Advanced Equity Derivatives E-Book

Sebastien Bossu

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Editions
• Sprache: Englisch

Seitenzahl: 162

Veröffentlichungsjahr: 2014

Beschreibung

In Advanced Equity Derivatives: Volatility and Correlation, Sébastien Bossu reviews and explains the advanced concepts used for pricing and hedging equity exotic derivatives. Designed for financial modelers, option traders and sophisticated investors, the content covers the most important theoretical and practical extensions of the Black-Scholes model. Each chapter includes numerous illustrations and a short selection of problems, covering key topics such as implied volatility surface models, pricing with implied distributions, local volatility models, volatility derivatives, correlation measures, correlation trading, local correlation models and stochastic correlation. The author has a dual professional and academic background, making Advanced Equity Derivatives: Volatility and Correlation the perfect reference for quantitative researchers and mathematically savvy finance professionals looking to acquire an in-depth understanding of equity exotic derivatives pricing and hedging.

Leseprobe

Table of Contents

Cover

Series Page

Title Page

Copyright

Dedication

Foreword

Preface

Acknowledgments

Chapter 1: Exotic Derivatives

1.1 Single-Asset Exotics

1.2 Multi-Asset Exotics

1.3 Structured Products

References

Problems

Chapter 2: The Implied Volatility Surface

2.1 The Implied Volatility Smile and Its Consequences

2.2 Interpolation and Extrapolation

2.3 Implied Volatility Surface Properties

2.4 Implied Volatility Surface Models

References and Bibliography

Problems

Chapter 3: Implied Distributions

3.1 Butterfly Spreads and the Implied Distribution

3.2 European Payoff Pricing and Replication

3.3 Pricing Methods for European Payoffs

3.4 Greeks

References

Problems

Chapter 4: Local Volatility and Beyond

4.1 Local Volatility Trees

4.2 Local Volatility in Continuous Time

4.3 Calculating Local Volatilities

4.4 Stochastic Volatility

References

Problems

Chapter 5: Volatility Derivatives

5.1 Volatility Trading

5.2 Variance Swaps

5.3 Realized Volatility Derivatives

5.4 Implied Volatility Derivatives

Problems

Chapter 6: Introducing Correlation

6.1 Measuring Correlation

6.2 Correlation Matrices

6.3 Correlation Average

6.4 Black-Scholes with Constant Correlation

6.5 Local Volatility with Constant Correlation

References

Problems

Chapter 7: Correlation Trading

7.1 Dispersion Trading

7.2 Correlation Swaps

Problems

Chapter 8: Local Correlation

8.1 The Implied Correlation Smile and Its Consequences

8.2 Local Volatility with Local Correlation

8.3 Dynamic Local Correlation Models

8.4 Limitations

References

Problems

Chapter 9: Stochastic Correlation

9.1 Stochastic Single Correlation

9.2 Stochastic Average Correlation

9.3 Stochastic Correlation Matrix

References

Problems

Appendix 9.A: Sufficient Condition for Lower Bound Unattainability

Appendix 9.B: Necessary Condition for Upper Bound Unattainability

Appendix A: Probability Review

A.1 Standard Probability Theory

A.2 Random Variables, Distribution, and Independence

A.3 Conditioning

A.4 Random Processes and Stochastic Calculus

Appendix B: Linear Algebra Review

B.1 Euclidean Spaces

B.2 Square Matrix Decompositions

Solutions Manual

Chapter 1: Exotic Derivatives

Chapter 2: The Implied Volatility Surface

Chapter 3: Implied Distributions

Chapter 4: Local Volatility and Beyond

Chapter 5: Volatility Derivatives

Chapter 6: Introducing Correlation

Chapter 8: Local Correlation

Chapter 9: Stochastic Correlation

Author's Note

About the Author

Index

End User License Agreement

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Guide

Table of Contents

List of Illustrations

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 5.1

Figure 5.2

Figure 5.3

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 7.1

Figure 7.2

Figure 8.1

Figure 8.2

Figure 8.3

Figure 9.1

Figure 9.2

Figure S.1

Figure S.2

Figure S.3

Figure S.4

Figure S.5

Figure S.6

List of Tables

Table 5.1

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Advanced Equity Derivatives

Volatility and Correlation

SÉBASTIEN BOSSU

Cover image: MGB (artmgb.com)

Cover design: Wiley

Copyright © 2014 by Sébastien Bossu. All rights reserved.

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

Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Bossu, Sébastien.

Advanced equity derivatives : volatility and correlation / Sébastien Bossu.

pages cm. – (Wiley finance series)

Includes bibliographical references and index.

ISBN 978-1-118-75096-4 (cloth); ISBN 978-1-118-77484-7 (ePDF); ISBN 978-1-118-77471-7 (ePub)

1. Derivative securities. I. Title.

HG6024.A3B67 2014

332.64′57–dc23

2013046823

“As for the expense,” gravely declared the deputy Haffner who never opened his mouth except on great occasions, “our children will pay for it, and nothing will be more just.”

