Analytical Finance_ Volume II_ The Mathematics of Interest Rate Derivatives, Markets, Risk and Valua - Analytical Finance Volume II Jan R M R\u00f6man

Analytical Finance_ Volume II_ The Mathematics of Interest Rate Derivatives, Markets, Risk and Valua

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Unformatted text preview: Analytical Finance: Volume II Jan R. M. Röman Analytical Finance: Volume II The Mathematics of Interest Rate Derivatives, Markets, Risk and Valuation Jan R. M. Röman Västerås Sweden ISBN 978-3-319-52583-9 ISBN 978-3-319-52584-6 (eBook) Library of Congress Control Number: 2016956452 © The Editor(s) (if applicable) and The Author(s) 2017 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: Tim Gainey / Alamy Stock Photo Printed on acid-free paper This Palgrave Macmillan imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To my son and traveling partner – Erik Håkansson Acknowledgments I like to thank all my students for all their comments and questions during my lectures. A special thanks goes to Mai Xin who asked me to translate my notes to English many years ago. I also like to thank Professor Dmitrii Silvestrov, who asked me to teach Analytical Finance and Professor Anatoly Malyarenko for his assistance and advice. Finally I will also give a special thanks to Thomas Gustafsson for all his comments, and great and deep discussion about financial mathematics. vii Preface This book is based upon my lecture notes for the course Analytical Finance II at Mälardalen University in Sweden. It’s the second course in analytical finance in the program Engineering Finance given by the Mathematics department. The previous book, Analytical Finance – The Mathematics of Equity Derivatives, Markets, Risk and Valuation, covers the equity market, including some FX derivatives. Both books are also a perfect choice for masters and graduate students in physics, astronomy, mathematics or engineering, who already know calculus and want to get into the business of finance. Most financial instruments are described succinctly in analytical terms so that the mathematically trained student can quickly get the expert knowledge she or he needs in order to become instantly productive in the business of derivatives and risk management. The books are also useful for managers and economists who do not need to dwell on the mathematical details. All the latest market practices concerning risk evaluation, hedging and counterparty risks are described in separate sections. This second volume covers the most central topics needed for the valuation of derivatives on interest rates and fixed income instruments. This also includes the mathematics needed to understand the theory behind the pricing of interest rate instruments, for example basic stochastic processes and how to bootstrap interest rate yield curves. The yield curves are used to generate and discount future cash-flows and value financial instruments. We include pricing with discrete time models as well as models in continuous time. ix x Preface First we will give a short introduction to financial instruments in the interest rate markets. We also discuss the parameters needed to classify the instruments and how to perform day counting according to market conventions. Day counting is important when dealing with interest rate instruments since their notional amounts can be huge, millions or even billions of USD in one trade. One or a few missing days of discounting will change the total price with thousands of USD. We also discuss the most common types of interest rate quoting conventions used in the markets. In Chapter 2 we present many of the different interest rates used in the market. We continue with swap interest rates in Chapter 3, where we also present details for several widely used interest rates such as LIBOR, EURIBOR and overnight rates in different currencies. In Chapter 4, many of the common instruments are presented. This includes the basic instruments, such as bonds, notes and bills of different kinds, including some with embedded options. Then we introduce floating rate notes, forward rate agreements, forwards and futures, including cheapest to deliver clauses. We then discuss different kinds of interest rate swaps and the derivatives related to these swaps, like swaptions, caps and floors. This also includes some credit derivatives, such as credit default swaps. For swaptions, caps and floors we explicitly discuss recent changes in these models due to negative nominal interest rates and derive a