insurance11.2 Principle behind Insurance11.2.1 Characteristics of Insurance and Insurable Risk . . . .11.2.1.1 Law of Large Numbers11.2.1.2 Requirement of Insurable Risk . . . . . . . .11.3 Types of Insurance. . . . . . . . . . . . . . . . . . . . . . .11.3.1 Benefits and Cost of Insurance to Society . . . . . . .11.4 Risk Management Framework for Pure Risk11.4.1 Pure Risk Evaluation11.4.2 Risk Management Strategies for Pure Risk11.4.3 Modeling Individual Mortality Risk11.5 Risk Management by Insurers11.5.1 Pricing, Investment, and Asset-Liability Management11.5.2 Risk Management, Securitization, and Reinsurance . .11.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.7 Questions and Exercises . . . . . . . . . . . . . . . . . . . . .405407409410410413414416417420423426427427431433434. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . .. . . . . . . . . . . . . . . . .IVAdvanced Simulation12 Advanced Simulation Topics12.1 Variance Reduction Techniques . . . . . . . . . . . . . . . . .12.1.1 Control Variates . . . . . . . . . . . . . . . . . . . . .12.1.2 Antithetic Variables . . . . . . . . . . . . . . . . . . .12.1.3 Stratified Sampling . . . . . . . . . . . . . . . . . . . .12.1.4 Latin Hypercube Sampling12.1.5 Importance Sampling12.2 Simulation-Based Optimization . . . . . . . . . . . . . . . . .12.2.1 Challenges of Simulation-Based Optimization . . . . .12.2.2 Simulation Optimization Methodologies . . . . . . . .12.2.2.1 Gradient-Based Methods . . . . . . . . . . .12.2.2.2 Simulated Annealing . . . . . . . . . . . . . .12.2.2.3 Tabu Search . . . . . . . . . . . . . . . . . .12.2.2.4 Scatter Search . . . . . . . . . . . . . . . . .12.2.2.5 Evolutionary Strategies . . . . . . . . . . . .12.2.2.6 Particle Swarm Optimization . . . . . . . . .12.3 MATLAB Tools for Variance Reduction and Optimization12.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12.5 Questions and Exercises . . . . . . . . . . . . . . . . . . . . .441442444447450453454455458460463464466467467469471471472. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . ..Bibliography479Index485 List ofFigures1.11.2Classification structure for types of risk. . . . . . . . . . . . .(a) Probability density function for normal distribution. (b)Cumulative distribution function for normal distribution.(a) Probability density function for uniform distribution. (b)Cumulative distribution function for uniform distribution. . .Display of Central Limit Theorem. (a) N = 1,000 (b) N = 5,000(c) N = 10,000 (d) N = 100,000.(a) Probability mass function for binomial distribution. (b) Cu-mulative distribution function for binomial distribution. . . .(a) Probability mass function for Poisson distribution. (b) Cu-mulative distribution function for Poisson distribution. . . . .(a) Probability density function for exponential distribution.(b) Cumulative distribution function for exponential distribu-tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(a) Probability density function for Weibull distribution. (b)Cumulative distribution function for Weibull distribution. . .(a) Probability density function for lognormal distribution. (b)Cumulative distribution function for lognormal distribution. .1.10 (a) Probability density function for Chi-square distribution. (b)Cumulative distribution function for Chi-square distribution.1.11 (a) Probability density function for gamma distribution. (b)Cumulative distribution function for gamma distribution.7. .201.3211.4. . . . . . . . . . . . . . . .221.5231.6241.7251.8261.92728. .292.12.2The overall flowchart for the Risk Management Process. . . .(a) Plot of the exponential, constant absolute risk aversion(CARA) utility function. (b) Plot of the power, constant rela-tive risk aversion (CRRA) utility function. . . . . . . . . . . .Plot of the loss-aversion utility, an example of behavioral utilityfunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Display of Value-at-Risk and Conditional Value-at-Risk. . . .(a) Plot of mean and standard deviation of combined risk for arange of weights on two individual risks. (b) Plot of mean andfirst percentile of combined risk for a range of weights on twoindividual risks, assuming normal distribution of combined risk.38422.344472.42.552xv xviList of Figures2.6(a) Unimodal distribution of risk and its central tendencies. (b)Bimodal distribution of risk. . . . . . . . . . . . . . . . . . . .Probability density plot displaying light-tail and heavy-tail. .