FX Barrier Options (eBook)

Zareer Dadachanji is a quantitative analysis consultant with nearly two decades of corporate experience, mostly in financial quantitative modelling across a range of asset classes. He has spent 14 years working as a front-office quant at banks and hedge funds, including NatWest/RBS, Credit Suisse and latterly Standard Chartered Bank, where he held the position of Global Head of FX Quants. Zareer's specialist areas of expertise are the modelling of FX and equity derivatives. He combines these specialist areas with substantial knowledge of general quantitative modelling, gained through years of senior-level engagement in the activities of global cross-asset quant teams. Zareer is the founder and director of Model Quant Solutions, an independent consultancy providing bespoke quantitative analysis and training on a range of financial subjects. The consultancy serves a variety of clients and client types across the finance industry. Zareer holds a triple first in Natural Sciences and a PhD in Computational and Theoretical Physics, both from the University of Cambridge.

Cover 1
Half-Title 2
Tltle 4
Copyright 5
Dedication 6
Contents 8
List of Figures 13
List of Tables 20
Preface 21
Acknowledgements 25
Foreword 26
Glossary of Mathematical Notation 28
Contract Types 29
1 Meet the Products 31
1.1 Spot 31
1.1.1 Dollars per euro or euros per dollar? 33
1.1.2 Big figures and small figures 34
1.1.3 The value of Foreign 34
1.1.4 Converting between Domestic and Foreign 36
1.2 Forwards 36
1.2.1 The FX forward market 37
1.2.2 A formula for the forward rate 38
1.2.3 Payoff of a forward contract 40
1.2.4 Valuation of a forward contract 42
1.3 Vanilla options 42
1.3.1 Put–call parity 45
1.4 European digitals 46
1.5 Barrier-contingent vanilla options 46
1.6 Barrier-contingent payments 53
1.7 Rebates 55
1.8 Knock-in-knock-out (KIKO) options 55
1.9 Types of barriers 56
1.10 Structured products 57
1.11 Specifying the contract 58
1.12 Quantitative truisms 59
1.12.1 Foreign exchange symmetry and inversion 59
1.12.2 Knock-out plus knock-in equals no-barrier contract 59
1.12.3 Put–call parity 60
1.13 Jargon-buster 60
2 Living in a Black–Scholes World 63
2.1 The Black–Scholes model equation forspot price 63
2.2 The process for ln S 65
2.3 The Black–Scholes equation for option pricing 68
2.3.1 The lagless approach 68
2.3.2 Derivation of the Black–Scholes PDE 69
2.3.3 Black–Scholes model: hedging assumptions 72
2.3.4 Interpretation of the Black–Scholes PDE 73
2.4 Solving the Black–Scholes PDE 75
2.5 Payments 75
2.6 Forwards 77
2.7 Vanilla options 77
2.7.1 Transformation of the Black–Scholes PDE 78
2.7.2 Solution of the diffusion equation for vanilla options 82
2.7.3 The vanilla option pricing formulae 87
2.7.4 Price quotation styles 89
2.7.5 Valuation behaviour of vanilla options 90
2.8 Black–Scholes pricing of barrier-contingent vanilla options 94
2.8.1 Knock-outs 95
2.8.2 Knock-ins 99
2.8.3 Quotation methods 100
2.8.4 Valuation behaviour of barrier-contingent vanilla options 100
2.9 Black–Scholes pricing of barrier-contingent payments 103
2.9.1 Payment in Domestic 104
2.9.2 Payment in Foreign 106
2.9.3 Quotation methods 106
2.9.4 Valuation behaviour of barrier-contingent payments 107
2.10 Discrete barrier options 110
2.11 Window barrier options 110
2.12 Black–Scholes numerical valuation methods 111
3 Black–Scholes Risk Management 112
3.1 Spot risk 113
3.1.1 Local spot risk analysis 113
3.1.2 Delta 114
3.1.3 Gamma 115
3.1.4 Results for spot Greeks 116
3.1.5 Non-local spot risk analysis 127
3.2 Volatility risk 127
3.2.1 Local volatility risk analysis 128
3.2.2 Non-local volatility risk 142
3.3 Interest rate risk 143
3.4 Theta 145
3.5 Barrier over-hedging 147
3.6 Co-Greeks 150
4 Smile Pricing 151
4.1 The shortcomings of the Black–Scholes model 151
4.2 Black–Scholes with term structure (BSTS) 153
4.3 The implied volatility surface 155
4.4 The FX vanilla option market 156
4.4.1 At-the-money volatility 159
4.4.2 Risk reversal 161
4.4.3 Butterfly 162
4.4.4 The role of the Black–Scholes model in the FX vanilla options market 163
4.5 Theoretical Value (TV) 163
4.5.1 Conventions for extracting market data for TV calculations 164
4.5.2 Example broker quote request 165
4.6 Modelling market implied volatilities 166
4.7 The probability density function 167
4.8 Three things we want from a model 171
4.9 The local volatility (LV) model 171
4.9.1 It’s the smile dynamics, stupid 185
4.10 Five things we want from a model 186
4.11 Stochastic volatility (SV) models 187
4.11.1 SABR model 187
4.11.2 Heston model 188
4.12 Mixed local/stochastic volatility (LSV) models 192
4.12.1 Term structure of volatility of volatility 200
4.13 Other models and methods 201
4.13.1 Uncertain volatility (UV) models 201
4.13.2 Jump–diffusion models 202
4.13.3 Vanna–volga methods 203
5 Smile Risk Management 205
5.1 Black–Scholes with term structure 205
5.2 Local volatility model 209
5.3 Spot risk under smile models 210
5.4 Theta risk under smile models 212
5.5 Mixed local/stochastic volatility models 212
5.6 Static hedging 213
5.7 Managing risk across businesses 214
6 Numerical Methods 216
6.1 Finite-difference (FD) methods 216
6.1.1 Grid geometry 217
6.1.2 Finite-difference schemes 219
6.2 Monte Carlo (MC) methods 223
6.2.1 Monte Carlo schedules 224
6.2.2 Monte Carlo algorithms 225
6.2.3 Variance reduction 227
6.2.4 The Brownian Bridge 229
6.2.5 Early termination 230
6.3 Calculating Greeks 230
6.3.1 Bumped Greeks 230
6.3.2 Greeks from finite-difference calculations 232
6.3.3 Greeks from Monte Carlo 233
7 Further Topics 235
7.1 Managed currencies 235
7.2 Stochastic interest rates (SIR) 236
7.3 Real-world pricing 240
7.3.1 Bid–offer spreads 240
7.3.2 Rules-based pricing methods 242
7.4 Regulation and market abuse 243
A Derivation of the Black–Scholes Pricing Equations for Vanilla Options 245
B Normal and Lognormal Probability Distributions 250
B.1 Normal distribution 250
B.2 Lognormal distribution 250
C Derivation of the Local Volatility Function 251
C.1 Derivation in terms of call prices 251
C.2 Local volatility from implied volatility 255
C.3 Working in moneyness space 257
C.4 Working in log space 258
C.5 Specialization to BSTS 259
D Calibration of Mixed Local/Stochastic Volatility (LSV) Models 260
E Derivation of Fokker–Planck Equation for the Local Volatility Model 262
Bibliography 264
Index 267