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Security Price Process Models:
Do These Have the Correct Properties For
Understanding Options Values?
ROBERT GEORGE TOMPKINS
4
Submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy
At the
University of Warwick
Standing on the Shoulders of Giants:
To my friends Michael Selby and Stewart Hodges
Table of Contents
VOLUME ONE
Table of Contents
List of Tables and Figures xv
Acknowledgments xxviii
Abstract xxxii
INTRODUCTION
1
CHAPTER ONE
THE ANALYSIS OF OBJECTIVE PROBABILITIES
IN FUTURES
MARKETS: LITERATURE
REVIEW AND EMPIRICAL DESIGN
8
1.1 INTRODUCTION 8
1.2 REVIEW OF THE LITERATURE ON VOLATILITY PROCESSES 9
Review of Research on Objective Dispersion Processes for Assets 10 Random Walk Processes, Stationary Series and Objective Volatility 13
Methods to Determine Objective Volatility 22 Problems in the Estimation of the Objective Volatility of Futures 27
1.3 VOLATILITY CONES - COMPARING VOLATILITIES OVER TIME 29
Determination of Volatility Cones 29 Problems with Overlapping and Non-Overlapping Data 32
A New Method for Onbiasing Overlapping Samples 36
1.4 AUTOCORRELATIONS OF ABSOLUTE RETURNS 39
1.5 MEAN REVERSION IN HISTORICAL VOLATILITY 42
1.6 MODELS TO EXPLAIN THE NATURE OF EMPIRICAL VOLATILITY 44 Constant Elasticity of Variance Model 45 Pricing Models which Incorporate Jumps in the Underlying Asset Price 46 Models which Incorporate Stochastic Volatility 48
1.7 CONCLUSION 56
APPENDIX 1.1 Conversion of the log volatility dispersion process to a simple
CHAPTER TWO
DATA, DESCRIPTIVE STATISTICS AND DERIVED
STATISTICS
58
2.1 INTRODUCTION 58
2.2 DATA SOURCES 58
Cleaning the Data Series 60
2.3 UNCONDITIONAL RETURN DISTRIBUTIONS FOR VARIOUS HORIZONS
62
Review of Other Work on the Determination of Objective Dispersion Processes 70
2.4 THE UNCONDITIONAL VOLATILITY SERIES FOR TWELVE MARKETS 73
2.5 AUTOCORRELATION TESTS FOR TWELVE MARKETS 2.6 COMPOSITE MEASURES OF AUTO CORRELATION
2.7 DETERMINATION OF VOLATILITY CONES
74
82 86
The 20-day Coefficient of Variation as a Measure of the Variability of Volatility 88
2.8 BIAS CORRECTION OF THE VOLATILITY CONE ANALYSIS 90
Comparison of the Unbiased Standard Deviations of Volatility to the Standard Deviations of Volatility
Consistent with an i. i. d. dispersion process. 93 2.9 A SIMPLE MODEL FOR THE STANDARD DEVIATION OF VOLATILITY AS
A FUNCTION OF THE TIME HORIZON 97
2.10 TARGET CONDITIONS THAT CAPTURE EMPIRICAL VOLATILITY
DYNAMICS 102
2.11 CONCLUSION
CHAPTER THREE
112
THE ANALYSIS OF OBJECTIVE PROBABILITIES
IN FUTURES
MARKETS: EXPLAINING THE EMPIRICAL DYNAMICS WITH
MODELS ASSUMING CONSTANT VARIANCE
3.1 INTRODUCTION
114
114
3.2 TESTING A GEOMETRIC BROWNIAN MOTION MODEL 114
GBM Test Results for Entire Period of Analysis 122 GBM Test Results for SubPeriods of Analysis 124
3.3 ALTERNATIVE MODELS TO EXPLAIN VOLATILITY DYNAMICS 126
3.4 TESTING A FAT-TAILED DISTRIBUTION MODEL 127
Student-t Test Results for Entire Period of Analysis 129 Student-t Test Results for SubPeriods of Analysis 134
3.5 CONCLUSION
CHAPTER FOUR
143
THE ANALYSIS OF OBJECTIVE PROBABILITIES
IN FUTURES
MARKETS: EXPLAINING THE EMPIRICAL DYNAMICS WITH
MODELS ASSUMING STOCHASTIC VARIANCE
146
4.1 INTRODUCTION 146
4.2 STOCHASTIC VOLATILITY MODELS USED IN THE ANALYSIS 147
4.3 CALIBRATION OF STOCHASTIC VOLATILITY MODELS 152
4.4 ANALYSIS OF STOCHASTIC VOLATILITY MODELS (ENTIRE PERIOD) 156
Optimal Parameters for the Three Stochastic Volatility Models (Entire Period) Results for the Three Stochastic Volatility Models (Entire Period)
156 157
4.5 ANALYSIS OF STOCHASTIC VOLATILITY MODELS (FIRST PERIOD)
- 161
Optimal Parameters for the Three Stochastic Volatility Models (First Period) Results for the Three Stochastic Volatility Models (First Period)
161 164
4.6 ANALYSIS OF STOCHASTIC VOLATILITY MODELS (SECOND PERIOD)
168
Optimal Parameters for the Three Stochastic Volatility Models (Second Period) 168 Results for the Three Stochastic Volatility Models (Second Period) 170
4.7 CONCLUSION 173
CHAPTER FIVE
THE ANALYSIS OF OBJECTIVE PROBABILITIES
IN FUTURES
MARKETS: EXPLAINING THE EMPIRICAL DYNAMICS WITH
MODELS ASSUMING STOCHASTIC VARIANCE AND THE
UNDERLYING ASSET PRICES FOLLOW A STUDENT-T
DISTRIBUTION
176
5.1 INTRODUCTION 176
5.2 ANALYSIS OF COMBINATION MODELS FOR THE ENTIRE PERIOD 176 Parameter Estimation for Combination Models for the Entire Period 177 Results for Combination Models for the Entire Period 179 Results of the Combination Models - S&P 500 (including/excluding the 1987 Crash) 183
5.3 ANALYSIS OF COMBINATION MODELS FOR THE FIRST PERIOD 185
Parameter Estimation for Combination Models for the First Period 185
Results for Combination Models for the First Period 187
5.4 ANALYSIS OF COMBINATION MODELS FOR THE SECOND PERIOD
- 192
Parameter Estimation for Combination Models for the Second Period 192 Results for Combination Models for the Second Period
5.5 CONCLUSION
VOLUME TWO
194
198
Table of Contents i
List of Tables and Figures V
CHAPTER SIX
THE ANALYSIS OF RISK NEUTRAL PROBABILITIES
IN
OPTIONS ON FUTURES: LITERATURE REVIEW AND
EMPIRICAL
DESIGN
201
6.1 INTRODUCTION 201
6.2 REVIEW OF THE LITERATURE ON RISK NEUTRAL EVALUATION
- 202
The Use of Risk Neutrality in Pricing Contingent Claims
6.3 REVIEW OF IMPLIED VOLATILITY
204
206
Determination of Implied Volatility 210 Weighting Implied Volatilities 215
