The volatility surface: a practitioner's guide
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Hoboken, NJ
Wiley
2006
|
| Schriftenreihe: | Wiley finance
|
| Schlagworte: | |
| Online-Zugang: | Table of contents Inhaltsverzeichnis |
| Beschreibung: | Includes index Includes bibliographical references (S. 163 - 167) and index |
| Beschreibung: | XXVII, 179 S. Ill., graph. Darst. 24 cm |
| ISBN: | 9780471792512 0471792519 |
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| 245 | 1 | 0 | |a The volatility surface |b a practitioner's guide |c Jim Gatheral |
| 264 | 1 | |a Hoboken, NJ |b Wiley |c 2006 | |
| 300 | |a XXVII, 179 S. |b Ill., graph. Darst. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Wiley finance | |
| 500 | |a Includes index | ||
| 500 | |a Includes bibliographical references (S. 163 - 167) and index | ||
| 650 | 7 | |a Finanzmathematik |2 swd | |
| 650 | 7 | |a Hedging |2 gtt | |
| 650 | 7 | |a Investment Banking |2 swd | |
| 650 | 7 | |a Opties |2 gtt | |
| 650 | 7 | |a Prijsberekening |2 gtt | |
| 650 | 7 | |a Risk management |2 gtt | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Options (Finance) |x Prices |x Mathematical models | |
| 650 | 4 | |a Stocks |x Prices |x Mathematical models | |
| 650 | 0 | 7 | |a Mathematisches Modell |0 (DE-588)4114528-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Volatilität |0 (DE-588)4268390-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Optionspreis |0 (DE-588)4115453-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Optionspreis |0 (DE-588)4115453-8 |D s |
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Datensatz im Suchindex
| _version_ | 1804136669433036800 |
|---|---|
| adam_text | Contents
List of Figures xiii
List of Tables xix
Foreword xxi
Preface xxiii
Acknowledgments xxvii
CHAPTER 1
Stochastic Volatility and Local Volatility 1
Stochastic Volatility 1
Derivation of the Valuation Equation 4
Local Volatility 7
History 7
A Brief Review of Dupire s Work 8
Derivation of the Dupire Equation 9
Local Volatility in Terms of Implied Volatility 11
Special Case: No Skew 13
Local Variance as a Conditional Expectation
of Instantaneous Variance 13
CHAPTER 2
The Heston Model 15
The Process 15
The Heston Solution for European Options 16
A Digression: The Complex Logarithm
in the Integration (2.13) 19
Derivation of the Heston Characteristic Function 20
Simulation of the Heston Process 21
Milstein Discretization 22
Sampling from the Exact Transition Law 23
Why the Heston Model Is so Popular 24
Vl
Vjji CONTENTS
CHAPTER 3
The Implied Volatility Surface 25
Getting Implied Volatility from Local Volatilities 25
Model Calibration 25
Understanding Implied Volatility 26
Local Volatility in the Heston Model 31
Ansatz 32
Implied Volatility in the Heston Model 33
The Term Structure of Black Scholes Implied Volatility
in the Heston Model 34
The Black Scholes Implied Volatility Skew
in the Heston Model 35
The SPX Implied Volatility Surface 36
Another Digression: The SVI Parameterization 37
A Heston Fit to the Data 40
Final Remarks on SV Models and Fitting
the Volatility Surface 42
CHAPTER 4
The Heston Nandi Model 43
Local Variance in the Heston Nandi Model 43
A Numerical Example 44
The Heston Nandi Density 45
Computation of Local Volatilities 45
Computation of Implied Volatilities 46
Discussion of Results 49
CHAPTER 5
Adding Jumps 50
Why Jumps are Needed 50
Jump Diffusion 52
Derivation of the Valuation Equation 52
Uncertain Jump Size 54
Characteristic Function Methods 56
Levy Processes 56
Examples of Characteristic Functions
for Specific Processes 57
Computing Option Prices from the
Characteristic Function 58
Proof of (5.6) 58
Contents jx
Computing Implied Volatility 60
Computing the At the Money Volatility Skew 60
How Jumps Impact the Volatility Skew 61
Stochastic Volatility Plus Jumps 65
Stochastic Volatility Plus Jumps in the Underlying
Only (SVJ) 65
Some Empirical Fits to the SPX Volatility Surface 66
Stochastic Volatility with Simultaneous Jumps
in Stock Price and Volatility (SVJJ) 68
SVJ Fit to the September 15, 2005, SPX Option Data 71
Why the SVJ Model Wins 73
CHAPTER 6
Modeling Default Risk 74
Merton s Model of Default 74
Intuition 75
Implications for the Volatility Skew 76
Capital Structure Arbitrage 77
Put Call Parity 77
The Arbitrage 78
