Praise for this new edition of Intr oduction to Option-Adjusted Spr ead Analy sis “O ption-adjusted spread analysis is as widely used as it is misunderstood Tom Miller’s update of th is fixed-in come classic is a paragon of clear thinking and clear writing T he r oad to under standing how OAS is computed and implemented begins her e ” steven v mann, phd Professor of Finance, Moore School of Business, University of South Carolina Coeditor, The Handbook of Fixed Income Securities And praise for the previous edition “L u cid an d w ell- w r itten , Option-Adjusted Spread Analysis focuses on th e evalu ation of p u t, callable, an d sin kin g fu n d bon d s … a top-quality job ” DERIVAT IVES ST RAT EGY “T his intr oductor y book boils each com plex concept down to the basics … For those who need to know the calculations behind the numbers on their Bloomberg screen, this is a neat, to-thepoint little tex tbook : How to determine fair value of bullet and nonbullet bonds, the role of volatility, the binomial tree of short rates, and applications of OAS analysis are some of the subjects covered.” FUT URES “Tom Win das open s th e black box of option -adjusted spreads P r actical ex am ples, ex cellent gr aphics, and clear ex planations guide the reader to an understanding of bond valuation Unlike academic discussions of valuation techniques, this book is not just for the ‘rocket scientist,’ but is for ev er y inv estor ” andrew davidson President, Andrew Davidson & Co., Inc New York Introduction to Option-Adjusted Spread Analysis Also available from Bloomberg Press The Credit Default Swap Basis by Moorad Choudhry Fixed-Income Securities and Derivatives Handbook: Analysis and Valuation by Moorad Choudhry Inside the Yield Book: The Classic That Created the Science of Bond Analysis by Sidney Homer and Martin L Leibowitz, PhD The Securitization Markets Handbook: Structures and Dynamics of Mortgage- and Asset-Backed Securities by Charles Austin Stone and Anne Zissu, PhD A complete list of our titles is available at www.bloomber g.com/books Attention Corporations This book is available for bulk purchase at special discount Special editions or chapter reprints can also be customized to specifications For information, please e-mail Bloomberg Press, press@ bloomberg.com , Attention: Director of Special Markets, or phone 212-617-7966 Introduction to Option-Adjusted Spread Analysis Revised and Expanded Third Edition of the OAS Classic by Tom Windas ◆ T OM M ILLER © 1993, 1996, 2007 by Bloomberg L.P All rights reserved Protected under the Berne Convention Printed in the United States of America No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher except in the case of brief quotations embodied in critical articles and reviews For information, please write: Permissions Department, Bloomberg Press, 731 Lexington Avenue, New York, NY10022 or send an e-mail to press@bloomberg.com BLO O MBERG, BLO O MBERG LEGAL, BLOOMBERG MARKET S, BLO O MBERG NEWS, BLOOMBERG PRESS, BLOOMBERG PROFESSIONAL, BLOOMBERG RADIO, BLOOMBERG TELEVISION, BLOOMBERG TERMINAL, and BLOOMBERG TRADEBOOK are trademarks and service marks of Bloomberg L.P All rights reserved This publication contains the author’s opinions and is designed to provide accurate and authoritative information It is sold with the understanding that the author, publisher, and Bloomberg L.P are not engaged in rendering legal, accounting, investment-planning, or other professional advice The reader should seek the services of a qualified professional for such advice; the author, publisher, and Bloomberg L.P cannot be held responsible for any loss incurred as a result of specific investments or planning decisions made by the reader First edition published 1993 Second edition published 1996 Third edition published 2007 10 Library of Congress Cataloging-in-Publication Data Miller, Tom Introduction to option-adjusted spread analysis : revised and expanded third edition of the OAS classic by Tom Windas / revised by Tom Miller ; foreword by Peter Wilson p cm Summary: “Explains option-adjusted spread analysis, a method for valuing bonds with options This book takes readers through each step of the calculation” Provided by publisher Includes bibliographical references and index ISBN-13: 978-1-57660-241-6 (alk paper) ISBN-10: 1-57660-241-9 