Volatility and Correlation: The Perfect Hedger and the Fox / Edition 2

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In Volatility and Correlation 2nd edition: The Perfect Hedger and the Fox, Rebonato looks at derivatives pricing from the angle of volatility and correlation. With both practical and theoretical applications, this is a thorough update of the highly successful Volatility & Correlation – with over 80% new or fully reworked material and is a must have both for practitioners and for students.

The new and updated material includes a critical examination of the ‘perfect-replication’ approach to derivatives pricing, with special attention given to exotic options; a thorough analysis of the role of quadratic variation in derivatives pricing and hedging; a discussion of the informational efficiency of markets in commonly-used calibration and hedging practices. Treatment of new models including Variance Gamma, displaced diffusion, stochastic volatility for interest-rate smiles and equity/FX options.

The book is split into four parts. Part I deals with a Black world without smiles, sets out the author’s ‘philosophical’ approach and covers deterministic volatility. Part II looks at smiles in equity and FX worlds. It begins with a review of relevant empirical information about smiles, and provides coverage of local-stochastic-volatility, general-stochastic-volatility, jump-diffusion and Variance-Gamma processes. Part II concludes with an important chapter that discusses if and to what extent one can dispense with an explicit specification of a model, and can directly prescribe the dynamics of the smile surface.

Part III focusses on interest rates when the volatility is deterministic. Part IV extends this setting in order to account for smiles in a financially motivated and computationally tractable manner. In this final part the author deals with CEV processes, with diffusive stochastic volatility and with Markov-chain processes.

Praise for the First Edition:

“In this book, Dr Rebonato brings his penetrating eye to bear on option pricing and hedging.… The book is a must-read for those who already know the basics of options and are looking for an edge in applying the more sophisticated approaches that have recently been developed.”
—Professor Ian Cooper, London Business School

“Volatility and correlation are at the very core of all option pricing and hedging. In this book, Riccardo Rebonato presents the subject in his characteristically elegant and simple fashion…A rare combination of intellectual insight and practical common sense.”
—Anthony Neuberger, London Business School

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Product Details

  • ISBN-13: 9780470091395
  • Publisher: Wiley
  • Publication date: 9/28/2004
  • Series: Wiley Finance Series, #278
  • Edition description: REV
  • Edition number: 2
  • Pages: 864
  • Sales rank: 1,166,533
  • Product dimensions: 6.90 (w) x 9.70 (h) x 2.10 (d)

Meet the Author

Riccardo Rebonato is Head of Group Market Risk for the Royal Bank of Scotland Group, and Head of The Royal Bank of Scotland Group Quantitative Research Centre. He is also a Visiting Lecturer at Oxford University for the Mathematical Finance Diploma and MSc. He holds Doctorates in Nuclear Engineering and Science of Materials/Solid State Physics. He sits on the Board of Directors of ISDA and on the Board of Trustees of GARP.
Prior to joining the Royal Bank of Scotland, he was Head of Complex Derivatives Trading Europe and Head of Derivatives Research at Barclays Capital (BZW), where he worked for nine years.
Before that he was a Research Fellow in Physics at Corpus Christi College, Oxford, UK. He is the author of three books, Modern Pricing of Interest-Rate Derivatives, Volatility and Correlation in Option Pricing and Interest-Rate Option Models. He has published several papers on finance in academic journals, and is on the editorial board of several journals. He is a regular speaker at conferences worldwide.

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Read an Excerpt

Volatility and Correlation

The Perfect Hedger and the Fox
By Riccardo Rebonato

John Wiley & Sons

ISBN: 0-470-09139-8

Chapter One

Theory and Practice of Option Modelling

1.1 The Role of Models in Derivatives Pricing

1.1.1 What Are Models For?

The idea that the price of a financial instrument might be arrived at using a complex mathematical formula is relatively new, and can be traced back to the Black-and-Scholes (1973) formula. Of course, formulae were used before then for pricing purposes, for instance in order to convert the price of a bond into its gross redemption yield. However, these early (pre Black-and-Scholes) formulae by and large provided a very transparent transformation from one set of variables to another, and did not carry along a heavy baggage of model assumptions. The Black-and-Scholes formula changed all that, and we now live in a world where it is accepted that the value of certain illiquid derivative securities can be arrived at on the basis of a model (the acceptance of this is the basis of the practice of marking-to-model).