quasi-analytical relationship between at-the-money lognormal and normal volatility. In Chapters 5 and 6 we continue with yield curves and the term structure of interest rates. We show how to bootstrap interest rate curves from prices of financial instruments. We also present the Nelson-Siegel model and the extension by Svensson. A detailed analysis of interpolation methods follows and the pros and cons of each method is clearly outlined. Spreads in the interbank market are discussed in Chapter 7. In Chapters 8 and 9, risk measures and some crucial features of modern risk management are discussed. In Chapter 10, a new method for valuing instruments with an embedded optionality is presented. This method, the option-adjusted spread (OAS) method, can also be used to value callable and putable bonds, cancellable swaps etc. The call (put) structure can also be of Bermudan exercise type. In Chapter 11 we begin to discuss the pricing theory and models based on stochastic processes. We continue with this, the continuous Preface xi time models through Chapters 12–17. We derive and solve the partial differential equation for interest rate instruments based on arbitrage and relative pricing. Several stochastic models are presented. Some have an affine term structure, such as Vasicek, Ho-Lee, Cox-IngersollRoss and Hull-White. Some models can be approximated by binomial or trinomial trees. These are Ho-Lee, Hull-White and Black-DermanToy. We also discuss the Heath-Jarrow-Morton framework and how to use forward measures in order to derive general option pricing formulas for interest rate instruments. After a short presentation on how to handle some exotic instruments in Chapter 18, we discuss in Chapter 19 how to deal with some standard derivative instruments, such as swaptions, caps and floors. This also includes the recent case of negative interest rates. In Chapter 20 is a brief introduction to convertible bonds. Finally, there are some chapters on modern pricing. These chapters describes the dramatic changes in the markets after the financial crises in 2008 – 2009. Before the crises, credit risk was more or less ignored when valuing financial instruments. But, after the crises, collateral agreements have become a way to minimize counterparty risk. Also the funding of the deals were changed as well as the views on riskfree interest rates. During the crises even LIBOR rated banks did default. Also the LIBOR rates were manipulated by some of the panel banks. With collateral agreements in several currencies we need to use a multi-curve framework and bootstrap several curves to find the cheapest to deliver curve. We also discuss credit value adjustment (CVA), debt value adjustment (DVA) and funding value adjustment (FVA). We also present the widely used LIBOR market model (LMM) and how to calibrate the LMM. Finally we present methods on how to manage exotic instruments by using linear Gaussian models (LGM). We also present something about the Stochastic Alpha Beta Rho (SABR) volatility model and how to convert between lognormal and normal distributed volatilities. Contents 1 Financial Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Money . . . . . . . . . . . . . . . . . . . . 1.1.2 Valuation of Interest Rate Instruments 1.1.3 Zero Coupon Pricing . . . . . . . . . . . 1.1.4 Day-Count Conventions . . . . . . . . . 1.1.5 Quote Types . . . . . . . . . . . . . . . . 2 1 ............ ............ ............ ............ ............ ............ 1 2 3 8 10 14 Interest Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Introduction to Interest Rates . . . . . . . . . . . . . . . . . . . . 2.1.1 Benchmark Rate, Base Rate (UK), Prime Rate (US) . 2.1.2 Deposit Rate . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Discount Rate, Capitalization Rate . . . . . . . . . . . 2.1.4 Simple Rate . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Effective (Annual) Rate . . . . . . . . . . . . . . . . . . 2.1.6 The Repo Rate . . . . . . . . . . . . . . . . . . . . . . . 2.1.7 Interbank Rate . . . . . . . . . . . . . . . . . . . . . . . 2.1.8 Coupon Rate . . . . . . . . . . . . . . . . . . . . . . . . 2.1.9 Zero Coupon Rate . . . . . . . . . . . . . . . . . . . . . 2.1.10 Real Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.11 Nominal Rate . . . . . . . . . . . . . . . . . . . . . . . . 2.1.12 Yield – Yield to Maturity (YTM) . . . . . . . . . . . . 2.1.13 Current Yield . . . . . . . . . . . . . . . . . . . . . . . . 2.1.14 Par Rate and Par Yield . . . . . . . . . . . . . . . . . . . 