Local and Implied Volatility in the Jump to Ruin Model 79
The Effect of Default Risk on Option Prices 82
The CreditGrades Model 84
Model Setup 84
Survival Probability 85
Equity Volatility 86
Model Calibration 86
CHAPTER 7
Volatility Surface Asymptotics 87
Short Expirations 87
The Medvedev Scaillet Result 89
The SABR Model 91
Including Jumps 93
Corollaries 94
Long Expirations: Fouque, Papanicolaou, and Sircar 95
Small Volatility of Volatility: Lewis 96
Extreme Strikes: Roger Lee 97
Example: Black Scholes 99
Stochastic Volatility Models 99
Asymptotics in Summary 100
X CONTENTS
CHAPTER 8
Dynamics of the Volatility Surface 101
Dynamics of the Volatility Skew under Stochastic Volatility 101
Dynamics of the Volatility Skew under Local Volatility 102
Stochastic Implied Volatility Models 103
Digital Options and Digital Cliquets 103
Valuing Digital Options 104
Digital Cliquets 104
CHAPTER 9
Happier Options 107
Definitions 107
Limiting Cases 108
Limit Orders 108
European Capped Calls 109
The Reflection Principle 109
The Lookback Hedging Argument 112
One Touch Options Again 113
Put Call Symmetry 113
QuasiStatic Hedging and Qualitative Valuation 114
Out of the Money Barrier Options 114
One Touch Options 115
Live Out Options 116
Lookback Options 117
Adjusting for Discrete Monitoring 117
Discretely Monitored Lookback Options 119
Parisian Options 120
Some Applications of Barrier Options 120
Ladders 120
Ranges 120
Conclusion 121
CHAPTER 10
Exotic Cliquets 122
Locally Capped Globally Floored Cliquet 122
Valuation under Heston and Local
Volatility Assumptions 123
Performance 124
Reverse Cliquet 125
Contents xj
Valuation under Heston and Local
Volatility Assumptions 126
Performance 127
Napoleon 127
Valuation under Heston and Local
Volatility Assumptions 128
Performance 130
Investor Motivation 130
More on Napoleons 131
CHAPTER 11
Volatility Derivatives 133
Spanning Generalized European Payoffs 133
Example: European Options 134
Example: Amortizing Options 135
The Log Contract 135
Variance and Volatility Swaps 136
Variance Swaps 137
Variance Swaps in the Heston Model 138
Dependence on Skew and Curvature 138
The Effect of Jumps 140
Volatility Swaps 143
Convexity Adjustment in the Heston Model 144
Valuing Volatility Derivatives 146
Fair Value of the Power Payoff 146
The Laplace Transform of Quadratic Variation under
Zero Correlation 147
The Fair Value of Volatility under Zero Correlation 149
A Simple Lognormal Model 151
Options on Volatility: More on Model Independence 154
Listed Quadratic Variation Based Securities 156
The VIX Index 156
VXB Futures 158
Knock on Benefits 160
Summary 161
Postscript 162
Bibliography 163
Index 169
Figures
1.1 SPX daily log returns from December 31, 1984, to December
31, 2004. Note the 22.9% return on October 19, 1987! 2
1.2 Frequency distribution of (77 years of) SPX daily log returns
compared with the normal distribution. Although the —22.9%
return on October 19, 1987, is not directly visible, the x axis
has been extended to the left to accommodate it! 3
1.3 Q Q plot of SPX daily log returns compared with the normal
distribution. Note the extreme tails. 3
3.1 Graph of the pdf of xt conditional on xj = log(K) for a 1 year
European option, strike 1.3 with current stock price = 1 and
20% volatility. 31
3.2 Graph of the SPX implied volatility surface as of the close on
September 15, 2005, the day before triple witching. 36
3.3 Plots of the SVI fits to SPX implied volatilities for each of the
eight listed expirations as of the close on September 15, 2005.
Strikes are on the x axes and implied volatilities on the y axes.
The black and grey diamonds represent bid and offer volatilities
respectively and the solid line is the SVI fit. 38
3.4 Graph of SPX ATM skew versus time to expiry. The solid line
is a fit of the approximate skew formula (3.21) to all empirical
skew points except the first; the dashed fit excludes the first three
data points. 39
3.5 Graph of SPX ATM variance versus time to expiry. The solid
line is a fit of the approximate ATM variance formula (3.18) to
the empirical data. 40
3.6 Comparison of the empirical SPX implied volatility surface with
the Heston fit as of September 15, 2005. From the two views
presented here, we can see that the Heston fit is pretty good
m
Xjy FIGURES
for longer expirations but really not close for short expirations.