Fixed-in come securities Option -adjusted spread an alysis I Win das, Tom Introduction to option-adjusted spread analysis II Title HG4650.M56 2007 332.63'2283 dc22 2006101298 Dedicated to the curious and those who teach us CONTENTS For ewor d ix by Peter Wilson Acknowledgments xi Intr oduction: W hy OAS Analy sis? PART ONE Y ield Analy sis Ver sus OAS Analy sis C HA P TE R Fatal Flaws in Tr aditional Y ield Calculations C HA P TE R T he Bond as a P or tfolio 15 PART T W O Valuing Options 27 C HA P TE R Intr insic Value 29 C HA P TE R T ime Value 33 PART T HREE Modeling Inter est Rates 41 C HA P TE R Implied Spot and For war d Rates 43 C HA P TE R Bey ond the L ognor mal Model 53 C HA P TE R Volatility and the Binomial Tr ee 57 C HA P TE R Matching the Model to the Mar ket 67 PART FOUR Measur ing the Spr ead 81 C HA P TE R Bullet Bonds 83 C HA P TE R Nonbullet Bonds 99 PART FIVE Applications of OAS Analy sis 113 C HA P TE R 1 Ev aluating P er for mance 115 C HA P TE R Estimating Fair Value 125 Conclusion: Building a Better OAS 135 Glossar y 145 Refer ences & Additional Reading 153 Index 155 FOREWORD thirty years ago, bonds were quoted by coupon and yield to maturity, with the latter widely regarded as a de facto total return expectation As the bond world evolved, new types of bonds like callables, putables, an d later, mortgage-backed securities were in ven ted Th is caused a dilemma for in vestors wh o were n ot exactly sure how to measure the risk and return of the new instruments Or, as a fixed-in come in vestor back in th e 1980s migh t h ave put it: “Those bonds I purchased in the late 1970s sure paid a wonderfully high coupon, but just when rates went down, the bond was called away, an d n ow I h ave to rein vest at much lower yields I wish I could put a value on the option that the issuer retains, to redeem the bonds at par.” The race was on to create methods to measure risk and return that took into consideration the new ways of structuring bonds One of those methods is called “option-adjusted spread analysis,” or “OAS analysis.” Three decades later, the fixed-income market is still known for constantly creating new ways to structure bonds that allow investors to take more types of bets an d h edge away more risks th an ever before Most of these new securities involve options, either implicitly or explicitly The need to understand options therefore grows steadily ix Introduction to Option-Adjusted Spread Analysis: Revised and Expanded Third Edition of the OAS Classic by Tom Windas Revised by Tom Miller © 1993, 1996, 2007 by Bloomberg L.P REFERENCES & ADDITIONAL READING Berger, Eric, PhD “Understanding Option-Adjusted Spread Duration.” Bloomberg Magazine, October 1992 A closer look at risk measures in general and OAS duration in particular ———, and William Gartland, CFA “The Bloomberg Corporate Bond OAS Model.” Unpublished paper An in-depth mathematical description of the Bloomberg lognormal model Black, Fisch er, Emman uel Dermam, an d William Toy “A O n e-Factor Model of Interest Rates and Its Application to Treasury Bond Options.” Financial Analysts Journal, Jan uary–February 1990 A straightforward description of a one-factor interest-rate model Black, Fisch er, an d Piotr Karasin ski “Bon d an d Option Pricin g Wh en Sh ort Rates Are Logn ormal.” Financial Analysts Journal, July–August 1991 A description of a one-factor model of bond prices, yields, and options Fabozzi, Frank J., and Steven V Mann Floating-Rate Securities New York: John Wiley & Sons, Inc., 2000 Gartlan d, William, CFA “Calculatin g Implied Spot Rates from Ben ch mark Yields.” Bloomberg Magazine, November 1992 A description of the various methodologies employed to derive implied spot rates from benchmark yields Ho, Th omas S Y “Evolution of In terest Rate Models: A Comparison ” The Journal of Derivatives, Summer 1995 A review of the evolution of interest-rate models 153 H u ll, Joh n , an d Alan Wh ite “New Ways with th e Yield Cu rve.” Risk, October 1990 A comparison of th ree in terest-rate models Koenigsberg, Mark, Janet Showers, and James Streit “The Term Structure of Volatility and Bond Option Valuation.” The Journal of Fixed Income, September 1991 An illuminating discussion of bond-option models, with particular attention paid to the use of a time-dependent volatility parameter