The models that developed from the family tree that has Black-and-Scholes at its roots shared the common assumptions that the estimation of the drift (growth rate, trend) component of the dynamics of the relevant financial driver was not relevant to arrive at the price of the derivative product. This insight directly follows from the concept of payoff replication, and is discussed in detail inthis book in Chapter 2.

In order to implement these models practitioners paid more and more attention to, and began to collect, direct empirical market data at a very 'atomistic' (often transactional) level. This was done for several reasons: for instance, for assessing the reasonableness of a model's assumptions, or for seeking guidance in the development of new models, or for estimating the inputs of existing models. The very availability of this wealth of information, however, suggested new opportunities. Perhaps, embedded in these data, there could be information about the market microstructure that could provide information not only about the 'volatility' of a price series, but also about its short-term direction.

Again, the practice was not strictly new, since the idea of predicting future price movements from their past history ('chartism' in a generalized sense) pre-dated Black-and-Scholes probably by decades. Yet these earlier approaches (which, incidentally, never won academic respectability), were typically based on, at most, daily observations, and purported to make predictions over time-scales of weeks and months. The new, transactional-level data, on the other hand, were made up of millions of observations, sometimes collected (as in the case of FX trades) minutes or seconds apart. The availability of these data made possible the calibration of predictive models, which try to anticipate stock price movements over time-scales sometimes as short as a few minutes.

This was just the type of data and models that many of the new, and, in the early 2000s, immensely popular, hedge funds required in order to try to 'get an edge' over an ever-growing competition (in 2001 one new hedge fund was being launched every week in continental Europe alone). These hedge funds and the proprietary trading desks of internationally active banks therefore become the users and developers of a second breed of models, which differed from the members of the Black-and-Scholes family because they were explicitly trying to have a predictive directional power. Unlike the early rather crude chartist approaches, these new models employed very complex and sophisticated mathematical techniques, and, if they were not being routinely published in academic journals, it had more to do with the secretive nature of the associated research than with any lack of intellectual rigour.

Two types of model had therefore developed and coexisted by the end of the 1990s: models as predictors (the 'hedge-fund models') and models as payoff-replication engines (the 'derivatives models'). With some caveats, the distinction was clear, valid and unambiguous. It is the derivatives models that are the subject of this book.

Having said that, recent developments in the derivatives industry have increasingly blurred this once-clear-cut distinction. Products have appeared whose payoff depends to first order on, say, the correlation between different equity indices (e.g. basket options), or on the correlation between FX rates and/or between different-currency yield curves (e.g. power-reverse-dual swaps), or on possibly discontinuous moves in credit spreads and default correlations (e.g. tranched credit derivatives). The Black-and-Scholes-inspired replication paradigm remains the prevalent approach when trying to price these new-breed models. Yet their value depends to first order on quantities poorly hedgeable and no easier to predict than directional market trends. The underlying model might well assume that these input quantities are deterministic (as is normally the case for correlations), but this does not take away the fact that they are difficult to estimate, that they render payoff replication very complex, if not impossible, and that their real-world realizations influence to a very large extent the variability of the option-plus-hedge portfolio. The result of this state of affairs is that, once the best hedging portfolio has been put in place, there remains an unavoidable variance of return at expiry from the complex option and its hedges.

Sure enough, perfect replication cannot be expected even for the simplest options: markets are not frictionless, trading cannot be continuous, bid-offer spreads do exist, etc. Yet the robustness of the Black-and-Scholes formula (discussed at several points in this book) ensures that the terminal variability of the overall total portfolio is relatively limited. The difference, however, between the early, relatively simple option payoffs and the new, more complex products, while in theory only a matter of degrees, is in practice large enough to question the validity of prices obtained on the basis of the replication approach (i.e. assuming that one can effectively hedge all the sources of uncertainty).