(a) Probability plot for a dataset that matches the light-tailednormal distribution model. (b) Probability plot for a datasetthat displays heavy-tail deviations from the normal distributionmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55572.72.8583.1The table shows the main regulatory agencies for the Groupof Eight (G8) countries for banking, investment banking, andinsurance industry, as of 2012. . . . . . . . . . . . . . . . . . .674.14.24.34.4The guideline for how to structure a simulation study. . . . .The stages to build the simulation model. . . . . . . . . . . .N-outcome discrete random variate generation. . . . . . . . .A pictorial depiction of the principle behind the inverse trans-form method. . . . . . . . . . . . . . . . . . . . . . . . . . . .A pictorial depiction of the principle behind the acceptance-rejection method. . . . . . . . . . . . . . . . . . . . . . . . . .A pictorial depiction of the construction of the Polar-Marsagliamethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Display of output from a linear congruential generator. (a) 1000numbers generated lie on three parallel lines. (b) The 1000 num-bers after implementing shuffling. . . . . . . . . . . . . . . . .Display of Probability Plots. (a) Lognormal probability plot.(b) Weibull probability plot. . . . . . . . . . . . . . . . . . . .Display of Validation Cost vs. Risk Cost Curve. . . . . . . . .9596991014.51024.61054.71084.81101164.95.1(a) A typical sample realization for a discrete-time stochas-tic process. (b) The binomial tree example of a discrete timestochastic process.. . . . . . . . . . . . . . . . . . . . . . . .A pictorial depiction of states of a Markov chain, transitionsfollowing Markovian property, and transition probabilities. . .(a) Three realizations of a simple random walk. (b) Three re-alizations of simple symmetric random walk. (c) Three realiza-tions of general random walk. (d) Three realizations of simplerandom walk with upper barrier set at 10. . . . . . . . . . . .Three sample path realizations of a Poisson process with variedlevels of λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(a) Three sample path realizations of an Ornstein-Uhlenbeckprocess. (b) Three sample path realizations of Ornstein-Uhlenbeck process with different risk levels. . . . . . . . . . .Three sample path realizations for the standard Brownian mo-tion or the Wiener process.. . . . . . . . . . . . . . . . . . .1295.21305.31345.41395.51435.6145 List of Figuresxvii5.7(a) Three sample path realizations for the standard Brownianmotion or Wiener process with drift. (b) Three sample pathrealizations for geometric Brownian motion. . . . . . . . . . .(a) Marginal CDF for first random variable, chosen to be betadistribution with parameters, a=2, b=2. (b) Marginal CDF forsecond random variable, chosen to be Weibull distribution withparameters, a=0.15, b=0.8. (c) Scatter plot of 1000 randomvariates generated by Gaussian copula with ρ = 0.7. (d) Scatterplot of 100 random variates generated using t-copula with ρ =0.7, ν = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1505.81556.1Applying different frequency of interest rate accrual for a risk-free investment. (a) Annual accrual applied for five years. (b)Monthly accrual applied for five years. (c) Daily accrual appliedfor five years. (d) Hourly accrual applied for five years. . . . .Comparison of exact and numerical solutions for an exampleordinary differential equation. . . . . . . . . . . . . . . . . . .Comparison of exact and numerical solution for an examplestochastic differential equation. . . . . . . . . . . . . . . . . .Comparison of distributional properties of the exact solution(in left panel) and numerical solution (in the right panel) forthe example stochastic differential equation. . . . . . . . . . .1636.21766.31786.41797.1Plot of risk-reward trade-off of individual stocks. The combi-nation of the individual stock helps mitigate the risk in thefrontier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Plot of mean and standard deviation of two stock returns. Thecorrelation of ρ = 1 and −1 define the right and left extents ofthe region, respectively.. . . . . . . . . . . . . . . . . . . . .Plot of mean and standard deviation space spanned by returnon portfolio of stocks. For a choice of expected portfolio returnthreshold, the optimum risk-return trade-off is made on theleft most feasible points. The dashed curve is the efficient risk-return trade-off points, or the efficient frontier.Simulation analysis of risk-reward of a portfolio based on equityreturns scenarios and parametric scenarios.(a) Display of pay-off and profit curve for a plain-vanilla Euro-pean call option with strike price, K=$80. (b) Display of pay-offand profit curve for a plain-vanilla European put option withstrike price, K=$80.. . . . . . . . . . . . . . . . . . . . . . .(a) Display of pay-off and profit curve for a short position ina plain-vanilla European call option with strike price, K=$80.