6.4 DIVERGENCES OF IMPLIED VOLATILITIES ACROSS STRIKE PRICES:
THE VOLATILITY SMILE 218
Review of the Literature on Volatility Smiles 222 Theoretical Reasons for the Existence of Volatility Smiles 226
6.5 APPROACHES FOR DEALING WITH VOLATILITY SMILES 241
Determination of Risk Neutral Probabilities Empirically 243
6.6 RATIONALE FOR EXAMINATION OF IMPLIED VOLATILITY SMILES RATHER THAN IMPLIED DISTRIBUTIONS
6.7 CONCLUSION
CHAPTER SEVEN
247 249
THE ANALYSIS OF RISK NEUTRAL PROBABILITIES
IN
OPTIONS ON FUTURES: STANDARDISATION
OF IMPLIED
VOLATILITY
SMILES
250
7.1 INTRODUCTION 250
7.2 DATA SOURCES 251
7.3 HISTORICAL RECORD OF IMPLIED VOLATILTIES 258
7.4 RESULTS OF THE IMPLIED VOLATILITY ANALYSIS 7.5 DETERMINATION OF VOLATILITY MATRICES
262 268
Constructing a Volatility Matrix for FTSE Index Options 270 Splitting the First Order and Second Order Strike Price Effects 278
7.6 IMPLICATIONS OF NONSTATIC IMPLIED VOLATILITY SURFACES FOR
DETERMINISTIC IMPLIED VOLATILITY MODELS
7.7 STANDARDISED IMPLIED VOLATILITY SMILES FOR ALL TWELVE
282
MARKETS 287
Examination of Standardised Smile Structures for the Twelve Markets in Sub Periods 292
7.8 CONCLUSION 293
CHAPTER EIGHT
THE ANALYSIS OF RISK NEUTRAL PROBABILITIES
IN
OPTIONS ON FUTURES. COMPARISON OF IMPLIED
VOLATILITY
SMILES WITHIN AND BETWEEN MARKETS:
ANALYSIS USING MULTIPLE REGRESSION /ANALYSIS OF
VARIANCE
296
8.1 INTRODUCTION 296
8.2 DETERMINISTIC IMPLIED VOLATILITY MODELS 297
8.3 MODELLING THE IMPLIED VOLATILITY SURFACE 298
Variables Used in the Analysis 300 Inclusion of Variables to Capture Shocks
Inclusion of Additional Variables
304 309
8.4 INTERPRETATION OF THE MODEL 311
8.5 ORDINARY LEAST SQUARES REGRESSION RESULTS 313
Modelling the First Order Strike Price Effect (SKEWNESS)
Modelling the Second Order Strike Price Effect (KURTOSIS)
316 324 The Impacts of Other Variables on Implied Volatility Surfaces 331
8.6 TESTING THE ROBUSTNESS OF THE REGRESSION MODELS 341
Testing for An Exclusion of Critical Variables Problem 342 Correction for Heteroscedasticity in the Regressions 346 Corrections for Serial Correlations in the Residuals of the Regressions
Assessing Multi-Collinearity Problems in the Regressions
348 351 Testing the Regression Models for Different Time Periods 353
8.7 COMPARISONS OF IMPLIED VOLATILTY MODELS WITHIN ASSET
CLASSES
Comparisons of Models for Stock Index Options
360 361
Implications of the Findings for Stock Index Options Smile Surfaces 366 Comparisons of Models for Fixed Income Options
Comparisons of Models for Foreign Exchange Options
368 370 8.8 COMPARISONS OF IMPLIED VOLATILTY MODELS FOR ALL MARKETS
374
8.9 CONCLUSION
CHAPTER NINE
378
THE ANALYSIS OF RISK NEUTRAL PROBABILITIES
IN
OPTIONS ON FUTURES COMPARISON OF IMPLIED
VOLATILITY
SMILES WITH SIMULATED SMILES DERIVED
FROM OBJECTIVE DISPERSION PROCESS MODELS
381
9.1 INTRODUCTION
9.2 SIMULATING IMPLIED VOLATILITY DYNAMICS CONSISTENT WITH
381
THE OBJECTIVE PROCESS MODELS 381
9.3 SIMULATING IMPLIED VOLATILITY DYNAMICS CONSISTENT WITH A
STUDENT-t DISTRIBUTION MODEL 384
9.4 TESTING THE EXPLANATORY POWER OF THEORICAL PRICING
MODELS FOR THE IMPLIED VOLATILITY SMILE PATTERNS 387 9.5 TESTING THE EXPLANATORY POWER OF A STUDENT-t DISTRIBUTION
MODEL FOR THE IMPLIED VOLATILITY PATTERNS 390
9.6 SIMULATING IMPLIED VOLATILITY DYNAMICS CONSISTENT WITH
GBM STOCHASTIC VOLATILITY MODELS 393
9.7 TESTING THE EXPLANATORY POWER OF STOCHASTIC VOLATILITY
MODELS FOR THE IMPLIED VOLATILITY PATTERNS 395
9.8 SIMULATING IMPLIED VOLATILITY DYNAMICS CONSISTENT WITH
STOCHASTIC VOLATILITY & STUDENT-t DISTRIBUTION MODELS 400 9.9 TESTING THE EXPLANATORY POWER OF STOCHASTIC VOLATILITY & STUDENT-t MODELS FOR THE IMPLIED VOLATILITY PATTERNS 403
9.10 COMPARISON OF THE THREE MODELS FOR EXPLAINING THE DYNAMICS OF THE IMPLIED VOLATILITY SMILES
Comparison by Minimised Least Squares
Comparison by Averaged Differences in Predicted and Actual Results
406 406 409 Implications of the Results 412
9.11 CONCLUSIONS
CHAPTER TEN
414
CONCLUSIONS AND SUGGESTIONS FOR FURTHER
RESEARCH
417
References
VOLUME THREE
423
List of Tables and Figures 1
Figure 2.1a Exponentially Weighted Unconditional Volatility for Four Stock Index Futures 1 Figure 2.1b Exponentially Weighted Unconditional Volatility for Four Fixed Income Futures 2 Figure 2.1c Exponentially Weighted Unconditional Volatility for Four Foreign Exchange Futures_
_3 Figure 2.2a Autocorrelogram for four Stock Index Futures absolute daily returns 4
Figure 2.2b Autocorrelogram for four Fixed Income Futures absolute daily returns 5 Figure 2.2c Autocorrelogram for four Foreign Exchange Futures absolute daily returns 6 Figure 2.2d Comparison of autocorrelograms on S&P-500 Futures 7
Figure 2.2e Autocorrelogram for DAX Index Futures and Cash using absolute daily returns 8 Figure 2.3a First period autocorrelogram for four Stock Index Futures absolute daily returns 9 Figure 2.3b First period autocorrelogram for four Fixed income Futures absolute daily returns 10 Figure 2.3c First period autocorrelogram for four Foreign exchange Futures absolute daily returns
- 11 Figure 2.4a Second period autocorrelogram for four Stock Index Futures absolute daily returns 12 Figure 2.4b Second period autocorrelogram for four Fixed Income Futures absolute daily returns
- 13 Figure 2.4c Second period autocorrelogram for four Foreign Exchange Futures absolute daily returns.
