Praise for Sadr Interest Rate Swaps · fm JWBT133-Sadr July 3, 2009 9:26 Printer: Yet to come Founded in 1807, John Wiley & Sons is the oldest independent publishing company in the

Interest Rate Swaps and Their Derivatives

EAN: 9780470443941 ISBN 978-0-470-44394-1

Amir Sadr

Interest RateSwaps and Their

DerivativesA Practitioner’s Guide

option product, the pricing and risk manage-ment of these requires dealing with volatility as the main risk factor—and he shows that one does not need to have a PhD in math to understand options. Sadr presents risk-neutral valuation as the fundamental pricing paradigm for derivatives, and illustrates the core idea of dynamic replication in a simple binomial setting. This unifi ed framework is used to derive industry-standard Black for-mula for fl ow products, and is developed into short-rate and full term-structure mod-els for more complex interest rate exotics including Bermudans.

For current or aspiring practitioners in in-terest rate products, Interest Rate Swaps and Their Derivatives provides a sound working knowledge and appreciation of the main features of these products and their pricing and risk management issues.

AMIR SADR, PHD, has experience as a quant, trader, fi nancial software developer, and academic in fi xed income markets. He traded options and exotics at HSBC in New York from 2005 to 2006 and traded at the proprietary desk for Greenwich Capital Markets (GCM) for four years prior to that. Sadr also has experience at Morgan Stan-ley as a vice president in the derivatives products group where he traded interest rate derivatives and exotics. Since 1996, Sadr has served as an adjunct professor at New York University in the Department of Finance and Accounting.

$85.00 USA/$102.00 CAN

I nterest rate swaps and their derivatives have become an integral part of the fi xed income market, but many of the pricing

and risk management issues for these now mainstream products can only be learned on a trading fl oor. While there are many books on fi xed income and interest rate derivatives, they generally suffer from be-ing either too elementary and bond-centric, mentioning swaps in passing, or too techni-cal and focused on exotics and the myriad implementation issues and algorithms used to tackle them.

Rather than focusing on exotics, Interest Rate Swaps and Their Derivatives thoroughly covers the mainstream products—swaps, fl ow options, Bermudans, semi-exotics—showing the common pricing techniques while also explaining how to generalize the concepts to more nuanced products.

Author Amir Sadr, experienced as a quant, trader, fi nancial software developer, and academic in the fi xed income fi eld, begins by presenting plain-vanilla swaps as an ex-tension of fi xed rate bonds—revealing how techniques for pricing these instruments are a generalization of similar methods used for pricing bonds and repos, and for the most part involve the concepts of fi nancing cost, discount factors, and projection of forward curves. He then moves on to cover the op-tions markets for fl ow products, including options on futures, caps and fl oors, and European swaptions—with detailed atten-tion to the actual trading practice of these products. Sadr explains how, as with any

SadrInterest Rate Swaps and Their Derivatives

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wileyfi nance.com

Praise for

“This is it! I have been looking for a practitioner’s guide to interest rate derivatives for over ten years! Most ‘new joiners’ on Wall Street only gain this knowledge over years of apprenticeship with seasoned professionals. In his book, Amir Sadr explains not only the math behind the products, but the street lingo and, most importantly, the mechanics of everything from overnight repos to Bermudans.”

—George Nunn, Head of Fixed Income Structuring, Americas, BNP Paribas

“Dr. Sadr has produced a single, comprehensive guide to the interest rate swap market. Bank dealers and corporate risk professionals already active in this market will fi nd the book to be one of the best trading fl oor reference guides out there. Students who want the real-world insights of a seasoned professional will also discover Dr. Sadr’s book to be an invaluable resource.”