So, in making the price for a complex derivative product, the trader will often have to take a directional market view on the realization of quantities such as correlations between FX rates and yield curves, default frequencies or correlations among forward rates in different currencies and/or equity indices, or sub-sectors thereof. As a consequence, the distinction between predictive models (that explicitly require the ability to predict future market quantities), and models as payoff replication machines (that are supposed to work whatever the future realizations of the market quantities will be) has recently become progressively blurred. This topic is revisited in the final section of this chapter, where I argue that one could make a case for re-thinking current derivatives pricing philosophy, which still implicitly heavily relies on the existence of a replication strategy.

1.1.2 The Fundamental Approach

In option pricing there are at least two prevalent approaches (which I call in what follows the 'fundamental' and 'instrumental') to dealing with models. The general philosophy that underlies the first can be described as follows. We begin by observing certain market prices for plain-vanilla options. We assume that these prices are correct, in the sense that they embody in the best and most complete possible way all the relevant information available about the stochastic process that drives the underlying (and possibly, other variables, such as the stochastic volatility; for simplicity I will confine the discussion to the underlying, which I will also call 'the stock'). We begin by positing that this true process is of a particular form (say, a jump-diffusion). We calculate what the prices should be if indeed our guess was correct. If the call prices derived using the model are not correct, we conclude that we have not discovered the true process for the underlying. If they are better than the prices produced by another model (say, a pure diffusion) we say that we have reason to believe that the new process (the jump-diffusion) could be a more accurate description of the real process for the underlying than the old one (the pure diffusion).

Alternatively, if the model has a large number of 'free parameters' and we believe that the underlying process is correctly specified, we use all of the parameters describing the dynamics of the underlying to recover the market prices of the options. This is what is implicitly done, for instance, with some implementations of the local volatility models (see Chapters 11 and 12).

Or again, if two models give a fit of similar quality to market prices of plain-vanilla options, the conclusion is often drawn that the model that implies the process for the underlying more similar to what is statistically observed in reality is the 'better' one. The relevance of this distinction is that, despite the similarity of the two models in reproducing the plain-vanilla prices, better prices for complex options (i.e. prices in better agreement with the market practice) would be obtained if the superior process were used.

The fundamental approach sounds very sensible. It is, however, underpinned by one very strong assumption: the trader who chooses and calibrates models this way is subscribing to the view that the market-created option prices must be fully consistent with the true, but a priori unknown, process for the underlying. The market, in other words, must be a perfect information-processing machine, which absorbs all the relevant information about the unknown process followed by the 'stock', and produces prices consistent with each other (no arbitrage) and with this information set (informational efficiency).

This implicit assumption is very widespread: take, for instance, the practice of recovering all the observable option prices using a local-volatility model, discussed in Chapter 11. Even if we knew the true process of the underlying to be exactly a diffusion with state-dependent (local) volatility, it would only make sense to determine the shape of the local volatility from the traded option prices if we also believed that these had been correctly created in the first place on the basis of this model. We will see, however (see Chapter 11), that the local-volatility modelling approach will recover by construction any exogenous set of market prices. Therefore, in carrying out the calibration we are implicitly making two assumptions:

1. that we know, from our knowledge of financial markets (as opposed to just from the market prices of options) that the true process for the underlying is indeed a local-volatility diffusion; and

2. that the market has fully incorporated this information in the price-making of plain-vanilla options.

In other words, by following this procedure we do not allow the possibility that the true process was a local-volatility diffusion, but that the market failed to incorporate this information in the prices of plain-vanilla options.

In reality option prices are not exogenous natural phenomena, nor are they made by omniscient demi-gods with supernatural knowledge of the 'true' processes for the stochastic state variables. Option prices are made by traders who, individually, might have little or no idea about the true stochastic process for the underlying; who might be using the popular 'model of the month'; who, for a variety of institutional constraints, might be afraid to use a model at odds with the current market practice (see Section 1.2.2); or who might be prevented from doing so by the limit structures in place at their trading houses.