The paler upper surface is the empirical SPX volatility surface
and the darker lower one the Heston fit. The Heston fit surface
has been shifted down by five volatility points for ease of visual
comparison. 41
4.1 The probability density for the Heston Nandi model with our
parameters and expiration T = 0.1. 45
4.2 Comparison of approximate formulas with direct numerical
computation of Heston local variance. For each expiration T,
the solid line is the numerical computation and the dashed line
is the approximate formula. 47
4.3 Comparison of European implied volatilities from application of
the Heston formula (2.13) and from a numerical PDE computa¬
tion using the local volatilities given by the approximate formula
(4.1). For each expiration T, the solid line is the numerical
computation and the dashed line is the approximate formula. 48
5.1 Graph of the September 16, 2005, expiration volatility smile as
of the close on September 15, 2005. SPX is trading at 1227.73.
Triangles represent bids and offers. The solid line is a nonlinear
(SVI) fit to the data. The dashed line represents the Heston skew
with SepO5 SPX parameters. 52
5.2 The 3 month volatility smile for various choices of jump diffu¬
sion parameters. 63
5.3 The term structure of ATM variance skew for various choices of
jump diffusion parameters. 64
5.4 As time to expiration increases, the return distribution looks
more and more normal. The solid line is the jump diffusion pdf
and for comparison, the dashed line is the normal density with
the same mean and standard deviation. With the parameters
used to generate these plots, the characteristic time T* = 0.67. 65
5.5 The solid line is a graph of the at the money variance skew
in the SVJ model with BCC parameters vs. time to expiration.
The dashed line represents the sum of at the money Heston and
jump diffusion skews with the same parameters. 67
5.6 The solid line is a graph of the at the money variance skew in
the SVJ model with BCC parameters versus time to expiration.
The dashed line represents the at the money Heston skew with
the same parameters. 67
Figures XV
5.7 The solid line is a graph of the at the money variance skew in the
SVJJ model with BCC parameters versus time to expiration. The
short dashed and long dashed lines are SVJ and Heston skew
graphs respectively with the same parameters. 70
5.8 This graph is a short expiration detailed view of the graph shown
in Figure 5.7. 71
5.9 Comparison of the empirical SPX implied volatility surface with
the SVJ fit as of September 15, 2005. From the two views
presented here, we can see that in contrast to the Heston case,
the major features of the empirical surface are replicated by
the SVJ model. The paler upper surface is the empirical SPX
volatility surface and the darker lower one the SVJ fit. The SVJ
fit surface has again been shifted down by five volatility points
for ease of visual comparison. 72
6.1 Three month implied volatilities from the Merton model assum¬
ing a stock volatility of 20% and credit spreads of 100 bp (solid),
200 bp (dashed) and 300 bp (long dashed). 76
6.2 Payoff of the 1x2 put spread combination: buy one put with
strike 1.0 and sell two puts with strike 0.5. 79
6.3 Local variance plot with X = 0.05 and a = 0.2. 81
6.4 The triangles represent bid and offer volatilities and the solid
line is the Merton model fit. 83
7.1 For short expirations, the most probable path is approximately
a straight line from spot on the valuation date to the strike at
expiration. It follows that ct|s (k, T) « [v)oc(0,0) + vioc(k, T)] /2
and the implied variance skew is roughly one half of the local
variance skew. 89
8.1 Illustration of a cliquet payoff. This hypothetical SPX cliquet
resets at the money every year on October 31. The thick solid
lines represent nonzero cliquet payoffs. The payoff of a 5 year
European option struck at the October 31, 2000, SPX level of
1429.40 would have been zero. 105
9.1 A realization of the zero log drift stochastic process and the
reflected path. 110
9.2 The ratio of the value of a one touch call to the value of
a European binary call under stochastic volatility and local
Xyi FIGURES
volatility assumptions as a function of strike. The solid line is
stochastic volatility and the dashed line is local volatility. Ill
9.3 The value of a European binary call under stochastic volatility
and local volatility assumptions as a function of strike. The solid
line is stochastic volatility and the dashed line is local volatility.