14
Table 2.5a Summary statistics of volatility cone for four Stock Index Futures 15
Table 2.5b Summary statistics of volatility cone for four Fixed Income Futures 16 Table 2.5c Summary statistics of volatility cone for four Foreign Exchange Futures 17 Figure 2.5a Volatility cones for four Stock Index Futures 18 Figure 2.5b Volatility cones for four Fixed Income Futures 19 Figure 2.5c Volatility cones for four Foreign Exchange Futures 20 Table 2.6a First period summary statistics of volatility cone for four Stock Index Futures 21 Table 2.6b First Period summary statistics of volatility cone for four Fixed Income Futures 22 Table 2.6c First period summary statistics of volatility cone for four Foreign Exchange Futures 23 Figure 2.6a First period volatility cones for four Stock Index Futures 24 Figure 2.6b First period volatility cones for four Fixed Income Futures 25 Figure 2.6c First period volatility cones for four Foreign Exchange Futures 26 Table 2.7a Second period summary statistics of volatility cone for four Stock Index Futures 27 Table 2.7b Second period summary statistics of volatility cone for four Fixed Income Futures 28 Table 2.7c Second period summary statistics of volatility cone for four Foreign Exchange Futures
- 29 Figure 2.7a Second period volatility cones for four Stock Index Futures 30 Figure 2.7b Second period volatility cones for four Fixed Income Futures 31 Figure 2.7c Second period volatility cones for four Foreign Exchange Futures 32 Figure 2.8a Comparison of volatility estimated using overlapping and non overlapping observations for
four Stock Index Futures 33
Figure 2.8b Comparison of volatility estimated using overlapping and non overlapping observations for
four Fixed Income Futures 34
Figure 2.8c Comparison of volatility estimated using overlapping and non overlapping observations for four Foreign Exchange Futures 35 Figure 2.9a First period comparison of volatility estimated using overlapping and non overlapping
observations for four Stock Index Futures 36 Figure 2.9b First period comparison of volatility estimated using overlapping and non overlapping
observations for four Fixed Income Futures 37 Figure 2.9c First period comparison of volatility estimated using overlapping and non overlapping
observations for four Foreign Exchange Futures 38 Figure 2.10a Second period comparison of volatility estimated using overlapping and non overlapping
observations for four Stock Index Futures 39
Figure 2.10b Second period comparison of volatility estimated using overlapping and non overlapping observations for four Fixed Income Futures 40 Figure 2.10c Second period comparison of volatility estimated using overlapping and non overlapping
observations for four Foreign Exchange Futures 41 Figure 2.11a Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for four Stock
Index Futures 42
Figure 2.11b Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for four Fixed
Income Futures 43
Figure 2.11c Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for four Foreign
Exchange Futures 44
Figure 2.12a First period Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for
four Stock Index Futures 45
Figure 2.12b First period Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for
four Fixed Income Futures 46
Figure 2.12c First period Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for
four Foreign Exchange Futures 47
Figure 2.13a Second period Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for
four Stock Index Futures 48
Figure 2.13b Second period Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for
four Fixed Income Futures 49
Figure 2.13c Second period Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for four Foreign Exchange Futures 50 Figure 2.14a Time decay factors for the volatility of volatility for four Stock Index Futures 51 Figure 2.14b Time decay factors for the volatility of volatility for four Fixed Income Futures 52 Figure 2.14c Time decay factors for the volatility of volatility for four Foreign Exchange Futures
- 53 Figure 2.15a First period time decay factors for the volatility of volatility for four Stock Index Futures
54
Figure 2.15b First period time decay factors for the volatility of volatility for four Fixed Income
Futures 55
Figure 2.15c First period time decay factors for the volatility of volatility for four Foreign Exchange
Futures 56
Figure 2.16a Second period time decay factors for the volatility of volatility for four Stock Index
Futures 57
Figure 2.16b Second period time decay factors for the volatility of volatility for four Fixed Income
Futures 58
Figure 2.16c Second period time decay factors for the volatility of volatility for four Foreign Exchange
Futures 59
Table 7.1a Summary statistics for the At-the-money Implied Volatilities for four Stock Index Options, estimated ona a Daily basis and at 5-days increments until expiration 60
Table 7.1b Summary statistics for the At-the-money Implied Volatilities for four Fixed Income
Options, estimated ona a Daily basis and at 5-days increments until expiration 61 Table 7.1c Summary statistics for the At-the-money Implied Volatilities for four Foreign Exchange
Options, estimated ona a Daily basis and at 5-days increments until expiration 62
Figure 7.1a Time Series Plots of At-the-money Implied Volatilities for Four Stock Index Options
- 63 Figure 7.1b Time Series Plots of At-the-money Implied Volatilities for Four Fixed Income Options 64 Figure 7.1c Time Series Plots of At-the-money Implied Volatilities for Four Foreign Exchange Options
65
Table 7.2 Option prices and implied volatilities for FTSE-100 as of May7th, 1996 66 Figure 7.2a Implied Volatility Smiles for Four Stock Index Options as of May 7,1996 67 Figure 7.2b Implied Volatility Smiles for Four Fixed Income Options as of May 7,1996 68 Figure 7.2c Implied Volatility Smiles for Four Foreign Exchange Options as of May 7,1996 69 Figure 7.4 Unstandardized volatility smiles for 1996 FTSE-100 contracts 70 Figure 7.5 Standardized volatility smiles for 1996 FTSE-100 contracts grouped by contracts 71 Figure 7.6 Standardized volatility smiles for 1996 FTSE-100 contracts grouped by days to expiration
72 Figure 7.7 Interpolated volatility smiles for 1996 FTSE-100 contracts 73 Figure 7.8a Skewness for 1996 FTSE-100 contracts 74 Figure 7.8b Skewness for 1996 FTSE-100 contracts 75
Figure 7.9a Kurtosis for 1996 FTSE-100 contracts Figure 7.9b Kurtosis for 1996 FTSE-100 contracts
76
77
Figure 7.1 Oa Scatterplots of minimum VSI strike price relative to underlying future price for the FTSE-
100 during 1996 78
Figure 7.10b Plot of the percentage difference between the Underlying Futures Price to the minimum
Strike Price for Four Options Cycles for the FTSE-100 during 1996 79
Figure 7.11a 1996 Volatility smiles for Four Stock Index Options 80 Figure 7.11b 1996 Volatility smiles for Four Fixed Income Options 81 Figure 7.11c 1996 Volatility smiles for Four Foreign Exchange Options 82 Figure 7.12a 1996 Skewness for Four Stock Index Options 83 Figure 7.12b 1996 Skewness for Four Fixed Income Options 84 Figure 7.12c 1996 Skewness for Four Foreign Exchange Options 85 Figure 7.13a 1996 Skewness for Four Stock Index Options 86 Figure 7.13b 1996 Skewness for Four Fixed Income Options 87 Figure 7.13c 1996 Skewness for Four Foreign Exchange Options 88 Figure 7.14a 1996 Kurtosis for Four Stock Index Options 89 Figure 7.14b 1996 Kurtosis for Four Fixed Income Options 90 Figure 7.14c 1996 Kurtosis for Four Foreign Exchange Options 91 Figure 7.15a 1996 Kurtosis for Four Stock Index Options 92 Figure 7.15b 1996 Kurtosis for Four Fixed Income Options 93 Figure 7.15c 1996 Kurtosis for Four Foreign Exchange Options 94
Figure 7.16a Standardized Volatility Smiles for Four Stock Index Options for the entire period of
analysis 95
Figure 7.16b Standardized Volatility Smiles for Four Fixed Income Options for the entire period of
analysis 96
Figure 7.16c Standardized Volatility Smiles for Four Foreign Exchange Options for the entire period of
analysis 97
Figure 7.17a Standardized Volatility Smiles for Four Stock Index Options for the first portion of the
available observations 98
Figure 7.17b Standardized Volatility Smiles for Four Fixed Income Options for the first portion of the
available observations 99
Figure 7.17c Standardized Volatility Smiles for Four Foreign Exchange Options for the first portion of
the available observations 100
Figure 7.18a Standardized Volatility Smiles for Four Stock Index Options for the second portion of the
available observations 101
Figure 7.18b Standardized Volatility Smiles for Four Fixed Income Options for the second portion of
the available observations 102
Figure 7.18c Standardized Volatility Smiles for Four Foreign Exchange Options for the second portion of the available observations 103
APPENDIX 7.1 104
Table 8.2a Ordinary Least Squares Regression Results for four Stock Index Options 120 Table 8.2b Ordinary Least Squares Regression Results for four Fixed Income Options 121 Table 8.2c Ordinary Least Squares Regression Results for four Foreign Exchange Options 122 Table 8.6a Ordinary Least Squares Regression Results for four Stock Index Options Including
