Global Risk Management Ren-Raw Chen, Ph.D. c Draft date February 11, 2021 i Library of Congress Control Number 1234567890 Chen, Ren-Raw Global Financial Risk Management { A Quantitative Guide / Ren-Raw Chen ISBN 1234567890123 All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. ii Contents Contents i Preface xv Acknowledgments xvii I Introduction 1 1 Math Primer 3 1.1 Continuous and Discrete Returns . .3 1.1.1 Definition . .3 1.1.2 Average Return . .4 1.1.3 Annualization and Deannualization . .5 1.2 Linear Algebra . .5 1.2.1 Addition/subtraction . .5 1.2.2 Multiplication/Division . .6 1.2.3 Scaling . .6 1.2.4 Power . .6 1.3 Calculus . .7 1.4 Statistics . .8 1.4.1 Random Variable . .8 1.4.2 Stochastic Process . .9 1.4.3 dS and ∆S ............................9 iii iv CONTENTS 1.4.4 Normal Distribution . .9 2 Overview 13 2.1 Introduction . 13 2.1.1 Business risks versus financial risks . 13 2.1.2 Financial risks . 14 2.1.3 Ways to measure and manage these risks . 17 2.2 Review of Simple Hedging . 17 2.3 Market Risk . 18 2.3.1 Value at Risk . 19 2.3.2 Stress Test . 21 2.4 Credit Risk . 21 2.4.1 Sources of credit risk . 22 2.4.2 Credit risk metrics . 24 2.5 Liquidity Risk . 25 2.6 Operational Risk (Wiki) . 26 2.7 Risk Management Modeling Building Blocks . 27 2.7.1 Basic Models by Asset Class . 27 2.7.2 Risk Management Tools available . 28 2.8 Basel Accords . 28 2.8.1 Basel I . 29 2.8.2 Basel II . 31 2.8.3 Basel III . 33 2.9 Types of Capital . 34 2.9.1 Regulatory capital . 34 2.9.2 Economic capital . 35 2.9.3 Risk-adjusted return on capital (RAROC) . 35 2.10 Appendix . 36 2.10.1 Report on Excite@Home's bankruptcy . 36 CONTENTS v 3 Hedging 37 3.1 What Is Hedging? . 37 3.2 Static versus Dynamic Hedging . 38 3.2.1 Static Hedging . 38 3.2.2 Dynamic Hedging . 39 3.2.3 Other Hedging Examples . 41 II Market Risk 43 4 Value At Risk 45 4.1 Introduction . 45 4.1.1 An Example . 47 4.1.2 Limitations of VaR . 48 4.1.3 Stress Testing . 48 4.1.4 Risk Reporting . 48 4.1.5 External Disclosure . 49 4.1.6 The Basel Accord II . 49 4.1.7 Capital Regulation . 50 4.2 Major Types of VaR . 51 4.2.1 Historical VaR . 52 4.2.2 Parametric VaR . 59 4.2.3 Factor Model Based VaR . 65 4.3 Marginal and Component VaR . 74 4.4 Decay . 76 4.5 Exercises . 77 5 Fixed Income Risk Management 79 5.1 Interest Rates . 79 5.1.1 US Treasuries . 79 5.1.2 LIBOR (London Interbank Offer Rate) . 81 vi CONTENTS 5.1.3 Agencies . 81 5.1.4 The Model . 82 5.2 Forward Expectation . 91 5.2.1 A Simple Concept . 91 5.2.2 More Formal Mathematics . 92 5.3 IR Risk Management . 93 5.3.1 Conventions . 93 5.3.2 Duration (delta) and Convexity (gamma) . 96 5.3.3 IR Swaps . 99 5.4 FX Risk Management . 102 5.4.1 FX Forward and Interest Rate Parity . 102 5.4.2 FX Swaps . 103 5.4.3 Quanto . 109 5.4.4 FX Option Formula . 110 5.4.5 FX Basis . 111 5.4.6 FX models . 111 5.5 Total Return Swap . 112 5.6 Residential Mortgage . 113 5.6.1 Refinance Modeling . 113 5.6.2 Prepayment Modeling . 114 5.6.3 Default Modeling . 116 5.6.4 OAS { Option Adjusted Spread . 116 5.7 Combined with Other Assets . 117 5.8 Appendix . 118 5.8.1 FX Swap Curve and Fixed-Fixed FX Swaps . 118 5.8.2 Commodities and Real Assets . 118 6 P&L Attribution 125 6.1 Introduction . 125 6.2 Taylor's Series Expansion and Explanatory Risk Factors . 125 CONTENTS vii 6.3 Pictorial P&L Attribution . 127 7 Value Adjustment 129 7.1 VA { Value Adjustment . 129 7.2 Various VAs . 130 7.2.1 CVA . 130 7.2.2 FVA . 130 7.2.3 KVA . 130 7.2.4 XVA . 130 8 Parameter Estimation 131 8.1 Introduction . 131 8.2 Regression estimation for the Vasicek model . 131 8.3 The Cox-Ingersoll-Ross Model . 135 8.4 The Ho-Lee Model . 136 8.5 More sophisticated econometric methods . 138 9 Simulation 139 9.1 Introduction . 139 9.2 Random Numbers . 139 9.2.1 Normal/log Normal . 139 9.2.2 Poisson . 140 9.2.3 Non-central Chi-square . 140 9.2.4 Multi-variate Gaussian . 142 9.3 Examples . 144 9.3.1 Black-Scholes . 144 9.3.2 Two stocks . 146 9.3.3 Stock with random interest rates . 147 9.4 Homework . 149 9.5 Appendix . 149 viii CONTENTS 10 Volatility and Extreme Value Theorem 151 10.1 Introduction . ..