The two lines are almost indistinguishable. Ill
9.4 The value of a one touch call under stochastic volatility and local
volatility assumptions as a function of barrier level. The solid
line is stochastic volatility and the dashed line is local volatility. 112
9.5 Values of knock out call options struck at 1 as a function of
barrier level. The solid line is stochastic volatility; the dashed
line is local volatility. 115
9.6 Values of knock out call options struck at 0.9 as a function of
barrier level. The solid line is stochastic volatility; the dashed
line is local volatility. 116
9.7 Values of live out call options struck at 1 as a function of barrier
level. The solid line is stochastic volatility; the dashed line is
local volatility. 117
9.8 Values of lookback call options as a function of strike. The solid
line is stochastic volatility; the dashed line is local volatility. 118
10.1 Value of the Mediobanca Bond Protection 2002 2005 locally
capped and globally floored cliquet (minus guaranteed redemp¬
tion) as a function of MinCoupon. The solid line is stochastic
volatility; the dashed line is local volatility. 124
10.2 Historical performance of the Mediobanca Bond Protection
2002 2005 locally capped and globally floored cliquet. The
dashed vertical lines represent reset dates, the solid lines coupon
setting dates and the solid horizontal lines represent fixings. 125
10.3 Value of the Mediobanca reverse cliquet (minus guaranteed
redemption) as a function of MaxCoupon. The solid line is
stochastic volatility; the dashed line is local volatility. 127
10.4 Historical performance of the Mediobanca 2000 2005 Reverse
Cliquet Telecommunicazioni reverse cliquet. The vertical lines
represent reset dates, the solid horizontal lines represent fixings
and the vertical grey bars represent negative contributions to the
cliquet payoff. 128
10.5 Value of (risk neutral) expected Napoleon coupon as a function
of MaxCoupon. The solid line is stochastic volatility; the dashed
line is local volatility. 129
Figures XVjj
10.6 Historical performance of the STOXX 50 component of the
Mediobanca 2002 2005 World Indices Euro Note Serie 46
Napoleon. The light vertical lines represent reset dates, the
heavy vertical lines coupon setting dates, the solid horizontal
lines represent fixings and the thick grey bars represent the
minimum monthly return of each coupon period. 130
11.1 Payoff of a variance swap (dashed line) and volatility swap
(solid line) as a function of realized volatility Sr Both swaps
are struck at 30% volatility. 143
11.2 Annualized Heston convexity adjustment as a function of T with
Heston Nandi parameters. 145
11.3 Annualized Heston convexity adjustment as a function of T with
Bakshi, Cao, and Chen parameters. 145
11.4 Value of 1 year variance call versus variance strike K with the
BCC parameters. The solid line is a numerical Heston solution;
the dashed line comes from our lognormal approximation. 153
11.5 The pdf of the log of 1 year quadratic variation with BCC
parameters. The solid line comes from an exact numerical
Heston computation; the dashed line comes from our lognormal
approximation. 154
11.6 Annualized Heston VXB convexity adjustment as a function of
t with Heston parameters from December 8, 2004, SPX fit. 160
Tables
3.1 At the money SPX variance levels and skews as of the close on
September 15, 2005, the day before expiration. 39
3.2 Heston fit to the SPX surface as of the close on September 15,
2005. 40
5.1 September 2005 expiration option prices as of the close on
September 15, 2005. Triple witching is the following day. SPX
is trading at 1227.73. 51
5.2 Parameters used to generate Figures 5.2 and 5.3. 63
5.3 Interpreting Figures 5.2 and 5.3. 64
5.4 Various fits of jump diffusion style models to SPX data. JD
means Jump Diffusion and SVJ means Stochastic Volatility plus
Jumps. 69
5.5 SVJ fit to the SPX surface as of the close on September 15, 2005. 71
6.1 Upper and lower arbitrage bounds for one year 0.5 strike options
for various credit spreads (at the money volatility is 20%). 79
6.2 Implied volatilities for January 2005 options on GT as of
October 20, 2004 (GT was trading at 9.40). Merton vols
are volatilities generated from the Merton model with fitted
parameters. 82
10.1 Estimated Mediobanca Bond Protection 2002 2005 coupons. 125
10.2 Worst monthly returns and estimated Napoleon coupons. Recall
that the coupon is computed as 10% plus the worst monthly
return averaged over the three underlying indices. 131
11.1 Empirical VXB convexity adjustments as of December 8, 2004. 159
xix
|
| adam_txt |
Contents
List of Figures xiii
List of Tables xix
Foreword xxi
Preface xxiii
Acknowledgments xxvii
CHAPTER 1
Stochastic Volatility and Local Volatility 1
Stochastic Volatility 1
Derivation of the Valuation Equation 4
Local Volatility 7
History 7
A Brief Review of Dupire's Work 8
Derivation of the Dupire Equation 9
Local Volatility in Terms of Implied Volatility 11
Special Case: No Skew 13
Local Variance as a Conditional Expectation
of Instantaneous Variance 13
CHAPTER 2
The Heston Model 15
The Process 15
The Heston Solution for European Options 16
A Digression: The Complex Logarithm
in the Integration (2.13) 19
Derivation of the Heston Characteristic Function 20
Simulation of the Heston Process 21
Milstein Discretization 22
Sampling from the Exact Transition Law 23