—Morris Sachs, Chief Risk Offi cer, 5:15 Capital

P1: a/b P2: c/d QC: e/f T1: g

fm JWBT133-Sadr July 3, 2009 9:26 Printer: Yet to come

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Derivatives

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DerivativesA Practitioner’s Guide

AMIR SADR

John Wiley & Sons, Inc.

iii



Copyright C© 2009 by Amir Sadr. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted inany form or by any means, electronic, mechanical, photocopying, recording, scanning, orotherwise, except as permitted under Section 107 or 108 of the 1976 United States CopyrightAct, without either the prior written permission of the Publisher, or authorization throughpayment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222Rosewood Drive, Danvers, MA 01923, (978)750-8400, fax (978) 646-8600, or on the Webat www.copyright.com. Requests to the Publisher for permission should be addressed to thePermissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030,(201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used theirbest efforts in preparing this book, they make no representations or warranties with respect tothe accuracy or completeness of the contents of this book and specifically disclaim any impliedwarranties of merchantability or fitness for a particular purpose. No warranty may be createdor extended by sales representatives or written sales materials. The advice and strategiescontained herein may not be suitable for your situation. You should consult with aprofessional where appropriate. Neither the publisher nor author shall be liable for any loss ofprofit or any other commercial damages, including but not limited to special, incidental,consequential, or other damages.

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Library of Congress Cataloging-in-Publication Data:

Sadr, Amir, 1963–Interest rate swaps and their derivatives: a practitioner’s guide / Amir Sadr.

p. cm. – (Wiley finance series)Includes bibliographical references and index.ISBN 978-0-470-44394-1 (cloth)

1. Interest rate swaps. 2. Interest rate futures. 3. Derivative securities. I. Title.HG6024.5.S32 2009332.63′23–dc22 2009008840

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

iv



Contents

Preface ix“Rates” Market ixBackground ixBook Structure xiAcknowledgments xvii

About the Author xix

List of Symbols and Abbreviations xxi

PART ONECash, Repo, and Swap Markets 1

CHAPTER 1Bonds: It’s All About Discounting 3

Time Value of Money: Future Value, Present Value 3Price-Yield Formula 5PV01, PVBP, Convexity 11Repo, Reverse Repo 16Forward Price/Yield, Carry, Roll-Down 19

CHAPTER 2Swaps: It’s Still About Discounting 25

Discount Factor Curve, Zero Curve 26Forward Rate Curve 27Par-Swap Curve 31Construction of the Swap/Libor Curve 34

CHAPTER 3Interest Rate Swaps in Practice 43

Market Instruments 43Swap Trading—Rates or Spreads 48

v



vi CONTENTS

Swap Spreads 51Risk, PV01, Gamma Ladder 56Calendar Rules, Date Minutiae 59

CHAPTER 4Separating Forward Curve from Discount Curve 67

Forward Curves for Assets 67Implied Forward Rates 69Float/Float Swaps 70Libor/Libor Basis Swaps 73Overnight Indexed Swaps (OIS) 75

PART TWOInterest-Rate Flow Options 77

CHAPTER 5Derivatives Pricing: Risk-Neutral Valuation 79

European-Style Contingent Claims 80One-Step Binomial Model 80From One Time-Step to Two 84From Two Time-Steps to . . . 90Relative Prices 91Risk-Neutral Valuation: All Relative Prices Must be

Martingales 92Interest-Rate Options Are Inherently Difficult to Value 93From Binomial Model to Equivalent Martingale Measures 94

CHAPTER 6Black’s World 97

A Little Bit of Randomness 97Modeling Asset Changes 103Black-Scholes-Merton/Black Formulae 104Greeks 112Digitals 116Call Is All You Need 117Calendar/Business Days, Event Vols 120

CHAPTER 7European-Style Interest-Rate Derivatives 123

Market Practice 124Interest-Rate Option Trades 124



Contents vii

Caplets/Floorlets: Options on Forward Rates 125European-Style Swaptions 129Skews, Smiles 137CMS Products 140Bond Options 147

PART THREEInterest-Rate Exotics 149

CHAPTER 8Short-Rate Models 151

A Quick Tour 152Dynamics to Implementation 153Lattice/Tree Implementation 154BDT Lattice Model 156Hull-White, Black-Karasinski Models 168Simulation Implementation 169