2.1.15 Prime Rate . . . . . . . . . . . . . . . . . . . . . . . . . . .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... 17 17 17 18 18 19 20 21 21 21 21 22 22 22 22 24 xiii xiv Contents ................ ................ ................ ................ ................ ................ ................ ................ ................ ................ ................ ................ ................ 24 24 25 26 26 26 27 27 27 27 27 28 29 Market Interest Rates and Quotes . . . . . . . . . . . . . . . . . . . . . . 31 2.1.16 2.1.17 2.1.18 2.1.19 2.1.20 2.1.21 2.1.22 2.1.23 2.1.24 2.1.25 2.1.26 2.1.27 2.1.28 3 Risk Free Rate . . . . . . . . . . . . Spot Rate . . . . . . . . . . . . . . . Forward Rate . . . . . . . . . . . . Swap Rate . . . . . . . . . . . . . . Term Structure of Interest Rates Treasury Rate . . . . . . . . . . . . Accrued Interest . . . . . . . . . . Dividend Rate . . . . . . . . . . . . Yield to Maturity (YTM) . . . . . Credit Rate . . . . . . . . . . . . . Hazard Rate . . . . . . . . . . . . . Rates and Discounting Summary Black-Scholes Formula . . . . . . 3.1 The Complexity of Interest Rates . . . . . . . . . . . . . . . 3.1.1 The LIBOR Rates . . . . . . . . . . . . . . . . . . . . 3.1.2 The EURIBOR Rates . . . . . . . . . . . . . . . . . . 3.1.3 The EONIA Rates . . . . . . . . . . . . . . . . . . . 3.1.4 The Euro Repurchase Agreement Rate – Eurepo 3.1.5 Sterling Overnight Index Average (SONIA) . . . 3.1.6 Federal Funds . . . . . . . . . . . . . . . . . . . . . . 3.1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 4 ...... ...... ...... ...... ...... ...... ...... ...... 31 31 33 39 40 43 44 44 Interest Rate Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1 Introduction to Interest Rate Instruments . . . . . . . . . . . 4.1.1 Bonds, Bills and Notes . . . . . . . . . . . . . . . . . 4.1.2 Bonds, Market Quoting Conventions and Pricing 4.1.3 Accrued Interest . . . . . . . . . . . . . . . . . . . . . 4.1.4 Floating Rate Notes . . . . . . . . . . . . . . . . . . . 4.1.5 FRA – Forward Rate Agreements . . . . . . . . . . . 4.1.6 Interest Rate Futures . . . . . . . . . . . . . . . . . . 4.1.7 Interest Rate Bond Futures and CTD . . . . . . . . 4.1.8 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.9 Overnight Index Swaps (OIS) . . . . . . . . . . . . . 4.1.10 Asset Swap and Asset Swap Spread . . . . . . . . . 4.1.11 Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.12 Credit Default Swaps . . . . . . . . . . . . . . . . . . 4.1.13 Hazard rate models . . . . . . . . . . . . . . . . . . . 4.1.14 Total Return Swaps . . . . . . . . . . . . . . . . . . . ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... 47 47 51 54 59 65 72 75 91 106 108 112 112 124 130 Contents 4.1.15 4.1.16 4.1.17 4.1.18 4.1.19 5 Caps, Floors and Collars . . . . . . . . . . . . . . . . Interest Rate Guarantees – IRG . . . . . . . . . . . . Repos and Reverses . . . . . . . . . . . . . . . . . . Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . CPPI – Constant-Proportions-Portfolio-Insurance . ..... ..... ..... ..... ..... ........................ ........................ 5.2 Zero-coupon Yield Curves . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 ISMA and Moosmüller . . . . . . . . . . . . . . . . . . . . . . . ........... ........... ........... ........... ........... ........... ........... ........... ........... 171 173 175 176 178 182 196 205 210 213 225 The Interbank Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.1 Spreads and the Interbank Market . . . . . . . . . . . . . . . . 7.1.1 TED-Spread and Other Spreads . . . . . . . . . . . 7.1.2 Overnight Indexed Swaps (OIS) and Basis Spread 7.1.3 Some Overnight Indices . . . . . . . . . . . . . . . . 7.1.4 Basis Swaps . . . . . . . . . . . . . . . . . . . . . . . . 8 165 170 Bootstrapping Yield Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.1 Constructing Zero-Coupon Yield Curves . . . . . . 6.1.1 The Matching Zero-Coupon Yield Curve 6.1.2 Implied Forward Rates . . . . . . . . . . . 6.1.3 Bootstrapping with Government Bonds 6.1.4 Bootstrapping a Swap Curve . . . . . . . 6.1.5 A More General Bootstrap . . . . . . . . . 6.1.6 Nelson-Siegel Parameterization . . . . . . 6.1.7 Interpolation Methods . . . . . . . . . . . 6.1.8 Spread and Spread Curves . . . . . . . . . 7 130 154 155 158 159 Yield Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.1 Introduction to Yield Curves 5.1.1 Credit Ratings . . . . 6 xv ..... ..... ..... ..... ..... 