Why the Heston Model Is so Popular 24
Vl
Vjji CONTENTS
CHAPTER 3
The Implied Volatility Surface 25
Getting Implied Volatility from Local Volatilities 25
Model Calibration 25
Understanding Implied Volatility 26
Local Volatility in the Heston Model 31
Ansatz 32
Implied Volatility in the Heston Model 33
The Term Structure of Black Scholes Implied Volatility
in the Heston Model 34
The Black Scholes Implied Volatility Skew
in the Heston Model 35
The SPX Implied Volatility Surface 36
Another Digression: The SVI Parameterization 37
A Heston Fit to the Data 40
Final Remarks on SV Models and Fitting
the Volatility Surface 42
CHAPTER 4
The Heston Nandi Model 43
Local Variance in the Heston Nandi Model 43
A Numerical Example 44
The Heston Nandi Density 45
Computation of Local Volatilities 45
Computation of Implied Volatilities 46
Discussion of Results 49
CHAPTER 5
Adding Jumps 50
Why Jumps are Needed 50
Jump Diffusion 52
Derivation of the Valuation Equation 52
Uncertain Jump Size 54
Characteristic Function Methods 56
Levy Processes 56
Examples of Characteristic Functions
for Specific Processes 57
Computing Option Prices from the
Characteristic Function 58
Proof of (5.6) 58
Contents jx
Computing Implied Volatility 60
Computing the At the Money Volatility Skew 60
How Jumps Impact the Volatility Skew 61
Stochastic Volatility Plus Jumps 65
Stochastic Volatility Plus Jumps in the Underlying
Only (SVJ) 65
Some Empirical Fits to the SPX Volatility Surface 66
Stochastic Volatility with Simultaneous Jumps
in Stock Price and Volatility (SVJJ) 68
SVJ Fit to the September 15, 2005, SPX Option Data 71
Why the SVJ Model Wins 73
CHAPTER 6
Modeling Default Risk 74
Merton's Model of Default 74
Intuition 75
Implications for the Volatility Skew 76
Capital Structure Arbitrage 77
Put Call Parity 77
The Arbitrage 78
Local and Implied Volatility in the Jump to Ruin Model 79
The Effect of Default Risk on Option Prices 82
The CreditGrades Model 84
Model Setup 84
Survival Probability 85
Equity Volatility 86
Model Calibration 86
CHAPTER 7
Volatility Surface Asymptotics 87
Short Expirations 87
The Medvedev Scaillet Result 89
The SABR Model 91
Including Jumps 93
Corollaries 94
Long Expirations: Fouque, Papanicolaou, and Sircar 95
Small Volatility of Volatility: Lewis 96
Extreme Strikes: Roger Lee 97
Example: Black Scholes 99
Stochastic Volatility Models 99
Asymptotics in Summary 100
X CONTENTS
CHAPTER 8
Dynamics of the Volatility Surface 101
Dynamics of the Volatility Skew under Stochastic Volatility 101
Dynamics of the Volatility Skew under Local Volatility 102
Stochastic Implied Volatility Models 103
Digital Options and Digital Cliquets 103
Valuing Digital Options 104
Digital Cliquets 104
CHAPTER 9
Happier Options 107
Definitions 107
Limiting Cases 108
Limit Orders 108
European Capped Calls 109
The Reflection Principle 109
The Lookback Hedging Argument 112
One Touch Options Again 113
Put Call Symmetry 113
QuasiStatic Hedging and Qualitative Valuation 114
Out of the Money Barrier Options 114
One Touch Options 115
Live Out Options 116
Lookback Options 117
Adjusting for Discrete Monitoring 117
Discretely Monitored Lookback Options 119
Parisian Options 120
Some Applications of Barrier Options 120
Ladders 120
Ranges 120
Conclusion 121
CHAPTER 10
Exotic Cliquets 122
Locally Capped Globally Floored Cliquet 122
Valuation under Heston and Local
Volatility Assumptions 123
Performance 124
Reverse Cliquet 125
Contents xj
Valuation under Heston and Local
Volatility Assumptions 126
Performance 127
Napoleon 127
Valuation under Heston and Local
Volatility Assumptions 128
Performance 130
Investor Motivation 130
More on Napoleons 131
CHAPTER 11
Volatility Derivatives 133
Spanning Generalized European Payoffs 133
Example: European Options 134
Example: Amortizing Options 135
The Log Contract 135
Variance and Volatility Swaps 136
Variance Swaps 137
Variance Swaps in the Heston Model 138
Dependence on Skew and Curvature 138
The Effect of Jumps 140
Volatility Swaps 143
Convexity Adjustment in the Heston Model 144
Valuing Volatility Derivatives 146
Fair Value of the Power Payoff 146
The Laplace Transform of Quadratic Variation under
Zero Correlation 147
The Fair Value of Volatility under Zero Correlation 149
A Simple Lognormal Model 151
Options on Volatility: More on Model Independence 154
Listed Quadratic Variation Based Securities 156
The VIX Index 156
VXB Futures 158
Knock on Benefits 160
Summary 161
Postscript 162
Bibliography 163
Index 169
Figures
1.1 SPX daily log returns from December 31, 1984, to December
31, 2004. Note the 22.9% return on October 19, 1987! 2
1.2 Frequency distribution of (77 years of) SPX daily log returns
compared with the normal distribution. Although the —22.9%
return on October 19, 1987, is not directly visible, the x axis
has been extended to the left to accommodate it! 3
1.3 Q Q plot of SPX daily log returns compared with the normal
distribution. Note the extreme tails. 3
3.1 Graph of the pdf of xt conditional on xj = log(K) for a 1 year
European option, strike 1.3 with current stock price = 1 and
20% volatility. 31
3.2 Graph of the SPX implied volatility surface as of the close on
September 15, 2005, the day before triple witching. 36
3.3 Plots of the SVI fits to SPX implied volatilities for each of the
eight listed expirations as of the close on September 15, 2005.
