{"product_id":"foreign-exchange-option-pricing-isbn-9780470683682","title":"Foreign Exchange Option Pricing","description":"This book covers foreign exchange options from the point of view of the finance practitioner. It contains everything a quant or trader working in a bank or hedge fund would need to know about the mathematics of foreign exchange—not just the theoretical mathematics covered in other books but also comprehensive coverage of implementation, pricing and calibration.   \u003cp\u003eWith content developed with input from traders and with examples using real-world data, this book introduces many of the more commonly requested products from FX options trading desks, together with the models that capture the risk characteristics necessary to price these products accurately. Crucially, this book describes the numerical methods required for calibration of these models – an area often neglected in the literature, which is nevertheless of paramount importance in practice. Thorough treatment is given in one unified text to the following features:\u003c\/p\u003e \u003cul type=\"disc\"\u003e \u003cli\u003eCorrect market conventions for FX volatility surface construction\u003c\/li\u003e \u003cli\u003eAdjustment for settlement and delayed delivery of options\u003c\/li\u003e \u003cli\u003ePricing of vanillas and barrier options under the volatility smile\u003c\/li\u003e \u003cli\u003eBarrier bending for limiting barrier discontinuity risk near expiry\u003c\/li\u003e \u003cli\u003eIndustry strength partial differential equations in one and several spatial variables using finite differences on nonuniform grids\u003c\/li\u003e \u003cli\u003eFourier transform methods for pricing European options using characteristic functions\u003c\/li\u003e \u003cli\u003eStochastic and local volatility models, and a mixed stochastic\/local volatility model\u003c\/li\u003e \u003cli\u003eThree-factor long-dated FX model\u003c\/li\u003e \u003cli\u003eNumerical calibration techniques for all the models in this work\u003c\/li\u003e \u003cli\u003eThe augmented state variable approach for pricing strongly path-dependent options using either partial differential equations or Monte Carlo simulation\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003eConnecting mathematically rigorous theory with practice, this is the essential guide to foreign exchange options in the context of the real financial marketplace.\u003c\/p\u003e \u003cp\u003eAcknowledgements xiii\u003c\/p\u003e \u003cp\u003eList of Tables xv\u003c\/p\u003e \u003cp\u003eList of Figures xvii\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Introduction 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 A Gentle Introduction to FX Markets 1\u003c\/p\u003e \u003cp\u003e1.2 Quotation Styles 2\u003c\/p\u003e \u003cp\u003e1.3 Risk Considerations 5\u003c\/p\u003e \u003cp\u003e1.4 Spot Settlement Rules 5\u003c\/p\u003e \u003cp\u003e1.5 Expiry and Delivery Rules 8\u003c\/p\u003e \u003cp\u003e1.5.1 Expiry and delivery rules – days or weeks 8\u003c\/p\u003e \u003cp\u003e1.5.2 Expiry and delivery rules – months or years 9\u003c\/p\u003e \u003cp\u003e1.6 