Swaps and Other Derivatives Wiley Finance Series Securities Operations: A Guide to Trade and Position Management Michael Simmons Monte Carlo Methods in Finance Peter Jackel Modeling, Measuring and Hedging Operational Risk Marcelo Cruz Building and Using Dynamic Interest Rate Models Ken Kortanek and Vladimir Medvedev Structured Equity Derivatives: The Definitive Guide to Exotic Options and Structured Notes Harry Kat Advanced Modelling in Finance using Excel and VBA Mary Jackson and Mike Staunton Operational Risk: Measurement and Modelling Jack King Advanced Credit Risk Analysis: Financial Approach and Mathematical Models to Assess, Price and Manage Credit Risk Didier Cossin and Hugues Pirotte Dictionary of Financial Engineering John F Marshall Pricing Financial Derivatives: The Finite Difference Method Domingo A Tavella and Curt Randall Interest Rate Modelling Jessica James and Nick Webber Handbook of Hybrid Instruments: Convertible Bonds, Preferred Shares, Lyons, ELKS, DECS and Other Mandatory Convertible Notes Izzy Nelken (ed.) Options on Foreign Exchange, Revised Edition David F DeRosa The Handbook of Equity Derivatives, Revised Edition Jack Francis, William Toy and J Gregg Whittaker Volatility and Correlation in the Pricing of Equity, FX and Interest-rate Options Riccardo Rebonato Risk Management and Analysis vol 1: Measuring and Modelling Financial Risk Carol Alexander (ed.) Risk Management and Analysis vol 2: New Markets and Products Carol Alexander (ed.) Implementing Value at Risk Philip Best Credit Derivatives: A Guide to Instruments and Applications Janet Tavakoli Implementing Derivatives Models Les Clewlow and Chris Strickland Interest-rate Option Models: Understanding, Analysing and Using Models for Exotic Interest-rate Options (second edition) Riccardo Rebonato Swaps and Other Derivatives Richard Flavell JOHN WILEY & SONS, LTD Copyright © 2002 John Wiley & Sons, Ltd, Baffins Lane, Chichester, West Sussex PO19 1UD, UK National 01243 779777 International (+ 44) 1243 779777 e-mail (for orders and customer service enquiries): cs-books @wiley.co.uk Visit our Home Page on http://www.wiley.co.uk All Rights Reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright 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paper production Contents Preface and Acknowledgements ix Introduction 1.1 Introduction 1.2 Applications of swaps 1.3 An overview of the swap market 1.4 The evolution of a swap market 1.5 Conclusion 1 10 Short-term interest rate swaps Objective 2.1 Discounting, the time value of money and other matters 2.2 Forward rate agreements and interest rate futures 2.3 Short-term swaps 2.4 Future valuing a swap 11 11 11 16 20 31 Generic interest rate swaps Objective 3.1 Generic interest rate swaps 3.2 Pricing through comparative advantage 3.3 The relative pricing of generic IRS 3.4 The relationship between the bond and swap markets 3.5 Implying a discount function 3.6 Building a blended curve 35 35 35 38 41 43 50 58 The pricing and valuation of non-generic swaps Objective 4.1 The pricing of simple non-generic swaps 4.2 Rollercoasters 4.3 A more complex example 4.4 An alternative to discounting 4.5 Swap valuation 65 65 65 72 75 85 85 More complex swaps Objective 5.1 Asset packaging 95 95 95 Contents 5.2 Credit swaps 5.3 Credit-adjusted swap pricing 5.4 Simple mismatch swaps 5.5 Average rate swaps 5.6 Overnight indexed swaps 5.7 Basis swaps 5.8 Yield curve swaps 5.9 Convexity effects of swaps 5.10 Inflation swaps 5.11 Equity and commodity swaps 5.12 Volatility swaps Appendix Measuring the convexity effect 106 121 128 129 131 137 142 152 156 165 175 184 Cross-currency swaps Objective 6.1 Floating-floating cross-currency swaps 6.2 Pricing and hedging of CCBS 6.3 CCBS and discounting 6.4 Fixed-floating cross-currency swaps 6.5 Floating-floating swaps continued 6.6 Fixed-fixed cross-currency swaps 6.7 Cross-currency swaps valuation 6.8 Dual currency swaps 6.9 Cross-currency equity swaps 6.10 Conclusion Appendix Adjustments to the pricing of a quanto diff swap 205 205 205 207 211 224 229 234 241 247 259 262 262 Interest rate OTC options Objective 7.1 Introduction 7.2 The Black option pricing model 7.3 Interest rate volatility 7.4 Par and forward volatilities 7.5 Caps, floors and collars 7.6 Digital options 7.7 Embedded structures 7.8 More complex structures 7.9 Swaptions 7.10 Structures with embedded