Emile Zola, La Curée (The Kill)

Foreword

I am pleased to introduce Sébastien Bossu's new book, Advanced Equity Derivatives, which is a great contribution to the literature in our field. Years of practical experience as an exotics structurer, combined with strong theoretical skills, allowed Sébastien to write a condensed yet profound text on a variety of advanced topics: volatility derivatives and volatility trading, correlation modeling, dispersion trading, local and stochastic volatility models, to name just a few.

This book not only reviews the most important concepts and recent developments in option pricing and modeling, but also offers insightful explications of great relevance to researchers as well as traders. For instance, readers will find formulas to overhedge convex payoffs, the derivation of Feller conditions for the Heston model, or an exposition of the latest local correlation models to correctly price basket options.

Perhaps the most exciting aspect of this book is its treatment of the latest generation of equity derivatives, namely volatility and correlation derivatives. Readers will find a wealth of information on these new securities, including original analyses and models to approach their valuation. The chapters on correlation are particularly commendable, as they shed light on an otherwise still obscure area.

The content quality, selection of topics, and level of insight truly set this book apart. I have no doubt that equity derivatives practitioners around the world, be they traders, quants or investors, will find it extremely pertinent, and I wish this book every success.

Peter Carr

Dr. Peter Carr has over 18 years of experience in the derivatives industry and is currently Global Head of Market Modeling at Morgan Stanley, as well as Executive Director of the Math Finance program at NYU's Courant Institute. He has over 70 publications in academic and industry-oriented journals and serves as an associate editor for eight journals related to mathematical finance. Dr. Carr is also the Treasurer of the Bachelier Finance Society, a trustee for the Museum of Mathematics in New York, and has received numerous awards, including Quant of the Year by Risk magazine in 2003, the ISA Medal for Science in 2008, and Financial Engineer of the Year in 2010.

Preface

In 2004, while working as an equity derivatives analyst at J.P. Morgan in London, I came upon an esoteric trade: someone was simultaneously selling correlation and buying it back for a (risky) profit using two different methods. I became obsessed with the rationale behind this trade, and, after writing down the math, I discovered with excitement that with some corrections this trade led to a pure dynamic arbitrage strategy—the kind you normally find only in textbooks.

I could see, however, that transaction costs and other market frictions made the strategy very hard to implement in practice, especially for price takers on the buy side. But the fact remained that correlation could be bought and sold at very different prices, and that didn't make sense to me. So I developed a simple “toy” model to see how this gap might be accounted for, and as I suspected I found that there should be little difference. What this meant is that one of the two correlation instruments involved in the trade, namely the correlation swap, was not priced at “fair value” according to my analysis.

Later on I refined my model, which I introduce in the last chapter of this book among other topics, and reached similar conclusions. I am very pleased that the topic of equity correlation has gained tremendous momentum since 2004, and it is one of this book's ambitions to introduce the work of others in this highly specialized field. I have no doubt that many new exciting results are yet to be discovered in the coming years.

I also wanted to cover other key advanced concepts in equity derivatives that are relevant to traders, quantitative analysts, and other professionals. Many of these concepts, such as implied distributions and local volatilities, are now well-known and established in the field, while others, such as local and stochastic correlation, lie at the forefront of current research.

To get the most out of this book, readers must already be familiar with the terminology and standard pricing theory of equity derivatives, which can be found in my textbook Introduction to Equity Derivatives: Theory & Practice, second edition, also published by John Wiley & Sons.

I relied on a fair amount of advanced mathematics, and therefore a graduate scientific education is a prerequisite here, especially for those readers who want to solve the problems included at the end of each chapter.

The book is made of nine chapters, which are meant to be read sequentially, starting with an exposition of the most widespread exotic derivatives and culminating with cutting-edge concepts on stochastic correlation, which are necessary to correctly price the next generation of equity derivatives such as correlation swaps.

Some simplifications, such as zero interest rates and dividends, were often necessary to avoid convoluted mathematical expressions. I strongly encourage readers to check the particular assumptions used for each formula before transposing it into another context.

I hope this book will prove insightful and useful to its target audience. I am always interested to hear feedback; please do not hesitate to contact me to share your thoughts.

Acknowledgments

I would like to thank Peter Carr for his foreword, and David Hait and his team at OptionMetrics for providing me with very useful option data. I am grateful to my team at Wiley—Bill Falloon, Meg Freeborn, and associates—for their guidance and professionalism throughout the publication process.

Many thanks also to a group of individuals who, directly or indirectly, made this book possible: Romain Barc, Martin Bertsch, Eynour Boutia, Jose Casino, Mauro Cesa, John Dattorro, Emanuel Derman, Jim Gatheral, Fabrice Rouah, Simone Russo, Roberto Silvotti, and Paul Wilmott.