Strikes are on the x axes and implied volatilities on the y axes.
The black and grey diamonds represent bid and offer volatilities
respectively and the solid line is the SVI fit. 38
3.4 Graph of SPX ATM skew versus time to expiry. The solid line
is a fit of the approximate skew formula (3.21) to all empirical
skew points except the first; the dashed fit excludes the first three
data points. 39
3.5 Graph of SPX ATM variance versus time to expiry. The solid
line is a fit of the approximate ATM variance formula (3.18) to
the empirical data. 40
3.6 Comparison of the empirical SPX implied volatility surface with
the Heston fit as of September 15, 2005. From the two views
presented here, we can see that the Heston fit is pretty good
m
Xjy FIGURES
for longer expirations but really not close for short expirations.
The paler upper surface is the empirical SPX volatility surface
and the darker lower one the Heston fit. The Heston fit surface
has been shifted down by five volatility points for ease of visual
comparison. 41
4.1 The probability density for the Heston Nandi model with our
parameters and expiration T = 0.1. 45
4.2 Comparison of approximate formulas with direct numerical
computation of Heston local variance. For each expiration T,
the solid line is the numerical computation and the dashed line
is the approximate formula. 47
4.3 Comparison of European implied volatilities from application of
the Heston formula (2.13) and from a numerical PDE computa¬
tion using the local volatilities given by the approximate formula
(4.1). For each expiration T, the solid line is the numerical
computation and the dashed line is the approximate formula. 48
5.1 Graph of the September 16, 2005, expiration volatility smile as
of the close on September 15, 2005. SPX is trading at 1227.73.
Triangles represent bids and offers. The solid line is a nonlinear
(SVI) fit to the data. The dashed line represents the Heston skew
with SepO5 SPX parameters. 52
5.2 The 3 month volatility smile for various choices of jump diffu¬
sion parameters. 63
5.3 The term structure of ATM variance skew for various choices of
jump diffusion parameters. 64
5.4 As time to expiration increases, the return distribution looks
more and more normal. The solid line is the jump diffusion pdf
and for comparison, the dashed line is the normal density with
the same mean and standard deviation. With the parameters
used to generate these plots, the characteristic time T* = 0.67. 65
5.5 The solid line is a graph of the at the money variance skew
in the SVJ model with BCC parameters vs. time to expiration.
The dashed line represents the sum of at the money Heston and
jump diffusion skews with the same parameters. 67
5.6 The solid line is a graph of the at the money variance skew in
the SVJ model with BCC parameters versus time to expiration.
