Volatility and Correlation : In the Pricing of Equity, FX, and Interest-Rate Options

by Riccardo Rebonato

• ISBN13: 9780471899983
• ISBN10: 0471899984
• Format: Hardcover
• Copyright: 2000-01-01
• Publisher: WILEY

Summary

In his new book, Riccardo Rebonato introduces financial professionals to the practical and subtle use of the concepts of volatility (the degree of randomness in a price movement) and correlation (the relationship between the changes in value of two financial assets) in the pricing of complex options. By explaining this approach in clear and accessible terms, the author provides traders, risk managers, financial professionals and students with the tools to undertake an effective investigation of option pricing models both at the qualitative and quantitative level. Dr Riccardo Rebonato is Head of Group Market Risk for the NatWest Group, London, UK. He holds Doctorates in Nuclear Engineering and Science of Materials/Solid State Physics. He has recently been appointed Lecturer in Mathematical Finance at Oxford University. Prior to joining NatWest, he was, at the same time, Head of the Complex Derivatives Trading desk and of the Complex Derivatives Research Group at Barclays Capital, where he worked for nine years. Before that he was a Research Fellow in Physics at Corpus Christi College, Oxford He is the author of the highly successful book Interest-Rate Option Models (Wiley, second edition 1998) and has published several papers on finance in academic journals. He is a regular speaker at conferences world-wide.

Author Biography

Dr Riccardo Rebonato is Head of Group Market Risk for the NatWest Group, London, UK. He holds Doctorates in Nuclear Engineering and Science of Materials/Solid State Physics. He has recently been appointed Lecturer in Mathematical Finance at Oxford University. Prior to joining NatWest, he was, at the same time, Head of the Complex Derivatives Trading Desk and of the Complex Derivatives Research Group at Barclays Capital, where he worked for nine years. Before that he was a Research Fellow in Physics at Corpus Christi College, Oxford. He is the author of the highly successful book Interest-Rate Option Models (Wiley, second edition 1998) and has published several papers on finance in academic journals. He is a regular speaker at conferences world-wide.