Contracts as Dummy Variables 123 Table 8.6b Ordinary Least Squares Regression Results for four Fixed Income Options Including
Contracts as Dummy Variables 124
Table 8.6c Ordinary Least Squares Regression Results for four Foreign Exchange Options Including
Contracts as Dummy Variables 125 Table 8.7a Weighted Least Squares Regression Results for four Stock Index Options 126 Table 8.7b Weighted Least Squares Regression Results for four Fixed Income Options 127 Table 8.7c Weighted Least Squares Regression Results for four Foreign Exchange Options 128 Table 8.8 Generalized Least Squares Regression Results fot three Selected Options Markets 129 Table 8.9a Ordinary Least Squares Regression Results for four Stock Index Options using the First
Half of Available Observations 130 Table 8.9b Ordinary Least Squares Regression Results for four Fixed Income Options using the First
Half of Available Observations 131 Table 8.9c Ordinary Least Squares Regression Results for four Foreign Exchange Options using the
First Half of Available Observations 132 Table 8.1Oa Ordinary Least Squares Regression Results for four Stock Index Options using the Second
Half of Available Observations 133
Table 8.10b Ordinary Least Squares Regression Results for four Fixed Income Options using the
Second Half of Available Observations 134 Table 8.10c Ordinary Least Squares Regression Results for four Foreign Exchange Options using the
Second Half of Available Observations 135 Table 8.13a Ordinary Least Squares Regression Results for four Stock Index Options compared to All
Stock Index Options 136
Table 8.13b Ordinary Least Squares Regression Results for four Fixed Income Options compared to
All Fixed Income Options 137
Table 8.13c Ordinary Least Squares Regression Results for four Foreign Exchange Options compared to All Foreign Exchange Options 138 Table 8.14 Ordinary Least Squares Regression Results for All Option Markets comaperd to the three
Asset Classes 139
Table 9.1 Monte Carlo generated Call Option Prices using a Student-t distribution with constant
volatility 140
Table 9.2 Volatilities implied by the Monte Carlo generated Call Option Prices using a Student-t
distribution with constant volatility 141 Table 9.3 Standardized implied volatilities using a Student-t distribution with constant volatility
- 142 Figure 9.1 Standardized Volatility Smiles using a Student-t distribution with constant volatility
_ 143 Figure 9.2a S&P-500 Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 144 Figure 9.2b FTSE-100 Whole Period Implied Volatility Smile Patterns that are consistent with the three Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 145 Figure 9.2c Nikkei-225 Whole Period Implied Volatility Smile Patterns that are consistent with the
three Security Price Models estimated using Unconditional Dispersion Process compared to the
Actual 146
Figure 9.2d DAX Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 147 Figure 9.3a Bund Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 148 Figure 9.3b Gilt Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 149 Figure 9.3c BTP Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 150 Figure 9.3d US T-Bond Whole Period Implied Volatility Smile Patterns that are consistent with the
three Security Price Models estimated using Unconditional Dispersion Process compared to the
Actual 151
Figure 9.4a D-Mark Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 152 Figure 9.4b B-Pound Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 153
Figure 9.4c J-Yen Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 154 Figure 9.4d S-Franc Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 155 Figure 9.5a S&P-500 First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 156 Figure 9.5b FTSE-100 First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 157 Figure 93c Nikkei-225 First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 158 Figure 9.5d DAX First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 159 Figure 9.6a Bund First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 160 Figure 9.6b Gilt First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 161 Figure 9.6c BTP First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 162 Figure 9.6d US T-Bond First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 163 Figure 9.7a D-Mark First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 164 Figure 9.7b B-Pound First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 165 Figure 9.7c 1-Yen First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 166 Figure 9.7d S-Franc First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 167 Figure 9.8a S&P-500 Second Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 168 Figure 9.8b FTSE-100 Second Period Implied Volatility Smile Patterns that are consistent with the
three Security Price Models estimated using Unconditional Dispersion Process compared to the
Actual 169
Figure 9.8c Nikkei-225 Second Period Implied Volatility Smile Patterns that are consistent with the three Security Price Models estimated using Unconditional Dispersion Process compared to the
Actual 170
Figure 9.8d DAX Second Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 171 Figure 9.9a Bund Second Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 172
Figure 9.9b Gilt Second Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 173 Figure 9.9c BTP Second Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 174 Figure 9.9d US T-Bond Second Period Implied Volatility Smile Patterns that are consistent with the
three Security Price Models estimated using Unconditional Dispersion Process compared to the
Actual 175
Figure 9.10a D-Mark Second Period Implied Volatility Smile Patterns that are consistent with the three Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 176 Figure 9.10b B-Pound Second Period Implied Volatility Smile Patterns that are consistent with the
three Security Price Models estimated using Unconditional Dispersion Process compared to the
Actual 177
Figure 9.1 Oc J-Yen Second Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 178 Figure 9.10d S-Franc Second Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 179
List of Tables and Figures
VOLUME ONE
CHAPTER TWO
Table 2.1 Markets Included in Research, Time Period of Data, Number of Observations 59 Table 2.2 Statistics of the Daily, Weekly and Monthly Returns for Twelve Financial Futures for the
Whole Period of Available Observations 64
Table 2.3 Periods & Observations for Markets Under Analysis, Broken into Two Sub-Periods 65 Table 2.4 Statistics of the Daily, Weekly and Monthly Returns for Twelve Financial Futures for the
First Half of the Observation Period
Table 2.5 Statistics of the Daily, Weekly and Monthly Returns for Twelve Financial Futures for the Second Half of the Observation Period
66
67
Table 2.6 Average Autocorrelations of Absolute Returns for the Lags 1-20 and 51-70 for Twelve
Markets for the Entire Period of Analysis 83 Table 2.7 Average Autocorrelations of Absolute Returns for the Lags 1-20 and 51-70 for Twelve
Markets for the First Half Period of Analysis
Table 2.8 Average Autocorrelations of Absolute Returns for the Lags 1-20 and 51-70 for Twelve Markets for the Second Half Period of Analysis
Table 2.9 Statistics of the 20 Day Volatility for Twelve Financial Futures
84
85 88 Table 2.10 Statistics of the 20 Day Volatility for Twelve Financial Futures for the first half of the
observation period 89
Table 2.11 Statistics of the 20 Day Volatility for Twelve Financial Futures for the second half of the
observation period 89
Table 2.12 Time Decay Factors for the Standard Deviation of Volatility for Twelve Financial Futures
Markets 99
Table 2.13 Time Decay Factors for the Standard Deviation of Volatility for Twelve Financial Futures
Markets For the First Half of the Available Observations 100 Table 2.14 Time Decay Factors for the Standard Deviation of Volatility for Twelve Financial Futures
Markets for the Second Half of the Available Observations
101
Table 2.15 Attributes That Describe the Empirical Dynamics of Twelve Financial Futures DispersionProcesses 106
Table 2.16 Attributes That Describe the Empirical Dynamics of Twelve Financial Futures Dispersion
Processes for the First Half of the Available Observations 107 Table 2.17 Attributes That Describe the Empirical Dynamics of Twelve Financial Futures Dispersion
Processes for the Second Half of the Available Observations. 108 Table 2.18 Correlation Matrix of Attributes for the 12 Financial Futures Markets for the Three Periods
of Analysis 111
CHAPTER THREE
Table 3.1 Estimated Values of the Attributes for a GBM price series
with Constant Variance 117
Table 3.2 Correlation Matrix of Attributes for the 100 simulations of a GBM price series with Constant variance
Table 3.3 Comparison of the Attributes for the Average of 100 GBM price series with Constant Variance to a Selected GBM price series with Constant variance.