The dashed line represents the at the money Heston skew with
the same parameters. 67
Figures XV
5.7 The solid line is a graph of the at the money variance skew in the
SVJJ model with BCC parameters versus time to expiration. The
short dashed and long dashed lines are SVJ and Heston skew
graphs respectively with the same parameters. 70
5.8 This graph is a short expiration detailed view of the graph shown
in Figure 5.7. 71
5.9 Comparison of the empirical SPX implied volatility surface with
the SVJ fit as of September 15, 2005. From the two views
presented here, we can see that in contrast to the Heston case,
the major features of the empirical surface are replicated by
the SVJ model. The paler upper surface is the empirical SPX
volatility surface and the darker lower one the SVJ fit. The SVJ
fit surface has again been shifted down by five volatility points
for ease of visual comparison. 72
6.1 Three month implied volatilities from the Merton model assum¬
ing a stock volatility of 20% and credit spreads of 100 bp (solid),
200 bp (dashed) and 300 bp (long dashed). 76
6.2 Payoff of the 1x2 put spread combination: buy one put with
strike 1.0 and sell two puts with strike 0.5. 79
6.3 Local variance plot with X = 0.05 and a = 0.2. 81
6.4 The triangles represent bid and offer volatilities and the solid
line is the Merton model fit. 83
7.1 For short expirations, the most probable path is approximately
a straight line from spot on the valuation date to the strike at
expiration. It follows that ct|s (k, T) « [v)oc(0,0) + vioc(k, T)] /2
and the implied variance skew is roughly one half of the local
variance skew. 89
8.1 Illustration of a cliquet payoff. This hypothetical SPX cliquet
resets at the money every year on October 31. The thick solid
lines represent nonzero cliquet payoffs. The payoff of a 5 year
European option struck at the October 31, 2000, SPX level of
1429.40 would have been zero. 105
9.1 A realization of the zero log drift stochastic process and the
reflected path. 110
9.2 The ratio of the value of a one touch call to the value of
a European binary call under stochastic volatility and local
Xyi FIGURES
volatility assumptions as a function of strike. The solid line is
stochastic volatility and the dashed line is local volatility. Ill
9.3 The value of a European binary call under stochastic volatility
and local volatility assumptions as a function of strike. The solid
line is stochastic volatility and the dashed line is local volatility.
The two lines are almost indistinguishable. Ill
9.4 The value of a one touch call under stochastic volatility and local
volatility assumptions as a function of barrier level. The solid
line is stochastic volatility and the dashed line is local volatility. 112
9.5 Values of knock out call options struck at 1 as a function of
barrier level. The solid line is stochastic volatility; the dashed
line is local volatility. 115
9.6 Values of knock out call options struck at 0.9 as a function of
barrier level. The solid line is stochastic volatility; the dashed
line is local volatility. 116
9.7 Values of live out call options struck at 1 as a function of barrier
level. The solid line is stochastic volatility; the dashed line is
local volatility. 117
9.8 Values of lookback call options as a function of strike. The solid
line is stochastic volatility; the dashed line is local volatility. 118
10.1 Value of the "Mediobanca Bond Protection 2002 2005" locally
capped and globally floored cliquet (minus guaranteed redemp¬
tion) as a function of MinCoupon. The solid line is stochastic
volatility; the dashed line is local volatility. 124
10.2 Historical performance of the "Mediobanca Bond Protection
2002 2005" locally capped and globally floored cliquet. The
dashed vertical lines represent reset dates, the solid lines coupon
setting dates and the solid horizontal lines represent fixings. 125
10.3 Value of the Mediobanca reverse cliquet (minus guaranteed
redemption) as a function of MaxCoupon. The solid line is
stochastic volatility; the dashed line is local volatility. 127
10.4 Historical performance of the "Mediobanca 2000 2005 Reverse
Cliquet Telecommunicazioni" reverse cliquet. The vertical lines
represent reset dates, the solid horizontal lines represent fixings
and the vertical grey bars represent negative contributions to the
cliquet payoff. 128
10.5 Value of (risk neutral) expected Napoleon coupon as a function
of MaxCoupon. The solid line is stochastic volatility; the dashed
line is local volatility. 129
Figures XVjj
10.6 Historical performance of the STOXX 50 component of the
"Mediobanca 2002 2005 World Indices Euro Note Serie 46"
Napoleon. The light vertical lines represent reset dates, the
heavy vertical lines coupon setting dates, the solid horizontal
lines represent fixings and the thick grey bars represent the
minimum monthly return of each coupon period. 130
11.1 Payoff of a variance swap (dashed line) and volatility swap
(solid line) as a function of realized volatility Sr Both swaps
are struck at 30% volatility. 143
11.2 Annualized Heston convexity adjustment as a function of T with
Heston Nandi parameters. 145
11.3 Annualized Heston convexity adjustment as a function of T with
Bakshi, Cao, and Chen parameters. 145
11.4 Value of 1 year variance call versus variance strike K with the
BCC parameters. The solid line is a numerical Heston solution;
the dashed line comes from our lognormal approximation. 153