CHAPTER 9Bermudan-Style Options 175

Bellman’s Equation—Backward Induction 176Bermudan Swaptions 177Bermudan Cancelable Swaps, Callable/Puttable Bonds 180Bermudan-Style Options in Simulation Implementation 183

CHAPTER 10Full Term-Structure Interest-Rate Models 185

Shifting Focus from Short Rate to Full Curve: Ho-Lee Model 186Heath-Jarrow-Morton (HJM) Full Term-Structure

Framework 186Discrete-Time, Discrete-Tenor HJM Framework 188Forward-Forward Volatility 191Multifactor Models 197HJM Framework Typically Leads to Nonrecombining Trees 199

CHAPTER 11Forward-Measure Lens 201

Numeraires Are Arbitrary 201Forward Measures 206BGM/Jamshidian Results 208



viii CONTENTS

Different Measures for Different Rates 210“Classic” or “New Improved”: Pick Your Poison! 212

CHAPTER 12In Search of “The” Model 215

Migration to Full-Term Structure Models 215Implementation Era 216Model versus Market: Liquidity and Concentration Risk 216Complexity Risk 217Remaining Challenges 218

APPENDIX ATaylor Series Expansion 219

Function of One Variable 219Function of Several Variables 220Ito’s Lemma: Taylor Series for Diffusions 220

APPENDIX BMean-Reverting Processes 223

Normal Dynamics 224Log-Normal Dynamics 226

APPENDIX CGirsanov’s Theorem and Change of Numeraire 229

Continuous-Time, Instantaneous-ForwardsHJM Framework 230

BGM Result 232

Notes 235

Index 239



Preface

The market for interest rate swaps and their derivatives has experiencedtremendous growth since its beginning in the early 1980s, and swaps are

now a key component of capital markets. While trading in swaps and theirderivatives was initially the domain of major money-center banks, mostinvestment and commercial banks these days run a swaps and options deskalongside their cash and repo desks.

“RATES” MARKET

The “rates” market consists of swaps, flow options (caps/floors, Europeanswaptions), Bermudan swaptions, some semi-exotics (CMS/CMT products),and exotics (structured notes, . . .). While at some point, Bermudan swap-tions were considered exotics, their popularity and volume has made theminto an integral part of the interest-rate options market.

While a newcomer to a typical broker-dealer trading floor can find amplebackground material on the bond and repo markets, he is often overwhelmedby the instruments and the technical requirements to understand swaps andtheir derivatives. For bonds and repos, a typical analyst can use a Bloombergterminal or the financial toolkit in Excel, or even an HP-12 calculator to getup and running. However, for swaps and options, he has to typically masterthe in-house derivatives system with many moving parts and nonstandardterms. The analyst can quickly become discouraged, and think of swapsand options to be the domain of quants and tech-savvy individuals whocan handle such seeming complexity. Some of this complexity is merelythe terms used in the swaps market: receiving/paying in swap lingo insteadof buying/selling cash bonds—economically the same things—while for theoptions and exotics markets the complexity is real. The goal of this book isto break down some of this complexity.

BACKGROUND

This book came about over the past 15 years as alongside my day time WallStreet job, I periodically taught an evening course on interest-rate swaps

ix



x PREFACE

and options at NYU’s School of Continuing and Professional Studies. Eachsemester, I was asked what a good book would be to accompany the course.My answer has always been, “I have yet to find a book,” and instead Iwould use notes gathered from various sources that I would pass out inthe class. Moreover, in my last trading job, I was assigned to mentor theentry-level analyst class in their rotation on the fixed income (repo, treasury,swaps, options) desks, and was asked to assemble a required reading list forthem. Again, I would have liked to recommend one book that would mostdirectly and expeditiously get the analysts up to speed both in theory andmore important the practice—so that we could extract the most work out ofthem!—but I still could not find such a book or training manual, and wouldend up recommending various chapters from a selection of books.