118
119
Table 3.4 Comparisons of the Empirical Dynamics of Twelve Financial Futures Dispersion Processes with the Dynamics of a GBM Price Series with Constant Variance for the Entire Period of Available
Observations 123
Table 3.5 Comparisons of the Empirical Dynamics of Twelve Financial Futures Dispersion Processes with the Dynamics of a GBM Price Series for the First Half of the Available Observations - 124-125 Table 3.6 Comparisons of the Empirical Dynamics of Twelve Financial Futures Dispersion Processes
with the Dynamics of a GBM Price Series for the Second Half of the Available Observations - 125 Table 3.7 Sample Moments of Eight Student-t Distributions 129 Table 3.8a Results for Student-t Distribution Models for Four Stock Index Futures Assuming Price
Series the Variance is Constant 130 Table 3.8b Results for Student-t Distribution Models for Four Fixed Income Futures Assuming Price
Series the Variance is Constant 131 Table 3.8c Results for Student-t Distribution Models for Four Foreign Exchange Futures Assuming
Price Series the Variance is Constant 132 Table 3.9 Comparisons of the Empirical Dynamics of Twelve Financial Futures Dispersion Processes
with the Dynamics of a GBM Price Series with Constant Variance and the Best Student-t distribution
with Constant Variance 133
Table 3.10a, Results for Student-t Distribution Models for Four Stock Index Futures Assuming Price
Series the Variance is Constant for the First Half of the Available Observations 135 Table 3.10b Results for Student-t Distribution Models for Four Fixed Income Futures Assuming Price
Series the Variance is Constant for the First Half of the Available Observations 136 Table 3.10c Results for Student-t Distribution Models for Four Foreign Exchange Futures Assuming
Price Series the Variance is Constant for the First Half of the Available Observations 137 Table 3.11 Comparisons of the Empirical Dynamics of Twelve Financial Futures Dispersion Processes
with the Dynamics of a GBM Price Series with Constant Variance and the Best t-distribution with
Constant Variance for the First Half of the Available Observations 138 Table 3.12a Results for Student-t Distribution Models for Four Stock Index Futures Assuming Price
Series the Variance is Constant for the Second Half of the Available Observations 139 Table 3.12b Results for Student-t Distribution Models for Four Fixed Income Futures Assuming Price
Series the Variance is Constant for the Second Half of the Available Observations 140 Table 3.12c Results for Student-t Distribution Models for Four Foreign Exchange Futures Assuming
Price Series the Variance is Constant for the Second Half of the Available Observations 141 Table 3.13 Comparisons of the Empirical Dynamics of Twelve Financial Futures Dispersion Processes
with the Dynamics of a GBM Price Series with Constant Variance and the Best t-distribution with
Constant Variance for the Second Half of the Available Observations 142
CHAPTER FOUR
Table 4.1 Initial Parameter Values for Hull & White Simulations 155 Table 4.2 Initial Parameter Values for Stein & Stein Simulations 155 Table 4.3 Initial Parameter Values for Heston Simulations 156 Table 4.4 Parameter values for the Best of the Three Stochastic Volatility
Models for the Twelve Financial Futures Markets 156-157 Table 4.5a Best Fitting Models for Four Stock Index Futures Assuming Price Series are Lognormally
Distributed and the Variance is Stochastic. 158 Table 4.5b Best Fitting Models for Four Fixed Income Futures Assuming Price Series are Lognormally
Distributed and the Variance is Stochastic. 158 Table 4.5c Best Fitting Models for Four Foreign Exchange Futures Assuming Price Series are
Lognormally Distributed and the Variance is Stochastic 159 Table 4.6 Comparisons of the Empirical Dynamics of Twelve Financial Futures Dispersion Processes
with the Dynamics of a GBM Price Series with Constant Variance versus the Best t-distribution with Constant Variance and the Best Stochastic Volatility Model that assumes the Underlying Price Series
is Lognormal. 159-160
Table 4.7 Parameter values for the Best of the Three Stochastic Volatility Models for the Twelve
Financial Futures Markets for the First Half of the Available Observations 161-162 Table 4.8a Best Fitting Models for Four Stock Index Futures Assuming Price Series are Lognormally
Distributed & the Variance is Stochastic (First Half of Available Observations). 165 Table 4.8b Best Fitting Models for Four Fixed Income Futures Assuming Price Series are Lognormally
Distributed & the Variance is Stochastic (First Half of Available Observations). 165 Table 4.8c Best Fitting Models for Four Foreign Exchange Futures Assuming Price Series Lognormally
Distributed & the Variance is Stochastic (First Half of Available Observations) 166 Table 4.9 Comparisons of the Empirical Dynamics of Twelve Financial Futures Dispersion Processes
with the Dynamics of a GBM Price Series with Constant Variance versus the Best t-distribution with Constant Variance and the Best Stochastic Volatility Model that assumes the Underlying Price Series
is Lognormal (Analysis Period included the First Half of the Available Observations) 166-167 Table 4.10 Parameter values for the Best of the Three Stochastic Volatility Models for the Twelve
Financial Futures Markets for the Second Half of the Available Observations 168-169 Table 4.11 a Best Fitting Models for Four Stock Index Futures Assuming Price Series are Lognormally
Distributed & the Variance is Stochastic (Second Half of Available Observations). 170 Table 4.11 b Best Fitting Models for Four Fixed Income Futures Assuming Price Series are
Lognormally Distributed & the Variance is Stochastic (Second Half of Available Observations)_ 171 Table 4.11 c Best Fitting Models for Four Foreign Exchange Futures Assuming Price Series
Lognormally Distributed & the Variance is Stochastic (Second Half of Available Observations)
_ 171 Table 4.12 Comparisons of the Empirical Dynamics of Twelve Financial Futures Dispersion Processes
with the Dynamics of a GBM Price Series with Constant Variance versus the Best t-distribution with Constant Variance and the Best Stochastic Volatility Model that assumes the Underlying Price Series is Lognormal (Analysis Period included the Second Half of the Available Observations) 172-173
CHAPTER FIVE
Table 5.1 a Parameters for the Best Fitting Models for Four Stock Index Futures Assuming Price Series follow a Student-t Distribution and the Variance is Stochastic. 177
Table 5.1b Parameters for the Best Fitting Models for Four Fixed Income Futures Assuming Price
Series follow a Student-t Distribution and the Variance is Stochastic 177 Table 5.1c Parameters for the Best Fitting Models for Four Foreign Exchange Futures Assuming Price
Series follow a Student-t Distribution and the Variance is Stochastic 178 Table 5.2a Results for the Best Fitting Models for Four Stock Index Futures Assuming Price Series
follow a Student-t Distribution and the Variance is Stochastic. 179 Table 5.2b Results for the Best Fitting Models for Four Fixed Income Futures Assuming Price Series
follow a Student-t Distribution and the Variance is Stochastic 179 Table 5.2c Results for the Best Fitting Models for Four Foreign Exchange Futures Assuming Price
Series follow a Student-t Distribution and the Variance is Stochastic 180 Table 5.3 Comparisons of the Empirical Dynamics of Twelve Financial Futures Dispersion Processes
with the Dynamics of a GBM Price Series with Constant Variance versus the Best t-distribution with Constant Variance
, the Best Stochastic Volatility Model that assume the Underlying Price Series is Lognormal and the Best Stochastic Volatility Model that assumes the Underlying Price Series follows a Student-t-distribution 181-182 Table 5.4 Comparison of the S&P 500 Futures Dynamics including/excluding 1987 crash 184 Table 5.5 Best Fitting Models for S&P 500 Futures excluding the 1987 crash 184