Kopprasch, Robert W “Option-Adjusted Spread Analysis: Going Down the Wrong Path.” Financial Analysts Journal, May–June 1994 A discussion of the pitfalls of OAS analysis Longstaff, Francis A., and Eduardo S Schwartz “Interest Rate Volatility an d Bon d Prices.” Financial Analysts Journal, July–August 1993 Th e effect of changes in bond-market volatility on default-free bonds Stigum, Marcia Money Market Calculations: Yields, Break-Evens, and Arbitrage Homewood, IL: Dow Jon es–Irwin , 1981 A detailed, compreh en sive treatment of short-term price–interest-rate calculations 154 Introduction to Option-Adjusted Spread Analysis: Revised and Expanded Third Edition of the OAS Classic by Tom Windas Revised by Tom Miller © 1993, 1996, 2007 by Bloomberg L.P GLOSSARY American option An option that may be exercised over a time period that begins sometime before, and terminates on, the option expiration date See European option arbitrage-free binomial tree of risk-free short rates See lognormal model at-the-money option An option whose strike price is equal or close to the underlying security’s current market price benchmark yield curve The yield curve described by benchmark, or riskfree, notes and bonds In the United States, the most recently issued Treasury bills, notes, and bonds are considered benchmark issues Bermudian option With regards to fixed-income securities, any embedded option whereby the bondholder has sold the issuer the right to repurchase the bond back from the investor, on interest payment dates only, from the time that the bond is first callable until its maturity date binomial distribution A probability distribution created by the results of a random process for which there are only two possible, mutually exclusive outcomes binomial process A sequen cin g of even ts wh ere on ly two possible outcomes develop from a preceding event bullet bond A bond with a simple cash-flow structure that provides coupon, or interest, payments at regular intervals over the life of the issue and repays the full principal amount to investors at maturity callable bond A bond containing a provision that allows the issuer to retire the debt before the scheduled maturity 145 call option An option tract in wh ich th e option buyer acquires th e right, but not the obligation, to buy a prescribed amount of an underlying security from the option seller at a specified price during a specific exercise period cheap security A security whose expected return is greater than that provided by issues with similar risks and is therefore considered undervalued See rich security continuously callable bonds A class of callable bon ds th at become eligible to be called sometime before maturity and remain callable until maturity Continuous call features are usually modeled as American call options convexity A measure of the extent to which a bond’s duration changes with its yield Positively convex bonds, such as bullet and put bonds, have prices th at rise more for a down ward ch an ge in yield th an th ey fall for an equal upward change Negatively convex bonds, such as callable bonds, have prices that rise less for a downward yield change than they fall for an equal upward change Mathematically, the convexity (CVX) of a bon d with h cash flows, each with a un ique time un til paymen t t j and a present value Pj, m compounding periods per year, yield y, and a total present value (price plus accrued interest) of P, is given as: CVX = × (1 + y/ m) h j =1 (j × ( j+1) × Pj) (m2 × P) The change in a bond’s price for a given change in yield, due to its convexity, is given by: P = 0.5 × ( y)2 × CVX × 100 P discretely callable bonds A class of bonds that are eligible to be called only on single dates or on a series of specific dates, such as coupon dates Single-date discrete call features are usually modeled as European call options, whereas multiple-date discrete calls are modeled as American calls with discrete exercise dates duration (effective) A type of modified duration that is solved for from observed changes in a bond’s price due to known interest-rate shifts, rather than from price-yield mathematics 146 duration (Macaulay) A measure of a bon d’s price volatility, devised by Frederick Macaulay in 1938 Macaulay duration is the present-valueweighted time to receipt of a bond’s cash flows, in