The final straw, so to speak, came when the head of trading asked me toorganize a couple of lectures to educate and familiarize our “cash” traders(treasury, swap, repo, agency) and generalist sales force on the workingsof options and exotics desk, where I was a trader. These lectures and theirorientation to front-line staff on a trading floor finalized the orientation ofthis book, where the goal is to demystify in the quickest way what actuallyhappens on the trading floor, and help people understand the concepts.

While there are many books on fixed-income and interest-rate deriva-tives, they generally suffer from being either too elementary and bond-centric, mentioning swaps in passing, or too technical and focused on exoticsand the myriad implementation issues and algorithms used to tackle them.The exotics area is the most challenging part of the market, and holds anunderstandable pull for quants, academics, and technically-oriented indi-viduals. Their pricing and risk issues remain challenging, although mostof the challenge is not in the theory, but in the efficient implementationof the theory. Rather than focusing on exotics, the goal of this book isthe more mundane task of adequately and thoroughly covering the main-stream products—swaps, flow options, Bermudans, semi-exotics—as theyare traded by showing the common pricing techniques, while showing howto generalize the concepts to other nuanced products.

The main audience for this book is the current or aspiring practitionersin interest-rate products. These would be traders, salespeople, marketers,structurers, and operations, finance, risk management, and IT profession-als involved in rates products. Indeed, this mix has usually been the mainaudience of my class at NYU, where people who are already involved insome aspect of interest-rate products want to have the theory and practicedemystified for them.

With this broad target audience in mind, the level of mathematics is keptto only what is needed, and special effort is made to not lose the audience



Preface xi

with overly technical discussions. Still, this being a book on rates deriva-tives, it requires a college-level of math with some calculus and probability,although most concepts are broken down as much as possible into theirintuitive and pictorial elements. Fundamental results are motivated and ex-pressed in simple settings, illustrated in examples for actively traded prod-ucts, and their generalized versions are simply stated without proof. As muchas possible, technical discussions are avoided and delayed. For example, Ito’sLemma is only introduced in the final two chapters when discussing HJM-type models. Even at this level, in Appendix A, we show how to think ofIto’s Lemma as a simple extension of Taylor Series with a quirky (but easy toremember) multiplication rule. This in my experience is the most productiveway of understanding rates products for the target audience.

BOOK STRUCTURE

The book is organized in three parts, following closely the layout of differenttrading desks on a typical USD fixed income trading floor: repo and cash(treasury) desks, swaps desk, options desk, with the exotics and structuredproducts desk alongside or part of the options desk.

Part 1. Cash, Repo, and Swap Markets

Part 1 deals with cash, repo, and swap products, namely all instrumentsthat are not volatility-based (Euro-dollar convexity adjustment being anexception) and do not require the options machinery. The techniques forpricing these instruments are quite similar, and are pretty much the idea offinancing cost, discount factors, and projection of forward curves.

Chapter 1 begins by a quick review of fixed-income basics: time value ofmoney, future and present value, price-yield formula, forward prices, andsensitivity measures such as PV01, PVBP, and convexity. This chapter canbe considered as “all you need to know about bonds” to proceed to swapsand options. The U.S. Treasury bond market is the main example used toillustrate these concepts, and many of its quote conventions and nuancesare explained in detail. As financing is the main driver of all fixed-incomeproducts, the repo market as it pertains to the U.S. treasuries is discussedin detail. We present popular trades such as Curve (slope of the treasurycurve), Curvature (weighted butterfly), and Carry trades.



xii PREFACE

Chapter 2 covers the pricing of plain-vanilla interest-rate swaps. The mainpricing difference between bonds and swaps is that swaps require the wholeterm structure of interest rates, while for bonds there is a single yield-to-maturity. We present the contractual cash flow structure of a typical swap,and show that pricing interest-rate swap requires access to a discount factorand projected forward curve. As long as the swap counterparties’ funding isthe same as the swap’s floating index, one can use the same discount curve forboth projection and discounting. Using this setup, we show how to extracta discount factor curve from the variety of quoted market instruments, anddiscuss the nuances of curve construction methods while highlighting themain tradeoff between smooth forwards versus having reasonable prescribedhedges.