227 228 228 232 233 Measuring the Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8.1 Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Delta . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Duration and Convexity . . . . . . . . . . . . . . 8.1.3 Modified Duration, Dollar Duration and DV01 8.1.4 Convexity . . . . . . . . . . . . . . . . . . . . . . . 8.1.5 Gamma . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.6 Accrued Interest . . . . . . . . . . . . . . . . . . . 8.1.7 Rho . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.8 Theta . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.9 Vega . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... 237 237 239 243 245 247 249 249 249 251 xvi Contents 8.1.10 8.1.11 8.1.12 8.1.13 8.1.14 9 YTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Portfolio Immunization Using Duration and Convexity The Fisher-Weil Duration and Convexity . . . . . . . . . Hedging with Duration . . . . . . . . . . . . . . . . . . . . Shifting the Zero-Coupon Yield Curve . . . . . . . . . . ............. ............. ............. ............. ................... ................... ................... ................... ................... 307 310 313 314 316 . . . . . . . . . . . . . . . . . . . 319 Pricing of Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 14.1 15 . . . . . . . . . . . . 291 . . . . . . . . . . . . 293 . . . . . . . . . . . . 297 Martingale Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 13.1 Introduction to Martingale Measures 14 279 280 281 284 286 288 289 Term Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 12.1 The Term Structure of Interest Rates 12.1.1 Yield- and Price Volatility . 12.1.2 The Market Price of Risk . . 12.1.3 Solutions to the TSE . . . . . 12.1.4 Relative Pricing . . . . . . . . 13 .. .. .. .. .. .. .. Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 11.1 Pricing Theory . . . . . . . . . . . . . . . . . . . . . 11.1.1 Interest Rates . . . . . . . . . . . . . . . . 11.1.2 Stochastic Processes for Interest Rates 12 261 263 266 276 Option-Adjusted Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 10.1 The OAS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Building the Binomial Tree . . . . . . . . . . . . . . . . . 10.1.3 Calibrate the Binomial Tree . . . . . . . . . . . . . . . . . 10.1.4 Calibrate the Tree With a Spread . . . . . . . . . . . . . . 10.1.5 Using the OAS Model to Value the Embedded Option . 10.1.6 Effective Duration and Convexity . . . . . . . . . . . . . 11 251 253 255 256 257 Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 9.1 Introduction to Risk Management . . . . . . . . 9.1.1 Capital Requirement . . . . . . . . . . 9.1.2 Risk Measurement and Risk Limits . . 9.1.3 Risk Control in Treasury Operations 10 .. .. .. .. .. Bond Pricing . . . . 14.1.1 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 Term-Structure Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 15.1 Martingale Models for the Short Rate . . . . . . . . . . . . . . . . . . . 333 Contents 15.1.1 15.1.2 15.1.3 15.1.4 15.1.5 15.1.6 15.1.7 15.1.8 15.1.9 16 The Q-Dynamics . . . . . . . . . . . . . . . . . . . . . Inverting the Yield Curve . . . . . . . . . . . . . . . Affine Term Structure . . . . . . . . . . . . . . . . . . Yield-Curve Fitting: For and Against . . . . . . . . . The BDT Model . . . . . . . . . . . . . . . . . . . . . The Black–Karasinski Model . . . . . . . . . . . . . . Two-Factor Models . . . . . . . . . . . . . . . . . . . Three-Factor Models . . . . . . . . . . . . . . . . . . Fitting Yield Curves with Maximum Smoothness . . . . . . . . . . . . . . . 463 . . . . . . . . . . . . . . 471 . . . . . . . . . . . . . . 474 .................. .................. .................. .................. 491 491 493 494 The Black Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 19.1 Pricing Interest Rate Options Using Black . . . . . . 19.1.1 Par and Forward Volatilities . . . . . . . . . 19.1.2 Caps and Floors . . . . . . . . . . . . . . . . 19.1.3 Swaps and Swaptions . . . . . . . . . . . . . 19.1.4 Swaps in the Multiple Curve Framework . 19.1.5 Swaptions with Forward Premium . . . . . 19.1.6 The Normal Black Model . . . . . . . . . . . 19.1.7 European Short-Term Bond Options . . . . 19.1.8 The Schaefer and Schwartz Model . . . . . 20 449 455 456 459 Exotic Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 18.1 Some Exotic Instruments . . . . . . . . . 18.1.1 Constant Maturity Contracts 18.1.2 Compound Options . . . . . . 18.1....
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