essence describing th e average len gth of time for a dollar to be paid to a bon dh older Th is measure acts as a proportion ality factor th at relates a percen tage change in the price of a bond to a percentage change in its yield Mathematically, the Macaulay duration (Dmacaulay) of a bond with h cash flows, each with a unique time until payment t j and a present value Pj, m compounding periods per year, yield y, and a total present value (price plus accrued interest) of P, is given as: Dmacaulay = h j=1 (Pj × t j ) P The Macaulay-duration-predicted price change resulting from a specified yield change is given by: P P = –Dmacaulay × y (1 + y/ m) duration (modified) Also referred to as adjusted duration, a measure of a bond’s percentage change in price for a given absolute change in its yield Mathematically, modified duration is given as: Dmodified = Dmacaulay × (1 + y/ m) The modified-duration-predicted price change from a specified absolute yield change is given by: P = –Dmodified × y P For positively convex bonds, modified duration underestimates price gains and overestimates price losses; it overestimates price gains and underestimates price losses for negatively convex bonds duration (option-adjusted spread) A type of effective duration determined by measuring the change in price of a bond caused by perturbing the term structure of interest rates in a given manner ( such as a parallel 147 yield-curve shift) and holding the assumed volatility and the OAS of the bond constant embedded option A hypothetical put or call option that models the earlyprincipal-redemption provision of a nonbullet bond European option An option that may be exercised only on its expiration date See American option exercise price The price at which an option holder has the right to buy (in a call option) or sell (in a put option) the underlying security expiration date The date on which the provisions of an option contract cease to be in effect historical volatility See volatility hypothetical options See embedded option implied benchmark forward rate A break-even interest rate for a future (forward) time period that is solved for from implied benchmark spot rates implied benchmark spot rate An interest rate, solved for from benchmark bon d yields, th at accurately prices a paymen t from an y ben ch mark bond occurring on a particular future date Such rates are sometimes referred to as theoretical zero-coupon rates implied credit rating Th e credit ratin g of a class of bon ds wh ose yield curve generates a price for a given bond that is in closest agreement with its observed market price implied spot curve A curve generated by plotting spot rates versus their maturities implied volatility See volatility incremental return The portion of a bond’s return that exceeds that provided by otherwise similar risk-free securities in-the-money option An option that possesses intrinsic value A call option is in the money when it is exercisable and the price of the underlying security is greater th an th e call’s strike price; a put option is in th e money when it is exercisable and the price of the underlying security is lower than the put’s strike price intrinsic value A measure of th e profitability of exercisin g an option immediately; th e differen ce between th e option ’s strike price an d the current market price of the underlying instrument In-the-money option s h ave in trin sic value; at-th e-mon ey option s h ave little or n o 148 intrinsic value; out-of-the-money options have no intrinsic value See time value lognormal model A fixed-in come option model th at gen erates in terest rates that are lognormally distributed—that is, rates whose logarithms describe a n ormal distribution A logn ormal bin omial in terest-rate model generates two mutually exclusive outcomes for an interest rate from any node; the distribution of rates from all nodes in a given time period is lognormally distributed long an option Pay a fee (premium) for the right, but not the obligation, to exercise a transaction at a predetermined price nonbullet bond A bond containing provisions allowing principal repayment, in whole or in part, before the stated maturity option A contract that gives the buyer the right, but not the obligation, to buy from, or sell to, the option seller a prescribed amount of an