Chapter 3 delves into actual swap market instruments (cash rates, futures,par-swap rates) using U.S. swap market as its prime example. USD swaps areusually quoted and traded as spreads to treasuries, and we present typicalbroker screens, trading mechanisms (rates/spreads), and different ways oftrading spreads: headline spread, matched-maturity spread, and asset-swapspreads. As swaps are priced off of a curve built from a blend of inputinstruments, we show how the bond sensitivity measures such as PV01and convexity have to get extended to bucket-PV01 and gamma-ladder inSwap-Land. An advantage of swaps is they are bilateral OTC contracts,with the ability to customize their cash flows, requiring the counterpartiesto negotiate and agree on all the cash flow details. We present some of thesedate minutiae and considerations when constructing a swap’s cash flows.

Chapter 4 discusses Basis Swaps and the need for the separate extractionof a forward curve distinct from the discount curve for their pricing andrisk management. In particular, one needs to first extract a discount curvefrom swaps keyed to one’s funding index, and then use this discount curveto extract the forward curve for other indices, from which one arrives atthe discount curve based on them. Historically, Libor rates have been usedsynonymously as risk-free rates, but with the recent banking turmoil, carefulattention is being paid on Libors of different tenors (1m, 3m, 6m). We discussthe issues related to Libor-Libor basis swaps, and show why this basis is notmerely driven by supply and demand dynamics.

We present OIS swaps in the final section and discuss their growingimportance and relevance to swap markets. Most credit support agreements(CSA) for swaps specify OIS rather than Libor rates for margining anymark-to-market value. We present the view that OIS rates should then beconsidered the funding index for plain-vanilla swaps, and swaps are in reality



Preface xiii

basis swaps. This view is still being debated in the markets, and has yet tobe universally adopted.

Part 2. Interest-Rate F low Opt ions

With the cash and swap products out of the way, Part 2 covers the optionsmarkets for flow products. These include options on futures (Euro-dollar)contracts, caps and floors, and European swaptions. As with any optionproduct, the pricing and risk management of these requires dealing withvolatility as the main risk factor.

Option trading in general has been delegated to the more technicallyoriented staff, and has enjoyed a mystique of being understood only bywhiz-kids and sophisticated traders who are able to understand the plethoraof the associated Greek letters. We will show that one does not need to havea Ph.D. in math, or be fully proficient in stochastic calculus and Ito’s Lemmato understand options. While the original Black-Scholes option pricing for-mula was derived using these advanced techniques, the modern approach isRisk-Neutral Valuation, which can be easily explained in a simple binomialsetting.

Chapter 5 introduces the basic concept of option replication in a simplebinomial setting suggested by Cox-Ross-Rubinstein, and follows its simpleextension into a two-period setting to illustrate the concept of dynamicreplication. This is surprisingly the main idea of option pricing, and it issaid (and shown in Chapter 5) that the binomial model is all you reallyneed to understand the theory of derivatives. The binomial model naturallyleads to the framework that has become known as risk-neutral valuation,which can be summarized as follows: Option prices are the prices of theirself-financing dynamic replicating portfolios, and can be obtained as the risk-neutral expected discounted value of their payoffs. This framework appliesto all options, be it equity, FX, or interest rates, and all option models are(simple or elaborate) specialized applications of it. Risk-neutral valuationremains the main option pricing framework for the rest of the book.

Chapter 6 presents Black-Scholes-Merton and Black’s Formulae, the mainpricing models used for European-style options. These formulae are shownto be the result of risk-neutral valuation when the uncertainty about theunderlying asset changes (absolute or percentage) are modeled via a Normaldistribution. For interest rate flow options, the main pricing model is Black’sFormula; originally the Log-Normal version, and in recent years supplantedby the Normal version. Black’s Formula is obtained by evaluating an integral(through some tedious algebra), and without resorting to stochastic calculus



xiv PREFACE

or the dreaded Ito’s Lemma. We present the standard formulas for calls/putsand digital options, and their sensitivities (Greeks) for both Normal andLog-Normal dynamics, and review general option concepts such as put-callparity, gamma versus theta, and the far reach of the call-formula.