Table 5.6a Parameters for the Best Fitting Models for Four Stock Index Futures Assuming Price Series follow a Student-t Distribution and the Variance is Stochastic for the First Half of the Available
Observations
Table 5.6b Parameters for the Best Fitting Models for Four Fixed Income Futures Assuming Price Series follow a Student-t Distribution and the Variance is Stochastic for the First Half of the
186
Available Observations 186
Table 5.6c Parameters for the Best Fitting Models for Four Foreign Exchange Futures Assuming Price Series follow a Student-t Distribution and the Variance is Stochastic for the First Half of the
Available Observations 187
Table 5.7a Results for the Best Fitting Models for Four Stock Index Futures Assuming Price Series
follow a Student-t Distribution and the Variance is Stochastic (First Half) 188 Table 5.7b Results for the Best Fitting Models for Four Fixed Income Futures Assuming Price Series
follow a Student-t Distribution and the Variance is Stochastic (First Half). 188 Table 5.7c Results for the Best Fitting Models for Four Foreign Exchange Futures Assuming Price
Series follow a Student-t Distribution and the Variance is Stochastic (First Half). 189 Table 5.8 Comparisons of the Empirical Dynamics of Twelve Financial Futures Dispersion Processes
with the Dynamics of a GBM Price Series with Constant Variance versus the Best t-distribution with Constant Variance, the Best Stochastic Volatility Models that assumes the Underlying Price Series is Lognormal or follows a Student-t distribution (First Half). 190-191
Table 5.9a Parameters for the Best Fitting Models for Four Stock Index Futures Assuming Price Series follow a Student-t Distribution and the Variance is Stochastic (Second Half). 192 Table 5.9b Parameters for the Best Fitting Models for Four Fixed Income Futures Assuming Price
Series follow a Student-t Distribution and the Variance is Stochastic (Second Halt). 193 Table 5.9c Parameters for the Best Fitting Models for Four Foreign Exchange Futures Assuming Price
Series follow a Student-t Distribution and the Variance is Stochastic (Second Half) 193 Table 5.10a Results for the Best Fitting Models for Four Stock Index Futures Assuming Price Series
follow a Student-t Distribution and the Variance is Stochastic (Second Half) 194 Table 5.10b Results for the Best Fitting Models for Four Fixed Income Futures Assuming Price Series
follow a Student-t Distribution and the Variance is Stochastic (Second Half) 195 Table 5.10c Results for the Best Fitting Models for Four Foreign Exchange Futures Assuming Price
Series follow a Student-t Distribution and the Variance is Stochastic (Second Half) 195 Table 5.11 Comparisons of the Empirical Dynamics of Twelve Financial Futures Dispersion Processes
with the Dynamics of a GBM Price Series with Constant Variance versus the Best t-distribution with Constant Variance
, the Best Stochastic Volatility Models that assume the Underlying Price Series is Lognormal or follows a t-distribution (Second Half) 196-197
VOLUME TWO
CHAPTER SEVEN
Table 7.3 Volatility Matrix for Options on the FTSE Index 268 Figure 7.3 Volatility Smiles of Options on the FTSE Index 269 Table 7.4 Quadratic Regression Results for Standardised Implied Volatility (VSI) as a function of the
Standardised Strike Price and Standardised Strike Price2 280 Table 7.5 Periods & Observations for Markets Under Analysis, Broken into Two Sub-Periods 293
CHAPTER EIGHT
Table 8.1 Dates on Which Two Major Shocks in Variance Occurred for the Twelve Markets Under
Examination 304
Table 8.3a Regression Results for the First Order Effect of the Strike Price for Four Stock Index
Options 317
Table 8.3b Regression Results for the First Order Effect of the Strike Price for Four Fixed Income
Options 320
Table 8.3c Regression Results for the First Order Effect of the Strike Price for Four Foreign Curren cy
Options 322
Table 8.4a Regression Results for the Second Order Effect of the Strike Price for Four Stock Index
Options 325
Table 8.4b Regression Results for the Second Order Effect of the Strike Price for Four Fixed Income
Options 327
Table 8.4c Regression Results for the Second Order Effect of the Strike Price for Four Foreign
Exchange Options 329
Table 8.5a Regression Results for the Remaining Independent Variables for Four Stock Index Options 332 Table 8.5b Regression Results for the Remaining Independent Variables for Four Fixed Income
Options 335
Table 8.5c Regression Results for the Remaining Independent Variables for Four Foreign Exchange
Options 338
Table 8.11 Comaprisons of the First Order Strike Price Effect for the First Period and the Actual First Order Strike Price Effect for the Second Period
Table 8.12 Comaprisons of the Second Order Strike Price Effect for the First Period and the Actual Second Order Strike Price Effect for the Second Period
356
358
CHAPTER NINE
Table 9.4 Results from Fitting the Student-t Model Theoretical Implied Volatility Surface with a
Polynomial Function 388
Table 9.5a Regression Results for the Predicted Smile Behaviour from a Constant Volatility Student-t Distribution model against the Actual Smile Behaviour for Twelve Option Markets for the Entire
Period of the Analysis 390
Table 9.5b Regression Results for the Predicted Smile Behaviour from a Constant Volatility Student-t Distribution model against the Actual Smile Behaviour for Twelve Option Markets for the First Half of the Available Observations 391 Table 9.5c Regression Results for the Predicted Smile Behaviour from a Constant Volatility Student-t
Distribution model against the Actual Smile Behaviour for Twelve Option Markets for the Second
Half of the Available Observations 391 Table 9.6a Regression Results for the Predicted Smile Behaviour from the Optimal Stochastic
Volatility Model assuming that Underlying asset follows GBM versus the Actual Smile Behaviour
for Twelve Option Markets for the Entire Period of the Analysis 396 Table 9.6b Regression Results for the Predicted Smile Behaviour from the Optimal Stochastic
Volatility Model assuming that Underlying asset follows GBM versus the Actual Smile Behaviour
for Twelve Option Markets for the First Half of the Available Observations 396 Table 9.6c Regression Results for the Predicted Smile Behaviour from the Optimal Stochastic
Volatility Model assuming that Underlying asset follows GBM versus the Actual Smile Behaviour
for Twelve Option Markets for the Second Half of the Available Observations 397 Table 9.7a Regression Results for the Predicted Smile Behaviour from the Optimal Stochastic
Volatility Model assuming that Underlying asset follows a Student-t Distribution against the Actual
Smile Behaviour for Twelve Option Markets for the Entire Period of the Analysis 404 Table 9.7b Regression Results for the Predicted Smile Behaviour from the Optimal Stochastic
Volatility Model assuming that Underlying asset follows a Student-t Distribution against the Actual
Smile Behaviour for Twelve Option Markets for the First Half of the Available Observations - 404 Table 9.7c Regression Results for the Predicted Smile Behaviour from the Optimal Stochastic
Volatility Model assuming that Underlying asset follows a Student-t Distribution against the Actual
Smile Behaviour for Twelve Option Markets for the Second Half of the Available Observations_ 405
Table 9.8a Comparisons of the Adjusted R-Squares Statistics for the Three Possible Models to explain the Dynamics of the Implied Volatility Smiles for Four Stock Index Options 406 Table 9.8b Comparisons of the Adjusted R-Squares Statistics for the Three Possible Models to explain
the Dynamics of the Implied Volatility Smiles for Four Fixed Income Options 407 Table 9.8c Comparisons of the Adjusted R-Squares Statistics for the Three Possible Models to explain