underlying instrument at a specified price during a specific exercise period See call option and put option option-adjusted spread (OAS) The constant basis-point spread that must be applied to the rates in a fixed-income option model to recover the price of the bond being analyzed When the analysis is conducted using risk-free rates, the OAS is viewed as the incremental return of the bond True adjustments to this spread occur only when the bond contains an embedded option option-adjusted spread (OAS) analysis A financial-analysis method that analyzes the impact of any options embedded in a bond’s structure and measures the issue’s expected incremental return option-free price The price of a bond after adjusting for the value of any embedded options In callable bonds, the option-free price is the sum of the bond price and the value of its embedded option; in putable bon ds, it is th e bon d price less th e embedded option ’s value See option-free yield option-free yield The yield to maturity associated with a bond’s optionfree price This is the yield a bond would have if it were a bullet See option-free price option-free yield curve A curve built from option-free yields option-free yield spread The yield spread associated with a bond’s optionfree yield to maturity 149 option premium The amount paid to purchase, or received from the sale of, an option option strike price See exercise price option value The worth of an option at a particular point in time, as computed by an option-valuation model option-valuation models The broad class of financial analysis tools designed to estimate the value of options out-of-the-money option An option that possesses no intrinsic value A call option is out of the money when the price of the underlying security is less than the call’s strike; a put option is out of the money when the price of the underlying is greater than the put’s strike overhedged A condition in which the risk exposure of a hedge position exceeds that of the primary position being hedged par When an instrument is trading at 100 percent of face value Also used to reference the total face value of a quantity of bonds percent volatility of short rates The relative uncertainty of future prices of an instrument It represents an annualized standard deviation of daily change in price price sensitivity The extent to which a bond’s price changes when a factor to which its price is linked, such as interest rates, changes putable bond A bond that contains a provision that allows the bondholder to receive the principal before the scheduled maturity put option An option tract in wh ich th e option buyer acquires th e right, but not the obligation, to sell a prescribed amount of an underlying security to the option seller at a specified price during a specific exercise period rich security A security whose expected return is less than that provided by issues with similar risks and is therefore considered overvalued See cheap security sector implied volatility The volatility implied by prices of nonbullet bonds in a given sector See volatility sector yield curve A yield curve composed of bonds sharing closely similar credit ratings and issuer types short an option Write or sell an option giving the buyer the right to exercise a transaction at a predetermined price range in exchange for paying a premium 150 short rate In th e logn orm al in terest-rate m od el, th e term u sed to describe any of the individual short-term interest rates composing a given period’s interest-rate distribution sinking-fund bond A bond containing a provision allowing the issuer to sink, or pay back to bondholders, portions of the bond’s principal periodically before maturity Sinks may be mandatory, in which case the issuer must sink the specified amounts on specified dates, or voluntary, in which the issuer has discretion as to whether a sink will take place stochastic Possessin g a ran dom or ch aotic ch aracteristic, like th e real world Risk managers tend to prefer such models, but the computational requirements are significantly higher strike price See exercise price swaptions Options on forward-starting interest-rate swaps term structure of interest rates The relationship between interest rates and time to maturity, as exhibited by closely similar bonds, such as riskfree Treasury issues