Chapter 7 shows how Black’s formulae as used for interest-rate flow op-tions such as ED options, cap/floors, and European swaptions. We use theliquid instruments in USD options markets and present in detail how theseare traded in practice. As most option quotes imply different volatilitiesfor different strikes, we present the SABR model, which is the most com-monly used model for capturing skews and smiles. Most flow options desksalso make markets in Constant-Maturity-Swap (CMS) products: CMSs,cap/floors, and curve cap/floors. While these products should properly bepriced within a term-structure model, one usually tries to resort to simpleanalytical approximation formulae for them, and we present the commonlyused analytical formulae and techniques. We discuss popular trades such asconditional curve trades using a pair of swaptions, and contingent spreadtrades implemented as a bond option and a swaption pair.

Part 3. Interest-Rate Exot ics

Black’s Formula and simple analytical formulas are all one needs to priceand risk-manage interest-rate flow options. For the next class of products—Bermudan options serving as a poster child—there are no simple analyticalformulae, and one has to resort to more complex computational algorithms.As these products depend on a variety of interest rates, one has to use modelsthat capture the dynamics of the whole term structure. The framework forpricing these options remains risk-neutral valuation, but applied to the wholeterm structure, and with multiple underlyings spanning the full maturityspectrum.

The first attempts to develop models for exotics were based on theevolution of the short rate as the state variable in a risk-neutral setting, andhave become known as short-rate models, with Hull-White and BDT/BKmodels the most commonly deployed ones. As the short rate spans the termof any longer rate, the dynamics of all rates could be related back to thefuture evolution of the short rates. The typical implementation of short-ratemodels admits a recombining tree (lattice) format, which lends itself easilyto the pricing of Bermudan options.

Chapter 8 introduces the common short-rate models used in practice andhighlights the main features of their dynamics. Using the BDT model—notthe most commonly used model in practice any more, but a good conduit to



Preface xv

expose the common ideas and techniques—we show the typical discretiza-tion of the process dynamics into a discrete-time setting, and how to satisfythe no-arbitrage requirement imposed in a risk-neutral setting, namely in-verting the yield curve. While brute-force trial and error works for invertingthe yield curve, we discuss the more computationally efficient Forward In-duction technique using Arrow-Debreu prices. Having inverted the yieldcurve, we can at each future node extract discount factors to compute op-tion payoffs that depend on par-swap rates, swap values, forward rates,and so on, and discount these back to today to arrive at option values. Mostshort-rate models have free parameters (local volatility, mean-reversion) thatcan be tweaked so that the market price of liquid options can be recovered.This process of tweaking the free parameters is called calibration, and ingeneral is a computationally hard problem. No matter, having constructeda calibrated arbitrage-free short-rate lattice, we are well on our way to priceany complex payoff.

The previous steps are common to all short-rate lattice models, andare particularly suited for non-path-dependent options. For Asian (path-dependent) options, we need to keep track of the evolution of interest ratesfrom today up to the time of its terminal payoff. While this can conceptu-ally be done in a lattice implementation, the computational burden quicklybecomes exorbitant, and one instead resorts to simulation methods, whichunfortunately suffer from run-to-run variability (simulation noise). Thereare many techniques for reducing simulation noise, and variance reductionis a specialized discipline with many implementation tricks. We present thesimplest such technique (antithetic) to provide a taste.

Chapter 9 presents Bermudan-style options, which confer to the holderthe option to choose an exercise time within an exercise window. We showthat this exercise option can still be handled within the risk-neutral valuationframework, but one has to search for the optimal exercise policy. As this classof problems has been studied in dynamic programming disciplines, thereexists an algorithmic way—backward induction—of extracting the optionvalue under this optimal exercise policy. We discuss the common Bermudancancelable swap structures and show how to use backward induction in alattice model to price them.