Last, a special mention to John Lyttle at Ogee Group for his help on many figures and problem solutions.

Chapter 1Exotic Derivatives

Strictly speaking, an exotic derivative is any derivative that is not a plain vanilla call or put. In this chapter we review the payoff and properties of the most widespread equity derivative exotics.

1.1 Single-Asset Exotics

1.1.1 Digital Options

A European digital or binary option pays off $1 if the underlying asset price is above the strike K at maturity T, and 0 otherwise:

In its American version, which is more uncommon, the option pays off $1 as soon as the strike level is hit.

The Black-Scholes price formula for a digital option is simply:

where F is the forward price of S for maturity T, r is the continuous interest rate, and σ is the volatility parameter. When there is an implied volatility smile this formula is inaccurate and a corrective term must be added (see Section 2-1.3).

Digital options are not easy to dynamically hedge because their delta can become very large near maturity. Exotic traders tend to overhedge them with a tight call spread whose range may be determined according to several possible empirical rules, such as:

Daily volatility rule

: Set the range to match a typical stock price move over one day. For example, if the annual volatility of the underlying stock is 32% annually; that is, 32%/√252 ≈ 2% daily, a digital option struck at $100 would be overhedged with $98–$100 call spreads.

Normalized liquidity rule

: Set the range so that the quantity of call spreads is in line with the market liquidity of call spreads with 5% range. The quantity of call spreads is

N

/

R

where

N

is the quantity of digitals and

R

is the call spread range. If the tradable quantity of call spreads with range 5% is

V

, the normalized tradable quantity of call spreads with range

R

would be

V

×

R

/ 0.05. Solving for

R

gives . In practice

V

is either provided by the option trader or estimated using the daily trading volume of the stock.

1.1.2 Asian Options

In an Asian call or put, the final underlying asset price is replaced by an average:

where for a set of pre-agreed fixing dates t1 < t2 < < tn ≤ T. For example, a one-year at-the-money Asian call on the S&P 500 index with quarterly fixings pays off , where S0 is the current spot price and S0.25,…, S1 are the future spot prices observed every three months.

On occasion, the strike may also be replaced by an average, typically over a short initial observation period.

Fixed-strike Asian options are always cheaper than their European counterparts, because AT is less volatile than ST.

There is no closed-form Black-Scholes formula for arithmetic Asian options. However, for geometric Asian options where , the Black-Scholes formulas may be used with adjusted volatility and dividend yield , as shown in Problem 1.3.3.

A common numerical approximation for the price of arithmetic Asian options is obtained by fitting a lognormal distribution to the actual risk-neutral moments of AT.

1.1.3 Barrier Options

In a barrier call or put, the underlying asset price must hit, or never hit, a certain barrier level H before maturity:

For a knock-in option, the underlying must hit the barrier, or else the option pays nothing.

For a knock-out option, the underlying must

never

hit the barrier, or else the option pays nothing.

Barrier options are always cheaper than their European counterparts, because their payoff is subject to an additional constraint. On occasion, a fixed cash “rebate” is paid out if the barrier condition is not met.

Similar to digital options, barrier options are not easy to dynamically hedge: their delta can become very large near the barrier level. Exotic traders tend to overhedge them by shifting the barrier a little in their valuation model.

Continuously monitored barrier options have closed-form Black-Scholes formulas, which can be found, for instance, in Hull (2012). The preferred pricing approach is the local volatility model (see Chapter 4).

In practice the barrier is often monitored on a set of pre-agreed fixing dates t1 < t2 < < tn ≤ T. Monte Carlo simulations are then commonly used for valuation.

Broadie, Glasserman, and Kou (1997) derived a nice result to switch between continuous and discrete barrier monitoring by shifting the barrier level H by a factor where β ≈ 0.5826, σ is the underlying volatility, and Δt is the time between two fixing dates.

1.1.4 Lookback Options

A lookback call or put is an option on the maximum or minimum price reached by the underlying asset until maturity:

Lookback options are always more expensive than their European counterparts: about twice as much when the strike is nearly at the money, as shown in Problem 1.3.5.

Continuously monitored lookback options have closed-form Black-Scholes formulas, which can be found, for instance, in Hull (2012). The preferred pricing approach is the local volatility model (see Chapter 4).

In practice the maximum or minimum is often monitored on a set of pre-agreed fixing dates t1 < t2 < < tn ≤ T. Monte Carlo simulations are then commonly used for valuation.

1.1.5 Forward Start Options

In a forward start option the strike is determined as a percentage k of the spot price on a future start date t0 > 0:

At t = t0 a forward start option becomes a regular option. Note that the forward start feature is not specific to vanilla options and can be added to any exotic option that has a strike.

Forward start options have closed-form Black-Scholes formulas. The preferred pricing approach is to use a stochastic volatility model (see Chapter 4).

1.1.6 Cliquet Options

A cliquet or ratchet option