term structure of volatility (TSOV) The relationship between interest-rate volatility and time to maturity, as exhibited by a particular market sector Fixed-in come option models th at require th e specification of a term structure of volatility as an input are called TSOV models theoretical fair-value analysis A method of financial analysis that estimates the theoretical value of a particular bond based on the values of closely similar issues theoretical zero-coupon rates See implied benchmark spot rate time value The difference between an option’s total value (option value) and its intrinsic value See intrinsic value underhedged A condition in which the risk exposure of a hedge position is less than that of the primary position being hedged volatility (assumed) The volatility specified as the uncertainty assumption in an option-evaluation model Most practitioners favor using implied volatilities derived from observed prices of relevant options as the basis for a volatility assumption Historical volatilities are used when implied values are unavailable or unknown volatility curve See term structure of volatility volatility (historical) A statistical measure of the variability or uncertainty of a security’s value, return , or oth er performan ce attribute over a 151 specified historical period This volatility measure may be computed for any security for which sufficient historical performance data exist and is a measure of past behavior volatility (implied) The volatility of an underlying security’s performance th at is implied by th e price an d exercise attributes of an option on that security This volatility measure may be computed only from an option’s price and is an indication of the market’s expectation about future behavior yield The single interest rate that present-values a designated series of future payments to a given present value For bonds, the yield is calculated to a designated redemption date, such as maturity or worst call Also known as “internal rate of return.” See EQUATION 1.1 on page yield curve A cu rve gen erated by p lottin g bon d yield s versu s th eir maturities yield spread The difference between the yields of two bonds yields-to-call analysis An analysis that calculates, for a given price, the yield to each possible future redemption date of a bond yield to workout The yield of a bond to a specified assumed redemption (workout) date yield-to-worst analysis An analysis that calculates, for a given price, the yield to each possible future redemption date of a bond, with the goal of identifying the redemption date with the lowest yield This is the worst-case yield from an investor’s standpoint 152 Introduction to Option-Adjusted Spread Analysis: Revised and Expanded Third Edition of the OAS Classic by Tom Windas Revised by Tom Miller © 1993, 1996, 2007 by Bloomberg L.P INDEX adjusting curves, 139–140 American options, 16 arbitrage-free binomial tree of riskfree short rates See binomialtree option model asset-backed securities, option-adjusted spread analysis and, 144 assumed volatility, 62 at the money, 30 benchmark forward rates See implied benchmark forward rates benchmarks, yield and risk and, 7–8 benchmark spot rates See implied benchmark spot rates benchmark yield curve defined, 44 hypothetical example, 45–46 benchmark yields, implied spot rates and implied forward rates and, 49–52 Bermudian option, 16 binomial distribution, 37 binomial process, 37 binomial tree of outcomes, 37–40 binomial-tree option model (lognormal model, arbitrage-free binomial tree of risk-free short rates), 40 See also price tree for bullet bond; price tree for callable bond; price tree for cash flow benchmark yield curve hypothetical example, 45–46 benchmark yields, implied spot rates, and implied forward rates, 49–52 bullet bonds and, 83–98 callable bonds and, 99–112 compound-interest equations, 46–49, 69–70 implied forward rates example, 57–58 interest-rate process example, 58–59 introduction to, 43–45 limitations of, 53–55 matching the model to the market, 67–79 nonbullet bonds and, 99–112 price equation, 70–71 price equations incorporating spread, 88–89, 93–94 risk-free short rates examples, 62–66, 75–79 volatility and, 57–66 bonds, as a portfolio, 15–25 See also under type of bond: e.g., bulet bonds, callable bonds, etc bullet bond(s) binomial-tree