the Dynamics of the Implied Volatility Smiles for Four Foreign Exchange Options 407 Table 9.9a Comparisons of An Average Difference Test for the Three Possible Models to explain the
Dynamics of the Implied Volatility Smiles for Four Stock Index Options 410 Table 9.9b Comparisons of An Average Difference Test for the Three Possible Models to explain the
Dynamics of the Implied Volatility Smiles for Four Fixed Income Options 410 Table 9.9c Comparisons of An Average Difference Test for the Three Possible Models to explain the
Dynamics of the Implied Volatility Smiles for Four Foreign Exchange Options 411
VOLUME THREE
Figure 2.1a Exponentially Weighted Unconditional Volatility for Four Stock Index Futures 1 Figure 2.1b Exponentially Weighted Unconditional Volatility for Four Fixed Income Futures 2
Figure 2.1c Exponentially Weighted Unconditional Volatility for Four Foreign Exchange Futures_ 3 Figure 2.2a Autocorrelogram for four Stock Index Futures absolute daily returns 4 Figure 2.2b Autocorrelogram for four Fixed Income Futures absolute daily returns 5 Figure 2.2c Autocorrelogram for four Foreign Exchange Futures absolute daily returns
Figure 2.2d Comparison of autocorrelograms on S&P-500 Futures
6
Figure 2.2e Autocorrelogram for DAX Index Futures and Cash using absolute daily returns 8 Figure 2.3a First period autocorrelogram for four Stock Index Futures absolute daily returns 9 Figure 2.3b First period autocorrelogram for four Fixed income Futures absolute daily returns 10 Figure 2.3c First period autocorrelogram for four Foreign exchange Futures absolute daily returns
_ 11 Figure 2.4a Second period autocorrelogram for four Stock Index Futures absolute daily returns 12 Figure 2.4b Second period autocorrelogram for four Fixed Income Futures absolute daily returns
_ 13 Figure 2.4c Second period autocorrelogram for four Foreign Exchange Futures absolute daily returns.
14 Table 2.5a Summary statistics of volatility cone for four Stock Index Futures 15 Table 2.5b Summary statistics of volatility cone for four Fixed Income Futures 16 Table 2.5c Summary statistics of volatility cone for four Foreign Exchange Futures 17
Figure 2.5a Volatility cones for four Stock Index Futures
Figure 2.5b Volatility cones for four Fixed Income Futures
18
19
Figure 2.5c Volatility cones for four Foreign Exchange Futures 20 Table 2.6a First period summary statistics of volatility cone for four Stock Index Futures 21 Table 2.6b First Period summary statistics of volatility cone for four Fixed Income Futures 22 Table 2.6c First period summary statistics of volatility cone for four Foreign Exchange Futures 23 Figure 2.6a First period volatility cones for four Stock Index Futures 24 Figure 2.6b First period volatility cones for four Fixed Income Futures 25 Figure 2.6c First period volatility cones for four Foreign Exchange Futures 26
Table 2.7a Second period summary statistics of volatility cone for four Stock Index Futures 27 Table 2.7b Second period summary statistics of volatility cone for four Fixed Income Futures 28 Table 2.7c Second period summary statistics of volatility cone for four Foreign Exchange Futures - 29
Figure 2.7a Second period volatility cones for four Stock Index Futures
Figure 2.7b Second period volatility cones for four Fixed Income Futures
30 31
Figure 2.7c Second period volatility cones for four Foreign Exchange Futures 32 Figure 2.8a Comparison of volatility estimated using overlapping and non overlapping observations for
four Stock Index Futures 33
Figure 2.8b Comparison of volatility estimated using overlapping and non overlapping observations for
four Fixed Income Futures 34
Figure 2.8c Comparison of volatility estimated using overlapping and non overlapping observations for four Foreign Exchange Futures 35 Figure 2.9a First period comparison of volatility estimated using overlapping and non overlapping
observations for four Stock Index Futures 36 Figure 2.9b First period comparison of volatility estimated using overlapping and non overlapping
observations for four Fixed Income Futures 37 Figure 2.9c First period comparison of volatility estimated using overlapping and non overlapping
observations for four Foreign Exchange Futures 38 Figure 2.10a Second period comparison of volatility estimated using overlapping and non overlapping
observations for four Stock Index Futures 39 Figure 2.10b Second period comparison of volatility estimated using overlapping and non overlapping
observations for four Fixed Income Futures 40 Figure 2.10c Second period comparison of volatility estimated using overlapping and non overlapping
observations for four Foreign Exchange Futures 41 Figure 2.11 a Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for four Stock
Index Futures 42
Figure 2.11 b Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for four Fixed
Income Futures 43
Figure 2.11c Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for four Foreign
Exchange Futures 44
Figure 2.12a First period Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for
four Stock Index Futures 45
Figure 2.12b First period Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for four Fixed Income Futures
Figure 2.12c First period Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for four Foreign Exchange Futures
46
47
Figure 2.13a Second period Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for
four Stock Index Futures 48
Figure 2.13b Second period Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for
four Fixed Income Futures 49
Figure 2.13c Second period Standard deviation of the volatility cone, Biased vs. Unbiased vs. I. I. D. for four Foreign Exchange Futures 50 Figure 2.14a Time decay factors for the volatility of volatility for four Stock Index Futures 51 Figure 2.14b Time decay factors for the volatility of volatility for four Fixed Income Futures 52 Figure 2.14c Time decay factors for the volatility of volatility for four Foreign Exchange Futures
- 53 Figure 2.15a First period time decay factors for the volatility of volatility for four Stock Index Futures
Figure 2.15b First period time decay factors for the volatility of volatility for four Fixed Income
Futures 55
Figure 2.15c First period time decay factors for the volatility of volatility for four Foreign Exchange
Futures 56
Figure 2.16a Second period time decay factors for the volatility of volatility for four Stock Index
Futures 57
Figure 2.16b Second period time decay factors for the volatility of volatility for four Fixed Income
Futures 58
Figure 2.16c Second period time decay factors for the volatility of volatility for four Foreign Exchange
Futures 59
Table 7.1a Summary statistics for the At-the-money Implied Volatilities for four Stock Index Options, estimated ona a Daily basis and at 5-days increments until expiration 60 Table 7.1b Summary statistics for the At-the-money Implied Volatilities for four Fixed Income
Options, estimated ona a Daily basis and at 5-days increments until expiration
54
61
Table 7.1c Summary statistics for the At-the-money Implied Volatilities for four Foreign Exchange
Options, estimated ona a Daily basis and at 5-days increments until expiration 62 Figure 7.1a Time Series Plots of At-the-money Implied Volatilities for Four Stock Index Options
- 63 Figure 7.1b Time Series Plots of At-the-money Implied Volatilities for Four Fixed Income Options 64 Figure 7.1c Time Series Plots of At-the-money Implied Volatilities for Four Foreign Exchange Options
65 Table 7.2 Option prices and implied volatilities for FTSE-100 as of May7th, 1996 66
Figure 7.2a Implied. Volatility Smiles for Four Stock Index Options as of May 7,1996 67 Figure 7.2b Implied Volatility Smiles for Four Fixed Income Options as of May 7,1996 68 Figure 7.2c Implied Volatility Smiles for Four Foreign Exchange Options as of May 7,1996 69 Figure 7.4 Unstandardized volatility smiles for 1996 FTSE-100 contracts 70 Figure 7.5 Standardized volatility smiles for 1996 FTSE-100 contracts grouped by contracts 71
