Austing Smile Pricing Explained

2014
ISBN: 978-1-137-33572-2
Verlag: Palgrave Macmillan UK
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 235 Seiten

Reihe: Financial Engineering Explained

Smile Pricing Explained provides a clear and thorough explanation of the concepts of smile modelling that are at the forefront of modern derivatives pricing. The key models used in practice are covered, together with numerical techniques and calibration.

Autoren/Hrsg.

Austing, P.

Inhaltsverzeichnis

1;Cover;1
2;Half-Title;2
3;Title;4
4;Copyright;5
5;Dedication;6
6;Contents;7
7;List of Symbols;11
8;Acknowledgements;13
9;Preface;14
10;1 Introduction to Derivatives;15
10.1;1.1 Hedging with Forward Contracts;15
10.2;1.2 Speculation with Forward Contracts;16
10.3;1.3 Arbitrage;16
10.4;1.4 Vanilla Options;17
10.5;1.5 Interest Rates;19
10.6;1.6 Valuing a Forward Contract;20
10.7;1.7 Key Points;23
10.8;1.8 Further Reading;23
11;2 Stochastic Calculus;24
11.1;2.1 Brownian Motion;24
11.2;2.2 Stochastic Model for Stock Price Evolution;27
11.3;2.3 Ito’s Lemma;28
11.4;2.4 The Product Rule;29
11.5;2.5 Log-Normal Stock Price Evolution;30
11.6;2.6 The Markov Property;31
11.7;2.7 Term Structure;32
11.8;2.8 Ito’s Lemma in More than One Dimension;33
11.9;2.9 Key Points;34
11.10;2.10 Further Reading;34
12;3 Martingale Pricing;35
12.1;3.1 Setting the Scene;35
12.2;3.2 Tradeable Assets;36
12.3;3.3 Zero Coupon Bond;36
12.4;3.4 Rolling Money Market Account;36
12.5;3.5 Choosing a Numeraire;37
12.6;3.6 Changing the Measure;37
12.7;3.7 Girsanov’s Theorem;38
12.8;3.8 Martingales;41
12.9;3.9 Continuous Martingales;42
12.10;3.10 Black–Scholes Formula for a Call Option;42
12.11;3.11 At-the-Money Options;46
12.12;3.12 The Black–Scholes Equation;46
12.13;3.13 An Elegant Derivation of the Black–Scholes Formula;48
12.14;3.14 Key Points;52
12.15;3.15 Further Reading;53
13;4 Dynamic Hedging and Replication;54
13.1;4.1 Dynamic Hedging in the Absence of Interest Rates;54
13.2;4.2 Dynamic Hedging with Interest Rates;56
13.3;4.3 Delta Hedging;57
13.4;4.4 The Greeks;57
13.5;4.5 Gamma, Vega and Time Decay;58
13.6;4.6 Vega and Volatility Trading;59
13.7;4.7 Key Points;60
13.8;4.8 Further Reading;60
14;5 Exotic Options in Black–Scholes;61
14.1;5.1 European Options;61
14.2;5.2 Asian Options;62
14.3;5.3 Continuous Barrier Options;64
14.3.1;5.3.1 The Reflection Principle;65
14.3.2;5.3.2 The Reflection Principle with Log-Normal Dynamic;67
14.3.3;5.3.3 Valuing Barrier Options in Black–Scholes;68
14.3.4;5.3.4 Discretely Monitored Barrier Options;70
14.4;5.4 Key Points;70
14.5;5.5 Further Reading;71
15;6 Smile Models;72
15.1;6.1 The Volatility Smile;72
15.2;6.2 Smile Implied Probability Distribution;76
15.3;6.3 The Forward Kolmogorov Equation;79
15.4;6.4 Local Volatility;80
15.5;6.5 Key Points;83
15.6;6.6 Further Reading;84
16;7 Stochastic Volatility;85
16.1;7.1 Properties of Stochastic Volatility Models;86
16.2;7.2 The Heston Model;87
16.2.1;7.2.1 What Makes the Heston Model Special;87
16.2.2;7.2.2 Solving for Vanilla Prices;90
16.2.3;7.2.3 The Feller Boundary Condition;94
16.3;7.3 The SABR Model;96
16.4;7.4 The Ornstein–Uhlenbeck Process;100