option model and, 83–98 calculating yield for, 8–9 defined, equivalency equations, 22–24 price tree for bond as a whole, 95–96 price tree for bond less coupon payments, 97–98 price tree for one-year, 85–86 price tree for one-year incorpo- 155 rating spread, 90–91 price tree for 1.5-year, 86–87 price tree for 1.5-year incorporating spread, 91, 92 price tree for six-month, 84–85 price tree for six-month incorporating spread, 89–90 callable bond(s), binomial-tree option model and, 99–112 cash-flow examples of, 19–21 components equation, 31–32 continuously, 19 defined, 2, 18–19 discretely, 19 equivalency equation, 24 intrinsic value of call option, 111–112 price tree for, less coupon payments, 104–105 price tree for, less coupon payments and incorporating spread, 107–109 price tree for embedded call option, 110–111 price tree for one-year incorporating spread, 106–107 price tree for 1.5-year, 103–104 price tree for 1.5-year incorporating spread, 107–108 price tree for six-month incorporating spread, 105–107 price tree for underlying bullet, 101–102 call option defined, 16 embedded, price tree for, 110– 111 intrinsic value of, 111–112 cash flow callable bonds, examples of, 19–21 price tree for one-year, 71–72 price tree for 1.5-year, 72–75 price tree for six-month, 68–69 156 putable bond, example of a, 17–18 cheap (undervalued), 32 compound-interest equations, 46–49, 69–70 continuously callable, 19 convertible bonds, option-adjusted spread analysis and, 144 convexity, 10 credit rating, implied, 133 curves adjusting, 139–140 reference, 136–139 discretely callable, 19 duration, effective, 120 modified, 119 option-adjusted spread, 120 effective duration, 120 embedded call option of a callable bond, price tree for, 110–111 European options, 16 exercise (strike) price, 16 expiration date, 16 fair value, using option-adjusted spread analysis to evaluate introduction to, 125–128 stage I of, 128 stage II of, 128–130 stage III of, 130–131 theoretical fair-value analysis, 126, 131–134 forward rates, implied benchmark, defined, 44 benchmark yields and implied spot rates and, 49–52 binomial-tree example, 57–58 historical volatility, 59, 141 implied benchmark forward rates benchmark yields and implied spot rates and, 49–52 defined, 44 implied benchmark spot rates benchmark yields and implied forward rates and, 49–52 defined, 46 implied credit rating, 133 implied forward rates, binomial-tree example, 57–58 implied spot curve, 94 implied volatility, 59, 141 sector, 130 incremental return See yield spread interest rate models, option valuation and, 140–143 in the money, 30 intrinsic value callable-bond components equation, 31–32 of call option, 111–112 defined, 29 examples of, 29–30 option-value components equation, 31 putable-bond components equation, 31–32 lognormal model See binomial-tree option model long an option, 16, 141 modified duration, 119 mortgage-backed securities, optionadjusted spread analysis and, 144 municipal bonds, option-adjusted spread analysis and, 144 nonbullet bonds, binomial-tree option model and, 99–112 intrinsic value of call option, 111–112 price tree for callable bond less coupon payments, 104–105 price tree for callable bond less coupon payments and incor- porating spread, 107–109 price tree for callable bond’s underlying bullet, 101–102 price tree for embedded call option of a callable bond, 110–111 price tree for one-year callable bond incorporating spread, 106–107 price tree for 1.5-year callable bond, 103–104 price tree for 1.5-year callable bond incorporating spread, 107–108 price tree for six-month callable bond incorporating spread, 105–107 time value, calculating, 34–40 time value, importance of, 33–34 yield for, calculating, 10–11 OAS (option-adjusted spread), 91 OAS analysis See option-adjusted spread analysis OAS (option-adjusted spread) duration, 120 option(s), American, 16 basic attributes of, 16 Bermudian, 16 call, 16, 110–112 equivalency equations, 22–24 European, 16 hypothetical, 15 long an, 16, 141 put, 16 short an, 16, 141 option-adjusted spread (OAS), 91 option-adjusted spread (OAS) analysis adjusting curves, 139–140 advances in, 135 fair value, estimating, 125–134 interest rate models and option valuation, 140–143 157 other securities, 144 performance, evaluating, 115– 123 reasons for using, 1–3 reference curves, 136–139 