Table of Contents

Foreword	xiii
Acknowledgements	xix
Case Studies	xxi
PART ONE FOUNDATIONS	1	(70)
Volatility: Fundamental Concepts and Definitions	3	(26)
Introduction and Plan of the Chapter	3	(1)
Fundamental Concepts and Definitions	4	(2)
Hedging Forward Contracts Using Spot Quantities	6	(2)
Hedging Options: Volatilities of Spot and Forward Processes	8	(6)
Definitions	14	(4)
A Series of Options on Futures Contracts	18	(1)
Hedging an Option with a Forward-Setting Strike	18	(4)
Switching from the Real World to the Pricing Measure	22	(7)
Variance and Mean Reversion in the Real and the Risk-Adjusted Worlds	29	(22)
Introduction and Plan of the Chapter	29	(1)
Hedging a Plain-Vanilla Option in the Presence of Constant Volatility	30	(4)
Hedging a Plain-Vanilla Option in the Presence of Time-Dependent Volatility	34	(7)
First View	35	(1)
Second View	36	(1)
Third View	36	(5)
Hedging a Plain-Vanilla Option When the Real-World Process is Mean Reverting	41	(3)
Hedging a Plain-Vanilla Option With Finite Re-Hedging Intervals	44	(7)
Instantaneous and Terminal Correlations	51	(20)
Introduction	51	(1)
The Stochastic Evolution of Imperfectly Correlated Variables	52	(5)
The Role of Terminal Correlation in the Joint Evolution of Stochastic Variables	57	(11)
Case 1: European Option, One Underlying Asset	58	(3)
Case 2: Path-Dependent Option, One Asset	61	(7)
Case 3: Path-Dependent Option, Two Assets
Generalising the Results	68	(3)
PART TWO DEALING WITH SMILES	71	(180)
Pricing Options in the Presence of Smiles	73	(24)
Introduction	73	(1)
Hedging With a Compensated Process: Plain-Vanilla and Binary Options	74	(4)
Smile Tale 1: `Sticky' Smiles	78	(2)
Smile Tale 2: `Floating' Smiles	80	(3)
Stylised Empirical Facts About Smiles	83	(4)
Equities	83	(2)
Interest Rates	85	(2)
Foreign Exchange Rates	87	(1)
General Features of the Smile-Modelling Approaches	87	(6)
Fully Stochastic Volatility Models	88	(1)
Complete-Markets Jump--Diffusion Models	89	(1)
Random-Amplitude Jump--Diffusion Models	90	(1)
Stochastic Volatility Functionally Dependent on the Underlying (Restricted-Stochastic-Volatility) Models	91	(2)
Risk Derivatives for Plain-Vanilla Options in the Presence of Smiles	93	(4)
Tree Methodologies for Smiley Option Prices	97	(32)
Introduction	97	(1)
General Considerations on Stochastic-Volatility Models	97	(3)
The Dupire, Rubinstein and Derman and Kani Approaches	100	(1)
Green's Function (Arrow--Debreu Prices) in the DK Construction	101	(3)
The Derman and Kani Tree Construction	104	(5)
Numerical Aspects of the Implementation of the DK Construction	109	(4)
Implementation Results	113	(16)
Efficient Extraction of the Future Local Volatility from Plain-Vanilla Option Prices	129	(60)
Introduction	129	(1)
The Computational Framework	130	(5)
Computational Results	135	(4)
The Link Between Implied and Local Volatility Surfaces	139	(14)
Symmetric (`FX') Smiles	140	(4)
Asymmetric (`Equity') Smile Surface	144	(6)
Monotonic (`Interest-Rate') Smile Surface	150	(3)
Gaining an Intuitive Understanding	153	(8)
No-Arbitrage Conditions on the Implied Volatility Smile Surface	161	(13)
A Worked-Out Example: Pricing Continous Double Barriers in the Presence of Smiles	174	(8)
Analysis of the Cost of Unwinding and Related Considerations About Option Pricing in the Presence of Smiles	182	(7)
Appendix 6.1: Proof that	186	(3)
Closed-Form Solutions for Smiley Option Prices via Direct Modelling of the Density	189	(26)
Introduction	189	(6)
Estimating the Risk-Neutral Density Function	195	(4)
Derivation of Analytic Formulae	199	(7)
Results and Applications	206	(7)
Conclusions and Range of Possible Applications	213	(2)
Appendix 7.1 Obtaining the Density of the Underlying from Quoted Option Prices	214	(1)
Explaining Smiles by Means of Mixed Jump---Diffusion Processes	215	(36)
Introduction	215	(1)
The Financial Model: Smile Tale 2 Revisited	216	(4)
Analytic Description of Mixed Jump--Diffusion Processes	220	(9)
A General Framework for Option Pricing in Complete or Incomplete Markets	229	(6)
Finding the Optimal Hedge	235	(1)
Numerical Implementation of the Britten-Jones-Neuberger Methodology	236	(7)
Computational Results	243	(6)
Discussion of the Results and Possible Developments	249	(2)
PART THREE INTEREST RATES	251	(78)
The Role of Mean Reversion in Interest-Rate Models	253	(18)
Introduction: Why Mean Reversion Matters in the Case of Interest-Rate Models	253	(3)
The BDT Mean-Reversion Paradox	256	(3)
The Unconditional Variance of the Short Rate in BDT---The Discrete Case	259	(2)
The Unconditional Variance of the Short Rate in BDT---The Continuous-Time Equivalent	261	(2)
Mean Reversion in Short-Rate Lattices: The Equi-Probable Binomial Versus the Bushy-Tree Approach	263	(4)
Extension to More General Interest-Rate Models: The `True' Role of Mean Reversion	267	(4)
Appendix 9.1: Evaluation of the Variance of the Logarithm of the Instantaneous Short Rate	269	(2)
Optimal Calibration of the Brace--Gatarek--Musiela Model	271	(32)
Introduction and Statement of the Problem	271	(2)
Constructing the Most General BGM (Market) Model	273	(5)
A Worked-Out Example: Caplets and a Two-Period Swaption	278	(2)
A Worked-Out Example: Serial Options	280	(1)
Reducing the Dimensionality of the BGM Model	281	(5)
Numerical Results	286	(12)
Fitting the Correlation Surface with a Three-Factor Model	286	(1)
Fitting the Correlation Surface with a Four-Factor Model	287	(11)
Conclusions	298	(5)
Specifying the Instantaneous Volatility of Forward Rates	303	(26)
The Link Between Instantaneous Volatility and the Future Term Structure of Volatilities	303	(3)
A Functional Form for the Instantaneous Volatility Function	306	(5)
Fitting the Instantaneous Volatility Function: Imposing Time-Homogeneity of the Term Structure of Volatilities	311	(7)
Fitting the Instantaneous Volatility Function: Information from the Swaption Market	318	(9)
Conclusions	327	(2)
References	329	(4)
Index	333