16.5;7.5 Mixture Models;102
16.6;7.6 Regime Switching Model;103
16.7;7.7 Calibrating Stochastic Volatility Models;106
16.8;7.8 Key Points;109
16.9;7.9 Further Reading;109
17;8 Numerical Techniques;110
17.1;8.1 Monte Carlo;111
17.1.1;8.1.1 Monte Carlo in One Dimension;111
17.1.2;8.1.2 Monte Carlo in More than One Dimension;114
17.1.3;8.1.3 Variance Reduction in Monte Carlo;116
17.1.4;8.1.4 Limitations ofMonte Carlo;118
17.2;8.2 The PDE Approach;119
17.2.1;8.2.1 Stable and Unstable Schemes;122
17.2.2;8.2.2 Choice of Scheme;127
17.2.3;8.2.3 OtherWays of Improving Accuracy;128
17.2.4;8.2.4 More Complex Contracts in PDE;128
17.2.5;8.2.5 Solving Higher Dimension PDEs;130
17.3;8.3 Key Points;133
17.4;8.4 Further Reading;134
18;9 Local Stochastic Volatility;135
18.1;9.1 The Fundamental Theorem of On-smile Pricing;136
18.2;9.2 Arbitrage in Implied Volatility Surfaces;137
18.3;9.3 Two Extremes of Smile Dynamic;140
18.3.1;9.3.1 Sticky Strike Dynamic;140
18.3.2;9.3.2 Sticky Delta Dynamic;141
18.4;9.4 Local Stochastic Volatility;142
18.5;9.5 Simplifying Models;145
18.5.1;9.5.1 Spot–Volatility Correlation;145
18.5.2;9.5.2 Term Structure Vega for a Barrier Option;148
18.5.3;9.5.3 Simplifying Stochastic Volatility Parameters;151
18.5.4;9.5.4 Risk Managing with Local Stochastic Volatility Models;152
18.6;9.6 Practical Calibration;154
18.7;9.7 Impact of Mixing on Contract Values;155
18.8;9.8 Key Points;161
18.9;9.9 Further Reading;162
19;10 Volatility Products;163
19.1;10.1 Overview;163
19.2;10.2 Variance Swaps;163
19.2.1;10.2.1 The Variance Swap Contract;163
19.2.2;10.2.2 Idealised Variance Swap Trade;164
19.2.3;10.2.3 Valuing the Idealised Trade;165
19.2.4;10.2.4 Beauty in Variance Swaps;167
19.2.5;10.2.5 Delta and Gamma of a Variance Swap;169
19.2.6;10.2.6 Practical Considerations;171
19.3;10.3 Volatility Swaps;172
19.3.1;10.3.1 Volatility Swap in Stochastic Volatility Models and LSV;173
19.3.2;10.3.2 Volatility Swap Versus Variance Swap;175
19.3.3;10.3.3 Valuing a Volatility Swap;176
19.3.4;10.3.4 Stochastic versus Local Volatility;177
19.4;10.4 Forward Volatility Agreements;178
19.4.1;10.4.1 Practicalities;182
19.5;10.5 Key Points;184
19.6;10.6 Further Reading;185
20;11 Multi-Asset;186
20.1;11.1 Overview;186
20.2;11.2 Local Volatility with Constant Correlation;186
20.3;11.3 Copulas;187
20.4;11.4 Correlation Smile;189
20.5;11.5 Marking Correlation Smile;189
20.5.1;11.5.1 Common Correlation Products;190
20.5.2;11.5.2 The Triangle Rule;193
20.6;11.6 Modelling;195
20.6.1;11.6.1 Local Correlation;196
20.6.2;11.6.2 Practicalities;197
20.6.3;11.6.3 Local Stochastic Correlation;198
20.7;11.7 Valuing European Contracts;201
20.7.1;11.7.1 Special Properties of Best-of Options;201
20.7.2;11.7.2 Valuing a Best-of Option in Black–Scholes;202
20.7.3;11.7.3 Construction of a Joint PDF;204
20.7.4;11.7.4 Using the Density Function for Pricing;205
20.8;11.8 Numeraire Symmetry;207
20.9;11.9 Baskets as Correlation Instruments;208
20.10;11.10 Summary;210
20.11;11.11 Key Points;211
20.12;11.12 Further Reading;211
21;Afterword;212
22;Appendix: Measure Theory and Girsanov’s Theorem;214
23;References;221
24;Further Reading;227
25;Index;230