things to keep in mind, 143–144 option-adjusted spread (OAS) duration, 120 option-free price, 117 option-free yield, 117 option-free yield curve, 130 option-free yield spread, 118 option premium, 16 option-valuation models, 15 interest rate models and, 140– 143 option (total) value, 29 callable-bond components equation, 31–32 option-value components equation, 31 putable-bond components equation, 31–32 out of the money, 30 overhedged, 14 overvalued (rich), 32 par, 17 percent volatility of short rates defined, 59 equation for, 60–61 performance, using option-adjusted spread analysis to evaluate, 115–123 portfolio, the bond as a, 15–25 premium, option, 16 price exercise (strike), 16 option-free, 117 price equation(s), 70–71 incorporating spread, 88–89, 93–94 price sensitivity, 10 price tree for bullet bond less coupon payments, 97–98 one-year, 85–86 158 one-year incorporating spread, 90–91 1.5-year, 86–87 1.5-year incorporating spread, 91, 92 six-month, 84–85 six-month bullet incorporating spread, 89–90 as a whole, 95–96 price tree for callable bond embedded call option, 110–111 less coupon payments, 104–105 one-year incorporating spread, 106–107 1.5-year, 103–104 1.5-year incorporating spread, 107–108 six-month incorporating spread, 105–107 underlying bullet, 101–102 price tree for cash flow one-year, 71–72 1.5-year, 72–75 six-month, 68–69 price/ yield equation, putable bond(s), cash-flow example of a, 17–18 components equation, 31–32 defined, 2, 17 equivalency equation, 23–24 put option, defined, 16 reference curves, 136–139 rich (overvalued), 32 risk, yield and benchmarks and, 7–8 risk-free short rates, binomial-tree examples, 62–66, 75–79 sector implied volatility, 130 sector yield curve, 128 sensitivity, price, 10 short an option, 16, 141 short rates, 53 percent volatility of, 59–61 risk-free, binomial-tree examples, 62–66, 75–79 sinking-fund bonds defined, 2, 21 treatment of, 21–22 spot curve, implied, 94 spot rates, implied benchmark, defined, 46 benchmark yields and implied forward rates and, 49–52 spread option-adjusted, 91 option-adjusted spread analysis, reasons for using, 1–3 option-adjusted spread duration, 120 option-free yield, 118 price equations incorporating, 88–89, 93–94 spread, price tree incorporating callable, less coupon payments, 107–109 one-year bullet bond, 90–91 one-year callable bond, 106–107 1.5-year, 107–108 six-month bullet bond, 89–90 six-month callable bond, 105–107 stochastic, 140 strike (exercise) price, 16 swaptions, 129 term structure of interest rates, term structure of volatility (TSOV), 54 theoretical fair-value analysis defined, 126 example of, 131–134 time value, 29 binomial tree of outcomes calculations, 37–40 calculating, 34–40 callable-bond components equation, 31–32 defined, 30 expected payout of a toss equation, 34–35 expected payout of n tosses equation, 35–37 importance of, to nonbullet bonds, 33–34 option-value components equation, 31 putable-bond components equation, 31–32 total value See option value undervalued (cheap), 32 U.S government agency bonds, option-adjusted spread analysis and, 144 value See intrinsic value; option value; time value volatility, 53 assumed, 62 binomial-tree option model and, 57–66 historical, 59, 141 implied, 59, 141 percent, of short rates, 59 sector-implied, 130 term structure of (TSOV), 54 volatility curve, 54 yield, benchmark yields, implied spot rates and implied forward rates, 49–52 calculating, 8–11 option-free, 117 risk and benchmarks and, 7–8 yield curve, 44 benchmark, 44–46 option-free, 130 sector, 128 yield spread (incremental return), option-free, 117 yields-to-call analysis, 11 yield to workout, yield-to-worst analysis, examples of, 11–13 limitations of, 3, 11–12, 14 159 ... come securities Option -adjusted spread an alysis I Win das, Tom Introduction to option- adjusted spread analysis II Title HG4650.M56 2007 332.63'2283 dc22 2006101298 Dedicated to the curious and... Cataloging-in-Publication Data Miller, Tom Introduction to option- adjusted spread analysis : revised and expanded third edition of the OAS classic by Tom Windas / revised by Tom Miller ; foreword by Peter... Wilson of Barclays Global Investors, and to Tom Windas for the care, attention, and craftsmanship of the first edition xi Introduction to Option- Adjusted Spread Analysis: Revised and Expanded