As there are models that don’t admit a lattice implementation due totheir non-Markovian dynamics—an up-down move does not end up in thesame state as a down-up move—and are usually implemented as a simulationmodel, we discuss the challenges of pricing Bermudan options for thesemodels. This is still an active area of research, and while certain algorithmshave become the standard (LSM, boundary extraction), there is still roomfor further improvement.



xvi PREFACE

Chapter 10 introduces the full term-structure models of HJM, which ex-plicitly evolve the full term-structure in a risk-neutral setting. We recall thata short-rate model implicitly evolves the full term-structure, but with HJMmodels, the term-structure is the explicit evolved quantity. We present thediscrete-time, discrete-tenor version of the HJM models, and show how toensure that they are arbitrage-free in a risk-neutral setting. This version,despite the seemingly complex notation, is relatively simple to understandand renders itself to algorithmic implementation as a computer program(simulation model). HJM models also offer an intuitive and flexible wayof modeling volatilities via the forward-forward volatility surface, and caneasily be extended to multifactor settings to drive the correlations of vari-ous forward rates. We show the volatility signature of flow products on theforward-forward volatility surface, and present approximations for swap-tion vols and correlations that can aid in calibration.

We show the continuous-time version of the HJM model for instan-taneous forward rates in Appendix C. While an elegant framework forinterest-rate products, HJM models are generally non-Markovian and needto get implemented as a simulation model, making them challenged forBermudan options.

Chapter 11 revisits the issue of numeraires, and shows in a simple binomialsetting how to change them. The technique of changing numeraires has beeninvoked to provide new insights (and justification) for using Black’s modelfor interest-rate flow options, the main objection to it being that it ignoresthe required stochastic discounting, and that it treats forward rates as assets(they are not). By switching numeraires to a discount bond with maturitycoinciding with the option payout, one arrives at the forward measure,under which both common practices are justified. Furthermore, by assumingNormal/Log-Normal terminal distribution under this forward measure, onecan fully recover Black’s formula as used for interest-rate flow options.

The forward measure lens provides theoretical justification—albeit atortured one—of market practice, and has given rise to the subclass of HJMmodels which focus on the evolution of discrete-tenor forward rates throughthis lens. These Brace-Gartarek-Musiela (BGM) Market models have gainedpopularity since they initially provided the hope for easy calibration tothe readily available cap or swaption vols. We discuss how these hopeswere somewhat premature, since when one needs to price the usual multi-rate exotics, the elegant forward-measure structure of various forward ratesbreaks down when considered under any unified measure, and the Marketmodel becomes non-Markovian, requiring simulation implementation withthe associated Bermudan pricing issues. In this way, they share the same



Preface xvii

gains and pains of a typical discrete-tenor HJM model viewed through themore intuitive money-market measure.

At this point, a typical reader should realize that modeling topics arebecoming more nuanced, and are best handled in other books that do propertechnical justice to them. At the same time, the discussions in this chaptershould allow one to become somewhat conversant and gain a workingknowledge and appreciation of the main features of these models and theirrelated issues.

ACKNOWLEDGMENTS

The learning has never stopped for me on Wall Street, and I am grateful tothe following individuals who have taught me new things: David Heath,Sean Hamidi, Joseph Langsam, James Tilley, Wei-Tong Shu, CharbelAburached, Andrew Gunstensen, Craig Gustafson, Sergio Kostek, PeterRitchken, Darrell Duffie, David Moore, Charles Henry, Morris Sachs,Stephen Siu, E.G. Fisher, Michael Sussman, John Mannion, Tim Dann,James Mather, Hongbing Hsu, Mitchell Stafman, Robert Wahl, LeslieHarris, George Nunn, Elan Ruggill, Tom Fitzmaurice, Gerald Cook, JosephVona, John Kuhn, Hugh Bush, Rohit Apte, Marc Braunstein, David Kwun,Joe Mastrocola, Steve Bredahl, Raymond Humiston, Brian Ciardi. Specialthanks to GCM’s ISD team for having put up with me for the past (and whoknows, next?) decade, and to all my NYU students over the years who havekept my feet to the ground. Final thanks to my editors at John Wiley & Sons:Bill Falloon, Laura Walsh, Meg Freeborn, Kevin Holm, and Jay Boggis forpatiently walking me through this project and correcting my many typos.All remaining errors are mine, and I welcome any corrections, suggestions,and comments sent to [email protected].