11.5 The pdf of the log of 1 year quadratic variation with BCC
parameters. The solid line comes from an exact numerical
Heston computation; the dashed line comes from our lognormal
approximation. 154
11.6 Annualized Heston VXB convexity adjustment as a function of
t with Heston parameters from December 8, 2004, SPX fit. 160
Tables
3.1 At the money SPX variance levels and skews as of the close on
September 15, 2005, the day before expiration. 39
3.2 Heston fit to the SPX surface as of the close on September 15,
2005. 40
5.1 September 2005 expiration option prices as of the close on
September 15, 2005. Triple witching is the following day. SPX
is trading at 1227.73. 51
5.2 Parameters used to generate Figures 5.2 and 5.3. 63
5.3 Interpreting Figures 5.2 and 5.3. 64
5.4 Various fits of jump diffusion style models to SPX data. JD
means Jump Diffusion and SVJ means Stochastic Volatility plus
Jumps. 69
5.5 SVJ fit to the SPX surface as of the close on September 15, 2005. 71
6.1 Upper and lower arbitrage bounds for one year 0.5 strike options
for various credit spreads (at the money volatility is 20%). 79
6.2 Implied volatilities for January 2005 options on GT as of
October 20, 2004 (GT was trading at 9.40). Merton vols
are volatilities generated from the Merton model with fitted
parameters. 82
10.1 Estimated "Mediobanca Bond Protection 2002 2005" coupons. 125
10.2 Worst monthly returns and estimated Napoleon coupons. Recall
that the coupon is computed as 10% plus the worst monthly
return averaged over the three underlying indices. 131
11.1 Empirical VXB convexity adjustments as of December 8, 2004. 159
xix |
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| author | Gatheral, Jim |
| author_facet | Gatheral, Jim |
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| dewey-search | 332.63/2220151922 |
| dewey-sort | 3332.63 102220151922 |
| dewey-tens | 330 - Economics |
| discipline | Wirtschaftswissenschaften |
| discipline_str_mv | Wirtschaftswissenschaften |
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| id | DE-604.BV022551431 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:13:42Z |
| indexdate | 2024-07-09T21:00:04Z |
| institution | BVB |
| isbn | 9780471792512 0471792519 |
| language | English |
| lccn | 2006009977 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015757739 |
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| physical | XXVII, 179 S. Ill., graph. Darst. 24 cm |
| publishDate | 2006 |
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| publisher | Wiley |
| record_format | marc |
| series2 | Wiley finance |
| spelling | Gatheral, Jim Verfasser aut The volatility surface a practitioner's guide Jim Gatheral Hoboken, NJ Wiley 2006 XXVII, 179 S. Ill., graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Wiley finance Includes index Includes bibliographical references (S. 163 - 167) and index Finanzmathematik swd Hedging gtt Investment Banking swd Opties gtt Prijsberekening gtt Risk management gtt Mathematisches Modell Options (Finance) Prices Mathematical models Stocks Prices Mathematical models Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Volatilität (DE-588)4268390-7 gnd rswk-swf Optionspreis (DE-588)4115453-8 gnd rswk-swf Optionspreis (DE-588)4115453-8 s Volatilität (DE-588)4268390-7 s Mathematisches Modell (DE-588)4114528-8 s b DE-604 http://www.loc.gov/catdir/toc/ecip0611/2006009977.html Table of contents HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015757739&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gatheral, Jim The volatility surface a practitioner's guide Finanzmathematik swd Hedging gtt Investment Banking swd Opties gtt Prijsberekening gtt Risk management gtt Mathematisches Modell Options (Finance) Prices Mathematical models Stocks Prices Mathematical models Mathematisches Modell (DE-588)4114528-8 gnd Volatilität (DE-588)4268390-7 gnd Optionspreis (DE-588)4115453-8 gnd |
| subject_GND | (DE-588)4114528-8 (DE-588)4268390-7 (DE-588)4115453-8 |
| title | The volatility surface a practitioner's guide |
| title_auth | The volatility surface a practitioner's guide |
| title_exact_search | The volatility surface a practitioner's guide |
| title_exact_search_txtP | The volatility surface a practitioner's guide |
| title_full | The volatility surface a practitioner's guide Jim Gatheral |
| title_fullStr | The volatility surface a practitioner's guide Jim Gatheral |
| title_full_unstemmed | The volatility surface a practitioner's guide Jim Gatheral |
| title_short | The volatility surface |
| title_sort | the volatility surface a practitioner s guide |
| title_sub | a practitioner's guide |
| topic | Finanzmathematik swd Hedging gtt Investment Banking swd Opties gtt Prijsberekening gtt Risk management gtt Mathematisches Modell Options (Finance) Prices Mathematical models Stocks Prices Mathematical models Mathematisches Modell (DE-588)4114528-8 gnd Volatilität (DE-588)4268390-7 gnd Optionspreis (DE-588)4115453-8 gnd |
| topic_facet | Finanzmathematik Hedging Investment Banking Opties Prijsberekening Risk management Mathematisches Modell Options (Finance) Prices Mathematical models Stocks Prices Mathematical models Volatilität Optionspreis |
| url | http://www.loc.gov/catdir/toc/ecip0611/2006009977.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015757739&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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