Figure 7.6 Standardized volatility smiles for 1996 FTSE-100 contracts grouped by days to expiration
72
Figure 7.7 Interpolated volatility smiles for 1996 FTSE-100 contracts Figure 7.8a Skewness for 1996 FTSE-100 contracts
Figure 7.8b Skewness for 1996 FTSE-100 contracts
73 74
75
Figure 7.9a Kurtosis for 1996 FTSE-100 contracts 76 Figure 7.9b Kurtosis for 1996 FTSE-100 contracts 77
Figure 7.1Oa Scatterplots of minimum VSI strike price relative to underlying future price for the FTSE-
100 during 1996 78
Figure 7.10b Plot of the percentage difference between the Underlying Futures Price to the minimum Strike Price for Four Options Cycles for the FTSE-100 during 1996
Figure 7.11a 1996 Volatility smiles for Four Stock Index Options
79 80
Figure 7.11b 1996 Volatility smiles for Four Fixed Income Options 81 Figure 7.1 Ic 1996 Volatility smiles for Four Foreign Exchange Options 82 Figure 7.12a 1996 Skewness for Four Stock Index Options 83 Figure 7.12b 1996 Skewness for Four Fixed Income Options 84 Figure 7.12c 1996 Skewness for Four Foreign Exchange Options 85 Figure 7.13a 1996 Skewness for Four Stock Index Options 86
Figure 7.13b 1996 Skewness for Four Fixed Income Options
Figure 7.13c 1996 Skewness for Four Foreign Exchange Options
87
88
Figure 7.14a 1996 Kurtosis for Four Stock Index Options 89 Figure 7.14b 1996 Kurtosis for Four Fixed Income Options 90 Figure 7.14c 1996 Kurtosis for Four Foreign Exchange Options 91 Figure 7.15a 1996 Kurtosis for Four Stock Index Options 92 Figure 7.15b 1996 Kurtosis for Four Fixed Income Options 93 Figure 7.15c 1996 Kurtosis for Four Foreign Exchange Options 94 Figure 7.16a Standardized Volatility Smiles for Four Stock Index Options for the entire period of
analysis 95
Figure 7.16b Standardized Volatility Smiles for Four Fixed Income Options for the entire period of
analysis 96
Figure 7.16c Standardized Volatility Smiles for Four Foreign Exchange Options for the entire period of
analysis 97
Figure 7.17a Standardized Volatility Smiles for Four Stock Index Options for the first portion of the
available observations 98
Figure 7.17b Standardized Volatility Smiles for Four Fixed Income Options for the first portion of the
available observations 99
Figure 7.17c Standardized Volatility Smiles for Four Foreign Exchange Options for the first portion of
the available observations 100
Figure 7.18a Standardized Volatility Smiles for Four Stock Index Options for the second portion of the
available observations 101
Figure 7.18b Standardized Volatility Smiles for Four Fixed Income Options for the second portion of
the available observations 102
Figure 7.18c Standardized Volatility Smiles for Four Foreign Exchange Options for the second portion of the available observations 103 Table 8.2a Ordinary Least Squares Regression Results for four Stock Index Options 120 Table 8.2b Ordinary Least Squares Regression Results for four Fixed Income Options 121 Table 8.2c Ordinary Least Squares Regression Results for four Foreign Exchange Options 122
Table 8.6a Ordinary Least Squares Regression Results for four Stock Index Options Including
Contracts as Dummy Variables 123 Table 8.6b Ordinary Least Squares Regression Results for four Fixed Income Options Including
Contracts as Dummy Variables 124 Table 8.6c Ordinary Least Squares Regression Results for four Foreign Exchange Options Including
Contracts as Dummy Variables 125 Table 8.7a Weighted Least Squares Regression Results for four Stock Index Options 126 Table 8.7b Weighted Least Squares Regression Results for four Fixed Income Options 127 Table 8.7c Weighted Least Squares Regression Results for four Foreign Exchange Options 128 Table 8.8 Generalized Least Squares Regression Results fot three Selected Options Markets 129
Table 8.9a Ordinary Least Squares Regression Results for four Stock Index Options using the First
Half of Available Observations 130 Table 8.9b Ordinary Least Squares Regression Results for four Fixed Income Options using the First
Half of Available Observations 131 Table 8.9c Ordinary Least Squares Regression Results for four Foreign Exchange Options using the
First Half of Available Observations 132 Table 8.10a Ordinary Least Squares Regression Results for four Stock Index Options using the Second
Half of Available Observations 133 Table 8.1Ob Ordinary Least Squares Regression Results for four Fixed Income Options using the
Second Half of Available Observations 134 Table 8.10c Ordinary Least Squares Regression Results for four Foreign Exchange Options using the
Second Half of Available Observations 135
Table 8.13a Ordinary Least Squares Regression Results for four Stock Index Options compared to All
Stock Index Options 136
Table 8.13b Ordinary Least Squares Regression Results for four Fixed Income Options compared to
All Fixed Income Options 137
Table 8.13c Ordinary Least Squares Regression Results for four Foreign Exchange Options compared
to All Foreign Exchange Options 138 Table 8.14 Ordinary Least Squares Regression Results for All Option Markets comaperd to the three
Asset Classes 139
Table 9.1 Monte Carlo generated Call Option Prices using a Student-t distribution with constant
volatility 140
Table 9.2 Volatilities implied by the Monte Carlo generated Call Option Prices using a Student-t
distribution with constant volatility 141 Table 9.3 Standardized implied volatilities using a Student-t distribution with constant volatility - 142 Figure 9.1 Standardized Volatility Smiles using a Student-t distribution with constant volatility - 143 Figure 9.2a S&P-500 Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 144 Figure 9.2b FTSE-100 Whole Period Implied Volatility Smile Patterns that are consistent with the three Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 145
Figure 9.2c Nikkei-225 Whole Period Implied Volatility Smile Patterns that are consistent with the three Security Price Models estimated using Unconditional Dispersion Process compared to the
Actual 146
Figure 9.2d DAX Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 147 Figure 9.3a Bund Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 148 Figure 9.3b Gilt Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 149 Figure 9.3c BTP Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 150 Figure 9.3d US T-Bond Whole Period Implied Volatility Smile Patterns that are consistent with the
three Security Price Models estimated using Unconditional Dispersion Process compared to the
Actual 151
Figure 9.4a D-Mark Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 152 Figure 9.4b B-Pound Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 153 Figure 9.4c J-Yen Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 154 Figure 9.4d S-Franc Whole Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 155 Figure 9.5a S&P-500 First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 156 Figure 9.5b FTSE-100 First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 157 Figure 9.5c Nikkei-225 First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 158 Figure 9.5d DAX First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 159 Figure 9.6a Bund First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 160 Figure 9.6b Gilt First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 161 Figure 9.6c BTP First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 162 Figure 9.6d US T-Bond First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 163 Figure 9.7a D-Mark First Period Implied Volatility Smile Patterns that are consistent with the three
Security Price Models estimated using Unconditional Dispersion Process compared to the Actual 164