swaptions 7.11 FX options 7.12 Hedging FX options 267 267 267 268 271 277 288 299 300 307 309 316 320 326 Traditional market risk management Objective 8.1 Introduction 8.2 Interest rate risk management 8.3 Gridpoint risk management — market rates 333 333 333 336 337 mtents vii Index 8.4 Equivalent portfolios 8.5 Gridpoint risk management — forward rates 8.6 Gridpoint risk management — zero coupon rates 8.7 Yield curve risk management 8.8 Swap futures 8.9 Theta risk 8.10 Risk management of IR option portfolios 8.11 Hedging of inflation swaps Appendix Analysis of swap curves 337 340 344 347 355 360 362 373 375 Imperfect risk management Objective 9.1 Introduction 9.2 A very simple example 9.3 A very simple example extended 9.4 Multifactor delta VaR 9.5 Choice of risk factors and cashflow mapping 9.6 Estimation of volatility and correlations 9.7 A running example 9.8 Simulation methods 9.9 Shortcomings and extensions to simulation methods 9.10 Delta-gamma and other methods 9.11 Spread VaR 9.12 Equity VaR 9.13 Stress testing Appendix Extreme value theory 379 379 379 380 386 388 394 399 401 405 414 427 433 439 441 444 447 This page intentionally left blank Preface and Acknowledgements This book is designed for financial professionals to understand how the vast bulk of OTC derivatives are structured, priced and hedged, and ultimately how to use such derivatives themselves A wide range of books already exist that describe in conceptual terms how and why such derivatives are used, and it is not the ambition of this book to supplant them There are also a number of books which describe the pricing and hedging of derivatives, especially exotic ones, primarily in mathematical terms Whilst exotics are an important and growing segment of the market, by far the majority of derivatives are still very much first generation, and as such relatively straightforward For example, interest rate swaps constitute over half of the $100 trillion OTC derivative market, and yet there have been few books published in the last decade that describe how they are created and valued in practical detail So how many of the professionals gain their knowledge? One popular way is "learning on the job", reinforced by the odd training course But swap structures can be quite complex, requiring more than just superficial knowledge, and probably every professional uses a computer-based system, certainly for the booking and regular valuation of trades, and most likely for their initial pricing and risk management These systems are complex, having to deal with real-world situations, and their practical inner details bear little resemblance to the idealized world of most books So often practitioners tend to treat the systems as black boxes, relying on some initial and frequently inadequate range of tests and hoping their intuition will guide them The greatest sources of comfort are often the existing customer list of the system (they can't all be wrong!) and, if the system is replacing an old one, comparative valuations The objective of this book is to describe how the pricing, valuation and risk management of generic OTC derivatives may be performed, in sufficient detail and with various alternatives, so that the approaches may be applied in practice It is based upon some 15 years of varying experience as a financial engineer for ANZ Merchant Bank in London, as a trainer and consultant to banks worldwide, and as Director of Financial Engineering at Lombard Risk Systems responsible for all the mathematics in the various pricing and risk management systems The audience for the book is firstly traders, sales people and front-line risk managers But increasingly such knowledge needs to be more widely spread within financial institutions, such as internal audit, risk control and IT Then there are the counterparties such as organizations using derivatives for risk management, who have frequently identified the need for transparent pricing This need has been exacerbated in recent years as many developed countries now require that these organizations demonstrate the effectiveness of risk management, and also perform regular (usually annual) mark-to- Worksheet 9.17 Market volatility and correlation information Market data between bond curve and three Libor curves Bond ly 3y 5y Libor ly 3y 5y Libor2 ly 3y 5y Libor3 ly 3y 5y Z-c rates Volatilities 4.688% 4.731% 