(b) Display of pay-off and profit curve for a short position in aplain-vanilla European put option with strike price, K=$80. .2017.22047.3. . . . . . . .2077.4. . . . . . . . . .2097.52127.6213 xviiiList of Figures7.7(a) Display of pay-off and price curve for a plain-vanilla Euro-pean call option with strike price, K=$35, σ = 23%, T−t = 1/2year, and short-term interest rate of r = 2%. (b) Display of pay-off and price curve for a plain-vanilla European put option withthe same set of parameters as the call option. . . . . . . . . .(a) Single period binomial tree model for stock price evolution.(b) Multi-period binomial tree model for stock price evolution.Implied volatility obtained from the Black-Scholes option pric-ing formula for plain-vanilla European call option with stockprice, St =$35, σ = 23%, T − t = 1/2 year, and short-terminterest rate of r = 2%.. . . . . . . . . . . . . . . . . . . . .7.10 The chart marks the dependence of European and Americanvanilla call and put option prices on parameters that determinethe price.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.11 Trajectories for valuation of a compound option.7.12 Pay-offs of an up-and-out barrier call option and a down-and-inbarrier call option. . . . . . . . . . . . . . . . . . . . . . . . .7.13 Pictorial display of algorithm to determine the price of anAmerican option using the binomial tree model. . . . . . . . .7.14 Monthly observations of VIX index from January 2005 throughmid-2012. The variability in the stock market is captured inthis index through the financial crises of 2008 and euro crisisevolving through 2011-2012. . . . . . . . . . . . . . . . . . . .7.15 (a) Profit and individual positions ofa protective put. (b) Profitand individual positions of a reverse protective put. . . . . . .7.16 (a) Profit and individual positions of a covered call. (b) Profitand individual positions of a reverse covered call. . . . . . . .7.17 (a) Profit and individual positions of a bull spread using calloptions. (b) Profit and individual positions of a bear spreadusing put options.. . . . . . . . . . . . . . . . . . . . . . . .7.18 (a) Profit and individual positions of a butterfly spread. . . .7.19 (a) Profit and individual positions of a straddle. (b) Profit andindividual positions of a strangle. . . . . . . . . . . . . . . . .7.20 (a) Profit and individual positions of a strip. (b) Profit andindividual positions of a strap.7.21 The points of time along the life of an option when trades mustbe made to cover the naked short call position. A margin aroundthe strike, K, is created of width 2ϵ to avoid rapid trades whenthe option is near at-the-money range. . . . . . . . . . . . . .7.22 Delta hedge strategy takes advantage of the fact that the slopeof the option price curve will converge to the terminal pay-offlevel as option reaches its maturity. . . . . . . . . . . . . . . .2187.82217.9225226229. . . . . . .230234240248248249251252. . . . . . . . . . . . . . . . .2522552568.1Cash flow from a bond with maturity, T years, and annualcoupon of c%. . . . . . . . . . . . . . . . . . . . . . . . . . . .266 List of Figuresxix8.2Different shapes of the term structure of interest rate by matu-rity. (a) Constant (b) Upward sloping (c) Inverted. . . . . . .Relation of the forward curve to the spot curve for differentshapes of the term structure of interest rates. . . . . . . . . .Bond price as a function of increasing yield. . . . . . . . . . .Value at Risk (VaR) and Conditional Value at Risk (CVaR)display for bond price. . . . . . . . . . . . . . . . . . . . . . .Volume of over-the-counter (OTC) interest rate derivatives in2008-2010 period (Courtesy Bank for International Settlements(BIS) Report).. . . . . . . . . . . . . . . . . . . . . . . . . .Prices for some commodities of different type, from January2002 through 2012. . . . . . . . . . . . . . . . . . . . . . . . .Level of volatility in commodity indices (Courtesy ReserveBank for Australia (RBA) Bulletin, June 2011). . . . . . . . .Participation in commodities markets for diversification bene-fits (Courtesy Reserve Bank for Australia (RBA) Bulletin, June2011). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.10 Sample of key exchanges for developed and emerging economycountries from all continents, as of March 2012. . . . . . . . .8.11 United States turnover of foreign exchange, all currencies.(Courtesy Federal Reserve Bank of New York (FRB NY) Re-port, April 2010). . . . . . . . . . . . . . . . . . . . . . . . . .8.12 USD and euro daily foreign exchange volume by currency.(Courtesy Federal Reserve Bank of New York (FRB NY) Re-port, April 2010). . . . . . . . . . . . . . . . . . . . . . . . . .8.13 Spot and forward exchange rates hold a key relationship. . . .8.14 Daily turnover comparison for foreign exchange spot, forwardand swaps. (Courtesy Federal Reserve Bank ofNew York (FRBNY) Report, April 2010).. . . . . . . . . . . . . . . . . . . .8.15 Distribut...

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