Cutoff Times 10\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Mathematical Preliminaries 13\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 The Black–Scholes Model 13\u003c\/p\u003e \u003cp\u003e2.1.1 Assumptions of the Black–Scholes model 13\u003c\/p\u003e \u003cp\u003e2.2 Risk Neutrality 13\u003c\/p\u003e \u003cp\u003e2.3 Derivation of the Black–Scholes equation 14\u003c\/p\u003e \u003cp\u003e2.3.1 Equity derivatives (without dividends) 14\u003c\/p\u003e \u003cp\u003e2.3.2 FX derivatives 15\u003c\/p\u003e \u003cp\u003e2.3.3 Terminal conditions and present value 17\u003c\/p\u003e \u003cp\u003e2.4 Integrating the SDE for S T 17\u003c\/p\u003e \u003cp\u003e2.5 Black–Scholes PDEs Expressed in Logspot 18\u003c\/p\u003e \u003cp\u003e2.6 Feynman–Kac and Risk-Neutral Expectation 18\u003c\/p\u003e \u003cp\u003e2.7 Risk Neutrality and the Presumption of Drift 20\u003c\/p\u003e \u003cp\u003e2.7.1 Equity derivatives (without dividends) 20\u003c\/p\u003e \u003cp\u003e2.7.2 FX derivatives – domestic risk-neutral measure 21\u003c\/p\u003e \u003cp\u003e2.7.3 FX derivatives – foreign risk-neutral measure 22\u003c\/p\u003e \u003cp\u003e2.8 Valuation of European Options 23\u003c\/p\u003e \u003cp\u003e2.8.1 Forward 26\u003c\/p\u003e \u003cp\u003e2.9 The Law of One Price 27\u003c\/p\u003e \u003cp\u003e2.10 The Black–Scholes Term Structure Model 28\u003c\/p\u003e \u003cp\u003e2.11 Breeden–Litzenberger Analysis 30\u003c\/p\u003e \u003cp\u003e2.12 European Digitals 31\u003c\/p\u003e \u003cp\u003e2.12.1 Static replication for bid\/offer digital pricing 32\u003c\/p\u003e \u003cp\u003e2.13 Settlement Adjustments 32\u003c\/p\u003e \u003cp\u003e2.14 Delayed Delivery Adjustments 33\u003c\/p\u003e \u003cp\u003e2.14.1 Delayed delivery adjustments – digitals 33\u003c\/p\u003e \u003cp\u003e2.14.2 Delayed delivery adjustments – Europeans 34\u003c\/p\u003e \u003cp\u003e2.15 Pricing using Fourier Methods 35\u003c\/p\u003e \u003cp\u003e2.15.1 European option pricing involving one numerical integral 37\u003c\/p\u003e \u003cp\u003e2.16 Leptokurtosis – More than Fat Tails 38\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Deltas and Market Conventions 41\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Quote Style Conversions 41\u003c\/p\u003e \u003cp\u003e3.2 The Law of Many Deltas 43\u003c\/p\u003e \u003cp\u003e3.2.1 Pips spot delta 44\u003c\/p\u003e \u003cp\u003e3.2.2 Percentage spot delta (premium adjusted) 45\u003c\/p\u003e \u003cp\u003e3.2.3 Pips forward delta 45\u003c\/p\u003e \u003cp\u003e3.2.4 Percentage forward delta (premium adjusted) 45\u003c\/p\u003e \u003cp\u003e3.2.5 Simple delta 45\u003c\/p\u003e \u003cp\u003e3.2.6 Equivalence between pips and percentage deltas 46\u003c\/p\u003e \u003cp\u003e3.2.7 Premium adjustment 46\u003c\/p\u003e \u003cp\u003e3.2.8 Summary 47\u003c\/p\u003e \u003cp\u003e3.3 FX Delta Conventions 47\u003c\/p\u003e \u003cp\u003e3.3.1 To premium adjust or not? 47\u003c\/p\u003e \u003cp\u003e3.3.2 Spot delta or forward delta? 