Smile Pricing Explained (eBook)

In Smile Pricing Explained, Peter Austing draws on a decade of experience building the mathematical models for derivatives trading at major investment banks, most recently Barclays Capital where he was a Director in Quantitative Analytics. Before moving to finance, Dr Austing held research positions in theoretical physics studying candidate theories of quantum gravity. It was at Oxford University, while teaching mathematics at St John's College, that he developed his love for teaching and accessible style.A seasoned quantitative practitioner, and regular conference speaker, Peter Austing is best known for his work on multi-asset derivative pricing with correlation smile. He is currently engaged in research at Imperial College, London.

Cover 1
Half-Title 2
Title 4
Copyright 5
Dedication 6
Contents 7
List of Symbols 11
Acknowledgements 13
Preface 14
1 Introduction to Derivatives 15
1.1 Hedging with Forward Contracts 15
1.2 Speculation with Forward Contracts 16
1.3 Arbitrage 16
1.4 Vanilla Options 17
1.5 Interest Rates 19
1.6 Valuing a Forward Contract 20
1.7 Key Points 23
1.8 Further Reading 23
2 Stochastic Calculus 24
2.1 Brownian Motion 24
2.2 Stochastic Model for Stock Price Evolution 27
2.3 Ito’s Lemma 28
2.4 The Product Rule 29
2.5 Log-Normal Stock Price Evolution 30
2.6 The Markov Property 31
2.7 Term Structure 32
2.8 Ito’s Lemma in More than One Dimension 33
2.9 Key Points 34
2.10 Further Reading 34
3 Martingale Pricing 35
3.1 Setting the Scene 35
3.2 Tradeable Assets 36
3.3 Zero Coupon Bond 36
3.4 Rolling Money Market Account 36
3.5 Choosing a Numeraire 37
3.6 Changing the Measure 37
3.7 Girsanov’s Theorem 38
3.8 Martingales 41
3.9 Continuous Martingales 42
3.10 Black–Scholes Formula for a Call Option 42
3.11 At-the-Money Options 46
3.12 The Black–Scholes Equation 46
3.13 An Elegant Derivation of the Black–Scholes Formula 48
3.14 Key Points 52
3.15 Further Reading 53
4 Dynamic Hedging and Replication 54
4.1 Dynamic Hedging in the Absence of Interest Rates 54
4.2 Dynamic Hedging with Interest Rates 56
4.3 Delta Hedging 57
4.4 The Greeks 57
4.5 Gamma, Vega and Time Decay 58
4.6 Vega and Volatility Trading 59
4.7 Key Points 60
4.8 Further Reading 60
5 Exotic Options in Black–Scholes 61
5.1 European Options 61
5.2 Asian Options 62
5.3 Continuous Barrier Options 64
5.3.1 The Reflection Principle 65
5.3.2 The Reflection Principle with Log-Normal Dynamic 67
5.3.3 Valuing Barrier Options in Black–Scholes 68
5.3.4 Discretely Monitored Barrier Options 70
5.4 Key Points 70
5.5 Further Reading 71
6 Smile Models 72
6.1 The Volatility Smile 72
6.2 Smile Implied Probability Distribution 76
6.3 The Forward Kolmogorov Equation 79
6.4 Local Volatility 80
6.5 Key Points 83
6.6 Further Reading 84
7 Stochastic Volatility 85
7.1 Properties of Stochastic Volatility Models 86
7.2 The Heston Model 87
7.2.1 What Makes the Heston Model Special 87
7.2.2 Solving for Vanilla Prices 90
7.2.3 The Feller Boundary Condition 94
7.3 The SABR Model 96
7.4 The Ornstein–Uhlenbeck Process 100
7.5 Mixture Models 102
7.6 Regime Switching Model 103
7.7 Calibrating Stochastic Volatility Models 106
7.8 Key Points 109
7.9 Further Reading 109
8 Numerical Techniques 110
8.1 Monte Carlo 111
8.1.1 Monte Carlo in One Dimension 111
8.1.2 Monte Carlo in More than One Dimension 114
8.1.3 Variance Reduction in Monte Carlo 116
8.1.4 Limitations ofMonte Carlo 118
8.2 The PDE Approach 119
8.2.1 Stable and Unstable Schemes 122
8.2.2 Choice of Scheme 127
8.2.3 OtherWays of Improving Accuracy 128
8.2.4 More Complex Contracts in PDE 128
8.2.5 Solving Higher Dimension PDEs 130
8.3 Key Points 133
8.4 Further Reading 134
9 Local Stochastic Volatility 135
9.1 The Fundamental Theorem of On-smile Pricing 136
9.2 Arbitrage in Implied Volatility Surfaces 137
9.3 Two Extremes of Smile Dynamic 140
9.3.1 Sticky Strike Dynamic 140
9.3.2 Sticky Delta Dynamic 141
9.4 Local Stochastic Volatility 142
9.5 Simplifying Models 145
9.5.1 Spot–Volatility Correlation 145
9.5.2 Term Structure Vega for a Barrier Option 148
9.5.3 Simplifying Stochastic Volatility Parameters 151
9.5.4 Risk Managing with Local Stochastic Volatility Models 152
9.6 Practical Calibration 154
9.7 Impact of Mixing on Contract Values 155
9.8 Key Points 161
9.9 Further Reading 162
10 Volatility Products 163
10.1 Overview 163
10.2 Variance Swaps 163
10.2.1 The Variance Swap Contract 163
10.2.2 Idealised Variance Swap Trade 164
10.2.3 Valuing the Idealised Trade 165
10.2.4 Beauty in Variance Swaps 167
10.2.5 Delta and Gamma of a Variance Swap 169
10.2.6 Practical Considerations 171
10.3 Volatility Swaps 172
10.3.1 Volatility Swap in Stochastic Volatility Models and LSV 173
10.3.2 Volatility Swap Versus Variance Swap 175
10.3.3 Valuing a Volatility Swap 176
10.3.4 Stochastic versus Local Volatility 177
10.4 Forward Volatility Agreements 178
10.4.1 Practicalities 182
10.5 Key Points 184
10.6 Further Reading 185
11 Multi-Asset 186
11.1 Overview 186
11.2 Local Volatility with Constant Correlation 186
11.3 Copulas 187
11.4 Correlation Smile 189
11.5 Marking Correlation Smile 189
11.5.1 Common Correlation Products 190
11.5.2 The Triangle Rule 193
11.6 Modelling 195
11.6.1 Local Correlation 196
11.6.2 Practicalities 197
11.6.3 Local Stochastic Correlation 198
11.7 Valuing European Contracts 201
11.7.1 Special Properties of Best-of Options 201
11.7.2 Valuing a Best-of Option in Black–Scholes 202
11.7.3 Construction of a Joint PDF 204
11.7.4 Using the Density Function for Pricing 205
11.8 Numeraire Symmetry 207
11.9 Baskets as Correlation Instruments 208
11.10 Summary 210
11.11 Key Points 211
11.12 Further Reading 211
Afterword 212
Appendix: Measure Theory and Girsanov’s Theorem 214
References 221
Further Reading 227
Index 230