A strong believer in market efficiency would counter that the errors of the individual uninformed traders do not matter, in that they will either cancel each other out (if uncorrelated), or will be eliminated (arbitraged away) by a superior and more unfettered trader with the best knowledge about the true process. I discuss the implications for option pricing of this strong form of market efficiency in the next sections, but this position must be squared with several empirical observations such as, for instance, the fact that steep equity smiles suddenly appeared after the equity market crash of 1987 - did traders not know before the event that the true process had a jump component? Another 'puzzling' fact for the believer of the fundamental view is that a close-to-perfect fit to the S&P500 smile in 2001 can be obtained with a cubic polynomial (see Ait-Sahalia (2002)): do we find it easier to believe that the market 'knows' about the true process for the index, prices options accordingly and when the prices obtained by using this procedure are converted into implied volatilities they magically lie on a cubic line? Or is it not simpler to speculate that traders quote prices of plain-vanilla options with a mixture of model use, trading views about the future volatility and cubic interpolation across strikes?

The matter cannot be settled with a couple of examples. For the moment I simply stress that the common and, prima facie, very sensible ('fundamental') approach to choosing and calibrating a model that I have just described can only be justified if one assumes that the observed prices of options reflect in an informationally efficient way everything that can be known about the true process. I will return to this topic in Section 1.4, which deals with the topic of calibration.

1.1.3 The Instrumental Approach

An alternative ('instrumental') way to look at the choice of process for the underlying is to regard a given process specification as a tool not so much for driving the underlying, but for creating present and future prices of options. In this approach it is therefore natural to compare how the prices of options (not of the underlying) move in reality and in the model. Typically this comparison will not be made in dollar terms, but using the 'implied volatility' language. Since, however, there is a one-to-one correspondence between implied volatilities and option prices (for a given value of the underlying) we can indifferently use either language.


Excerpted from Volatility and Correlation by Riccardo Rebonato Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Preface xxi

0.1 Why a Second Edition? xxi

0.2 What This Book Is Not About xxiii

0.3 Structure of the Book xxiv

0.4 The New Subtitle xxiv

Acknowledgements xxvii

I Foundations 1

1 Theory and Practice of Option Modelling 3

2 Option Replication 31

3 The Building Blocks 75

4 Variance and Mean Reversion in the Real and the Risk-Adjusted Worlds 101

5 Instantaneous and Terminal Correlation 141

II Smiles – Equity and FX 165

6 Pricing Options in the Presence of Smiles 167

7 Empirical Facts About Smiles 201

8 General Features of Smile-Modelling Approaches 237

9 The Input Data: Fitting an Exogenous Smile Surface 249

10 Quadratic Variation and Smiles 293

11 Local-Volatility Models: the Derman-and-Kani Approach 319

12 Extracting the Local Volatility from Option Prices 345

13 Stochastic-Volatility Processes 389

14 Jump–Diffusion Processes 439

15 Variance–Gamma 511

16 Displaced Diffusions and Generalizations 529

17 No-Arbitrage Restrictions on the Dynamics of Smile Surfaces 563

III Interest Rates – Deterministic Volatilities 601

18 Mean Reversion in Interest-Rate Models 603

19 Volatility and Correlation in the LIBOR Market Model 625

20 Calibration Strategies for the LIBOR Market Model 639

21 Specifying the Instantaneous Volatility of Forward Rates 667

22 Specifying the Instantaneous Correlation Among Forward Rates 687

IV Interest Rates – Smiles 701

23 How to Model Interest-Rate Smiles 703

24 (CEV) Processes in the Context of the LMM 729

25 Stochastic-Volatility Extensions of the LMM 751

26 The Dynamics of the Swaption Matrix 765

27 Stochastic-Volatility Extension of the LMM: Two-Regime Instantaneous Volatility 783

Bibliography 805

Index 813

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