48\u003c\/p\u003e \u003cp\u003e3.3.3 Notation 49\u003c\/p\u003e \u003cp\u003e3.4 Market Volatility Surfaces 49\u003c\/p\u003e \u003cp\u003e3.4.1 Sample market volatility surfaces 50\u003c\/p\u003e \u003cp\u003e3.5 At-the-Money 50\u003c\/p\u003e \u003cp\u003e3.5.1 At-the-money – ATMF 51\u003c\/p\u003e \u003cp\u003e3.5.2 At-the-money – DNS 51\u003c\/p\u003e \u003cp\u003e3.5.3 At-the-money strikes – summary 52\u003c\/p\u003e \u003cp\u003e3.5.4 Example – EURUSD 1Y 52\u003c\/p\u003e \u003cp\u003e3.5.5 Example – USDJPY 1Y 53\u003c\/p\u003e \u003cp\u003e3.6 Market Strangle 53\u003c\/p\u003e \u003cp\u003e3.6.1 Example – EURUSD 1Y 55\u003c\/p\u003e \u003cp\u003e3.7 Smile Strangle and Risk Reversal 55\u003c\/p\u003e \u003cp\u003e3.7.1 Smile strangle from market strangle – algorithm 56\u003c\/p\u003e \u003cp\u003e3.8 Visualisation of Strangles 57\u003c\/p\u003e \u003cp\u003e3.9 Smile Interpolation – Polynomial in Delta 59\u003c\/p\u003e \u003cp\u003e3.9.1 Example – EURUSD 1Y – polynomial in delta 59\u003c\/p\u003e \u003cp\u003e3.10 Smile Interpolation – SABR 60\u003c\/p\u003e \u003cp\u003e3.10.1 Example – EURUSD 1Y – SABR 61\u003c\/p\u003e \u003cp\u003e3.11 Concluding Remarks 62\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Volatility Surface Construction 63\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Volatility Backbone – Flat Forward Interpolation 65\u003c\/p\u003e \u003cp\u003e4.2 Volatility Surface Temporal Interpolation 67\u003c\/p\u003e \u003cp\u003e4.2.1 Volatility smile extrapolation 67\u003c\/p\u003e \u003cp\u003e4.2.2 Volatility smile interpolation 68\u003c\/p\u003e \u003cp\u003e4.2.3 Flat forward vol interpolation in smile strikes 69\u003c\/p\u003e \u003cp\u003e4.2.4 Example – EURUSD 18M from 1Y and 2Y tenors – SABR 70\u003c\/p\u003e \u003cp\u003e4.3 Volatility Surface Temporal Interpolation – Holidays and Weekends 70\u003c\/p\u003e \u003cp\u003e4.4 Volatility Surface Temporal Interpolation – Intraday Effects 73\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Local Volatility and Implied Volatility 77\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Introduction 77\u003c\/p\u003e \u003cp\u003e5.2 The Fokker–Planck Equation 78\u003c\/p\u003e \u003cp\u003e5.2.1 Derivation of the one-dimensional Fokker–Planck equation 79\u003c\/p\u003e \u003cp\u003e5.2.2 The multidimensional Fokker–Planck equation 82\u003c\/p\u003e \u003cp\u003e5.3 Dupire’s Construction of Local Volatility 83\u003c\/p\u003e \u003cp\u003e5.3.1 Dupire’s local volatility – the r\u003csup\u003ed\u003c\/sup\u003e = r\u003csup\u003ef \u003c\/sup\u003e= 0 case 84\u003c\/p\u003e \u003cp\u003e5.3.2 Dupire’s local volatility – with nonzero but constant interest rates 85\u003c\/p\u003e \u003cp\u003e5.4 Implied Volatility and Relationship to Local Volatility 86\u003c\/p\u003e \u003cp\u003e5.5 Local Volatility as Conditional Expectation 87\u003c\/p\u003e \u003cp\u003e5.6 Local Volatility for FX Markets 88\u003c\/p\u003e \u003cp\u003e5.7 Diffusion and PDE for Local Volatility 89\u003c\/p\u003e \u003cp\u003e5.8 The CEV Model 90\u003c\/p\u003e \u003cp\u003e5.8.1 Asymptotic expansion 91\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Stochastic Volatility 95\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Introduction 95\u003c\/p\u003e \u003cp\u003e6.2 Uncertain Volatility 95\u003c\/p\u003e \u003cp\u003e6.3 Stochastic Volatility Models 96\u003c\/p\u003e \u003cp\u003e6.3.1 The Heston model 98\u003c\/p\u003e \u003cp\u003e6.3.2 The Stein and Stein model 104\u003c\/p\u003e \u003cp\u003e6.3.3 Longstaff’s double square root model 105\u003c\/p\u003e \u003cp\u003e6.3.4 Scott’s exponential Ornstein–Uhlenbeck model 105\u003c\/p\u003e \u003cp\u003e6.3.5 The SABR model 106\u003c\/p\u003e \u003cp\u003e6.4 Uncorrelated Stochastic Volatility 107\u003c\/p\u003e \u003cp\u003e6.5 Stochastic Volatility Correlated with Spot 108\u003c\/p\u003e \u003cp\u003e6.6 The Fokker–Planck PDE Approach 111\u003c\/p\u003e \u003cp\u003e6.7 The Feynman–Kac PDE Approach 113\u003c\/p\u003e \u003cp\u003e6.7.1 Heston model – example 116\u003c\/p\u003e \u003cp\u003e6.7.2 Heston model – logspot coordinates 117\u003c\/p\u003e \u003cp\u003e6.8 Local Stochastic Volatility (LSV) Models 117\u003c\/p\u003e \u003cp\u003e6.8.1 Calibration of local volatility in LSV models 118\u003c\/p\u003e \u003cp\u003e6.8.2 Fokker–Planck equation for the LSV model 119\u003c\/p\u003e \u003cp\u003e6.8.3 Forward induction for local volatility calibration on LSV 120\u003c\/p\u003e \u003cp\u003e6.8.4 Calibrating stochastic and local volatilities 124\u003c\/p\u003e \u003cp\u003e6.8.5 The pricing PDE for LSV models 127\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Numerical Methods for Pricing and Calibration 129\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 One-Dimensional Root Finding – Implied Volatility Calculation 129\u003c\/p\u003e \u003cp\u003e7.2 Nonlinear Least Squares Minimisation 130\u003c\/p\u003e \u003cp\u003e7.3 Monte Carlo Simulation 131\u003c\/p\u003e \u003cp\u003e7.3.1 Handling large timesteps with local volatility 134\u003c\/p\u003e \u003cp\u003e7.3.2 Monte Carlo convergence goes as 1\/√N 135\u003c\/p\u003e \u003cp\u003e7.3.3 Finding a balance between simulations and timesteps 138\u003c\/p\u003e \u003cp\u003e7.3.4 Quasi Monte Carlo convergence can be as good as 1\/N 142\u003c\/p\u003e \u003cp\u003e7.3.5 Variance reduction 143\u003c\/p\u003e \u003cp\u003e7.4 Convection–Diffusion PDEs in Finance 147\u003c\/p\u003e \u003cp\u003e7.4.1 Visualising diffusion 149\u003c\/p\u003e \u003cp\u003e7.4.2 Visualising convection 151\u003c\/p\u003e \u003cp\u003e7.5 Numerical Methods for PDEs 153\u003c\/p\u003e \u003cp\u003e7.6 Explicit Finite Difference Scheme 155\u003c\/p\u003e \u003cp\u003e7.6.1 Boundary conditions 157\u003c\/p\u003e \u003cp\u003e7.6.2 Von Neumann stability and the dimensionless heat equation 159\u003c\/p\u003e \u003cp\u003e7.7 Explicit Finite Difference on Nonuniform Meshes 163\u003c\/p\u003e \u003cp\u003e7.7.1 Mixed partial derivative terms on nonuniform meshes 165\u003c\/p\u003e \u003cp\u003e7.8 Implicit Finite Difference Scheme 165\u003c\/p\u003e \u003cp\u003e7.9 The Crank–Nicolson Scheme 167\u003c\/p\u003e \u003cp\u003e7.10 Numerical Schemes for Multidimensional PDEs 168\u003c\/p\u003e \u003cp\u003e7.10.1 Two-dimensional Crank–Nicolson scheme 169\u003c\/p\u003e \u003cp\u003e7.10.2 An early ADI scheme – Peaceman–Rachford splitting 169\u003c\/p\u003e \u003cp\u003e7.10.3 Douglas–Rachford splitting 171\u003c\/p\u003e \u003cp\u003e7.10.4 Craig–Sneyd splitting 172\u003c\/p\u003e \u003cp\u003e7.11 Practical Nonuniform Grid Generation Schemes 173\u003c\/p\u003e \u003cp\u003e7.11.1 Uniform grid generation 173\u003c\/p\u003e \u003cp\u003e7.11.2 Uniform grid generation with required levels 173\u003c\/p\u003e \u003cp\u003e7.11.3 Spatial grid generation 174\u003c\/p\u003e \u003cp\u003e7.11.4 Temporal grid generation 175\u003c\/p\u003e \u003cp\u003e7.12 Further Reading 176\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 First Generation Exotics – Binary and Barrier Options 177\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 The Reflection Principle 179\u003c\/p\u003e \u003cp\u003e8.2 European Barriers and Binaries 180\u003c\/p\u003e \u003cp\u003e8.2.1 European barriers 180\u003c\/p\u003e \u003cp\u003e8.2.2 Barrier parity relationships 182\u003c\/p\u003e \u003cp\u003e8.2.3 European digitals 183\u003c\/p\u003e \u003cp\u003e8.3 Continuously Monitored Binaries and Barriers 183\u003c\/p\u003e \u003cp\u003e8.3.1 Domestic binaries 188\u003c\/p\u003e \u003cp\u003e8.3.2 Foreign binaries 189\u003c\/p\u003e \u003cp\u003e8.3.3 Instant one-touch products 190\u003c\/p\u003e \u003cp\u003e8.3.4 Barrier products 191\u003c\/p\u003e \u003cp\u003e8.3.5 KIKOs and ONTOs 194\u003c\/p\u003e \u003cp\u003e8.4 Double Barrier Products 194\u003c\/p\u003e \u003cp\u003e8.5 Sensitivity to Local and Stochastic Volatility 195\u003c\/p\u003e \u003cp\u003e8.6 Barrier Bending 197\u003c\/p\u003e \u003cp\u003e8.7 Value Monitoring 202\u003c\/p\u003e \u003cp\u003e8.7.1 Compounds 202\u003c\/p\u003e \u003cp\u003e8.7.2 Americans 203\u003c\/p\u003e \u003cp\u003e8.7.3 Bermudans 203\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Second Generation Exotics 205\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Chooser Options 206\u003c\/p\u003e \u003cp\u003e9.2 Range Accrual Options 206\u003c\/p\u003e \u003cp\u003e9.3 Forward Start Options 207\u003c\/p\u003e \u003cp\u003e9.3.1 Strike reset options 209\u003c\/p\u003e \u003cp\u003e9.4 Lookback Options 209\u003c\/p\u003e \u003cp\u003e9.4.1 Double lookback options 211\u003c\/p\u003e \u003cp\u003e9.5 Asian Options 212\u003c\/p\u003e \u003cp\u003e9.5.1 Notes on seasoned Asians and fixing at expiry 214\u003c\/p\u003e \u003cp\u003e9.6 Target Redemption Notes 214\u003c\/p\u003e \u003cp\u003e9.7 Volatility and Variance Swaps 214\u003c\/p\u003e \u003cp\u003e9.7.1 Volatility observation 215\u003c\/p\u003e \u003cp\u003e9.7.2 Product specification and value at expiry 216\u003c\/p\u003e \u003cp\u003e9.7.3 Variance swap product valuation 217\u003c\/p\u003e \u003cp\u003e9.7.4 Volatility swap product valuation 219\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Multicurrency Options 225\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Correlations, Triangulation and Absence of Arbitrage 226\u003c\/p\u003e \u003cp\u003e10.2 Exchange Options 229\u003c\/p\u003e \u003cp\u003e10.3 Quantos 229\u003c\/p\u003e \u003cp\u003e10.3.1 Self-quanto option 230\u003c\/p\u003e \u003cp\u003e10.3.2 Self-quanto forward 231\u003c\/p\u003e \u003cp\u003e10.3.3 