4.855% 5.438% 5.581% 5.805% 6.438% 6.781% 7.205% 8.438% 9.181% 10.205% 18.0% 16.5% 14.0% 20.0% 18.5% 16.0% 24.0% 22.5% 20.0% 29.0% 27.5% 25.0% Volatilities correlation matrix between bond and Libor curves Bond pa Liborl Libor2 Libor3 ly 3y 5y ly 3y 5y ly 3y 5y ly 3y 5y Bond ly 3y 5v 18% 17% 14% 0.9 0.8 0.9 0.95 0.8 0.95 0.7 0.65 0.5 0.65 0.7 0.65 0.45 0.55 0.6 0.3 0.3 0.2 0.2 0.3 0.3 0.2 0.2 0.25 0.2 0.2 0.15 0.15 0.25 0.25 0.15 0.15 0.2 Liborl ly 3y 5y Libor2 ly 3y 5y 20% 19% 16% 24% 23% 20% 0.7 0.65 0.45 0.3 0.2 0.2 0.65 0.7 0.55 0.85 0.7 0.2 0.15 0.1 0.85 0.9 0.7 0.9 0.2 0.2 0.15 0.15 0.25 0.25 0.1 0.17 0.2 0.25 0.3 0.2 0.2 0.3 0.3 0.2 0.15 0.25 0.3 0.3 0.2 0.5 0.65 0.6 0.2 0.3 0.25 0.2 0.25 0.17 0.15 0.25 0.2 0.9 0.75 0.9 0.9 0.75 0.9 0.6 0.7 0.65 0.5 0.65 0.75 Libor3 ly 3y 5y 29% 28% 25% 0.2 0.15 0.15 0.2 0.25 0.15 0.15 0.25 0.2 0.25 0.2 0.2 0.3 0.3 0.15 0.2 0.3 0.25 0.75 0.6 0.5 0.85 0.7 0.65 0.85 0.65 0.75 0.75 0.85 0.85 0.65 0.65 0.65 0.75 0.65 0.75 Worksheet 9.17 Volatility and correlation information (continued) Market data between bond curve and three spread curves Z-c rates Volatilities 4.688% 4.731% 4.855% 0.750% 0.850% 0.950% 1.000% 1.200% 1.400% 2.000% 2.400% 3.000% 18.0% 16.5% 14.0% 104.1% 87.0% 79.3% 170.2% 134.5% 110.7% 82.2% 75.8% 58.4% Bond ly 3y 5y Spread ly 3y 5y Spread2 ly 3y 5y Spread3 ly 3y 5y Correlation matrix between bond and spread curves Bl B2 B3 S211 Bl B2 B3 S11 S12 S13 0.90 0.80 0.90 0.95 0.80 0.95 –0.11 –0.07 –0.17 –0.04 –0.08 –0.10 –0.17 –0.18 –0.16 -0.17 -0.14 -0.14 0.74 0.75 0.74 0.92 0.75 0.92 -0.45 -0.35 -0.26 S11 S12 S13 -0.11 -0.04 -0.17 -0.07 -0.08 -0.18 -0.17 -0.10 -0.16 S211 S212 S213 -0.17 -0.23 -0.08 -0.14 -0.16 -0.14 -0.14 -0.13 -0.13 -0.45 -0.31 -0.36 -0.35 -0.39 -0.38 S212 S213 S321 S322 S323 -0.23 -0.16 -0.13 -0.08 -0.14 -0.13 0.02 0.02 0.04 0.04 0.10 0.10 0.05 0.05 0.09 –0.31 -0.39 -0.30 -0.36 -0.38 -0.39 0.24 0.34 0.16 0.17 0.19 0.17 0.23 0.05 0.17 -0.26 -0.30 -0.39 0.91 0.76 0.91 0.89 0.76 0.89 0.04 0.23 0.43 -0.03 -0.01 0.01 -0.03 0.14 0.13 0.16 0.17 0.17 0.04 -0.03 -0.03 0.23 -0.01 0.14 0.43 0.01 0.13 0.21 0.26 0.21 0.60 0.26 0.60 C/3 73 (/> V S321 S322 S323 0.02 0.04 0.05 0.02 0.10 0.05 0.04 0.10 0.09 0.24 0.17 0.23 0.34 0.19 0.05 O Imperfect Risk Management 439 9.12 EQUITY VaR Finally, a brief look at calculating VaR when there are equities (or indeed commodities) in the portfolio Equities may be handled quite simply by treating each one as a separate risk factor For example, consider the following simple USD portfolio: Holding 100,000 500,000 ,000.000 Stock Stock Index Equity forward Total (USD) Total (DEM) 3mo USD Libor USD -DEM spot rate Current price 10 Current value (USD) 1,000,000.00 2,000,000.00 5,000,000.00 -896,805.90 7,103,194.10 12,785.749.39 7% 1.8 It consists of two stocks, a holding of the index, and an equity forward contract to pay $12 per share on 500,000 shares of Stock in months' time Assuming (quite simplistically) zero growth in the share price, the value of the forward is 500,000 x [10-12xDF3] We wish to calculate day, 95% VaR in DEM We therefore have five risk factors: the two stock prices, the index price, 3mo Libor and the spot rate Given appropriate volatilities and correlations, it is straightforward to calculate VaR = $712,472 (see Worksheet 9.18) However using individual stocks may increase the data requirements significantly Consider for example a single portfolio replicating the S&P 500: the number of crosscorrelations is in excess of 100,000! For a bank in which equity constitutes a significant proportion of activity, the accuracy provided by modelling the individual stocks may well warrant the time and cost of collecting and cleansing the data But for many organizations, the effort is simply not worthwhile Beta analysis of the equity market is very common, with beta defined in: rs = a, + ßs r1 + ex where rs, and r1 are the return on a share s and on the index / respectively; as is excess return on shares (in theory, this should be zero); ßs is the coefficient linking share performance to the index; ss is an error term, assumed to be uncorrelated with either the market or the other stocks We can therefore write23: var(r s ) = (ß s ) var(r ) + var(s) If we can assume that var(es) is negligible, then cr(s) = ßs x a(I) We can therefore replace the individual stocks in the above example by the index, i.e.: VaR(equity) = {