xviii



About the Author

Amir Sadr received his Ph.D. in 1990 from Cornell University with thesiswork on the Foundations of Probability Theory. After working at AT&TBell Laboratories until 1993, he started his Wall Street career at MorganStanley, initially as a Vice President in Quantitative Modeling and devel-opment of exotic interest rate models, and later on as an exotics traderin Derivative Product Group. He founded Panalytix, Inc., in 1997 to de-velop financial software for pricing and risk management of interest-ratederivatives. In 2001, he joined Greenwich Capital as Managing Director forproprietary trading. He joined HSBC in 2005 as Senior Trader in chargeof CAD exotics and USD inflation trading. His latest role was the COO ofBrevan Howard US Asset Management in Connecticut.

xix



xx



List of Symbols and Abbreviations

α(T1, T2) accrual fraction for calculation period [T1, T2] accordingto some day-count

w generic random future pathσ, σN log-volatility, normal volatilitybp 1% of 1%, 0.0001CT[n] U.S. current n-year Treasuryd, d1,2 moneyness, d = F−K

σ√

te, d1,2 = ln(F/K)

σ√

te± 1/2σ

√te

D(T), D(t, T) discount factor at time t (today, if omitted) for maturity Tf (t, [T1, T2]) simple forward rate at time t (today if omitted) for the

forward deposit period [T1, T2]f (t, T) instantaneous forward rate at time t (today if omitted)

for the forward deposit period starting at TFA(t, T), F forward price of asset A at time t (today, if omitted) for

forward delivery date T > tfc(t, [T1, T2]) continuously-compounded forward rate at time t (today

if omitted) for the forward deposit period [T1, T2]FV, PV future value, present valueLN(µ, σ 2) Log-Normal random variableM(t, w) money market (unit currency rolled over at successive

short rates) account numeraire, M(t, w) = e∫ t

0 r (u,w)du

N(µ, σ 2) A Normal random variable with mean µ, and standarddeviation σ

N(x) cumulative distribution function of a standard (N(0, 1))Normal random variable

N′(x) 1√2π

e−x2/2

P(C, y, N, m), P price of a bond with coupon rate C, paid m times/year,with N remaining coupons, with yield y

PClean, PDirty clean, dirty price of a coupon bondPV01 present value change due to an “01” (1 bp) change in

yieldPVBP present value change due to a 1 bp change in coupon,

present value of a 1 bp annuityVar(X) variance of a random variable X

xxi



xxii LIST OF SYMBOLS AND ABBREVIATIONS

w accrued fraction (according to some daycount) of theperiodic coupon for bond calculation

y(T) zero rate (according to some payment frequency and day-count) for a zero-coupon bond maturing at T

yc(T) continuously-compounded zero rate for a zero-couponbond maturing at T

ATMF at the money forward option, K = FA

CDF cumulative distribution functionDF (probability) density functionMF modified following roll conventionP&L, P+L profit and lossr.v. random variableRMS root-mean-square averageRTP, RTR right-to-pay (payer), right-to-receive (receiver) swaptionYTM yield to maturity100M 100 million, sometimes written as 100MM, where M

stands for 1000’s100-nnm price quote convention for U.S. treasuries, 100 + nn/32 +

(m/8)/32 in percentage points. A ‘+’ for ‘m’ stands for 4.f ′(x), f ′′(x) first, second derivative of f with respect to x




Derivatives

xxiii



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Praise for Sadr Interest Rate Swaps · fm JWBT133-Sadr July 3, 2009 9:26 Printer: Yet to come Founded in 1807, John Wiley & Sons is the oldest independent publishing company in the