General quanto options 231\u003c\/p\u003e \u003cp\u003e10.4 Best-ofs and Worst-ofs 233\u003c\/p\u003e \u003cp\u003e10.4.1 Two-asset best-of call 234\u003c\/p\u003e \u003cp\u003e10.4.2 Three-asset best-of call 236\u003c\/p\u003e \u003cp\u003e10.4.3 N-asset best-of call 239\u003c\/p\u003e \u003cp\u003e10.5 Basket Options 239\u003c\/p\u003e \u003cp\u003e10.6 Numerical Methods 241\u003c\/p\u003e \u003cp\u003e10.7 A Note on Multicurrency Greeks 242\u003c\/p\u003e \u003cp\u003e10.8 Quantoing Untradeable Factors 243\u003c\/p\u003e \u003cp\u003e10.9 Further Reading 244\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Longdated FX 245\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Currency Swaps 245\u003c\/p\u003e \u003cp\u003e11.2 Basis Risk 247\u003c\/p\u003e \u003cp\u003e11.3 Forward Measure 249\u003c\/p\u003e \u003cp\u003e11.4 LIBOR in Arrears 250\u003c\/p\u003e \u003cp\u003e11.5 Typical Longdated FX Products 253\u003c\/p\u003e \u003cp\u003e11.5.1 Power reverse dual currency notes 253\u003c\/p\u003e \u003cp\u003e11.5.2 FX target redemption notes 254\u003c\/p\u003e \u003cp\u003e11.5.3 Effect on USDJPY volatility smile 255\u003c\/p\u003e \u003cp\u003e11.6 The Three-Factor Model 255\u003c\/p\u003e \u003cp\u003e11.7 Interest Rate Calibration of the Three-Factor Model 257\u003c\/p\u003e \u003cp\u003e11.7.1 Determination of drifts 257\u003c\/p\u003e \u003cp\u003e11.7.2 Determination of Hull–White volatilities 258\u003c\/p\u003e \u003cp\u003e11.8 Spot FX Calibration of the Three-Factor Model 259\u003c\/p\u003e \u003cp\u003e11.8.1 FX vanillas with lognormal spot FX 260\u003c\/p\u003e \u003cp\u003e11.8.2 FX vanillas with CEV local volatility 261\u003c\/p\u003e \u003cp\u003e11.8.3 FX vanillas with Dupire local volatility 262\u003c\/p\u003e \u003cp\u003e11.9 Conclusion 264\u003c\/p\u003e \u003cp\u003eReferences 265\u003c\/p\u003e \u003cp\u003eFurther Reading 271\u003c\/p\u003e \u003cp\u003eIndex 273\u003c\/p\u003e \u003cb\u003eDr Iain J. Clark\u003c\/b\u003e, (London, UK), is Head of Foreign Exchange Quantitative Analysis at Dresdner Kleinwort in London, where he set up and runs the team responsible for developing pricing libraries for the front office. Previously, he was Director of the Quantitative Research Group in Lehman Brothers, Fixed Income Quantitative Analyst at BNP Paribas and has also worked in FX Commodities Derivatives research at JP Morgan. He holds an MSc in Mathematics from the University of Edinburgh, and a PhD in Applied Mathematics from the University of Queensland, Australia. Dr Clark is a regular speaker at key finance events, and has presented at London Imperial College, The Bachelier Society Annual Conference, London Imperial College, world business Strategies annual Conference, Risk events, Marcus Evans events and many more.","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989231321317,"sku":"NP9780470683682","price":131.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9780470683682.jpg?v=1761783303","url":"https:\/\/k12savings.com\/es\/products\/foreign-exchange-option-pricing-isbn-9780470683682","provider":"K12savings","version":"1.0","type":"link"}