Part I Introduction to Option Pricing 1 Asset Pricing Basics... 3 1.1 Fundamental Concepts.................................. 3 1.2 State Prices in a One-Period Binomial Model.............. 11 1.3 Probabilities and Numeraires............................. 14 1.4 Asset Pricing with a Continuum of States................. 17 1.5 Introduction to Option Pricing........................... 21 1.6 An Incomplete Markets Example......................... 24 Problems... 25 2 Continuous-Time Models... 27 2.1 Simulating a Brownian Motion........................... 28 2.2 Quadratic Variation..................................... 29 2.3 ItôProcesses... 31 2.4 Itô sformula... 33 2.5 Multiple Itô Processes... 35 2.6 Examples of Itô s Formula... 37 2.7 Reinvesting Dividends................................... 38 2.8 Geometric Brownian Motion............................. 39 2.9 Numeraires and Probabilities............................. 41 2.10 Tail Probabilities of Geometric Brownian Motions.......... 44 2.11 Volatilities............................................. 46 Problems... 48 3 Black-Scholes... 49 3.1 Digital Options......................................... 49 3.2 Share Digitals.......................................... 51 3.3 Puts and Calls......................................... 52 3.4 Greeks... 53 3.5 DeltaHedging... 55
XII 3.6 Gamma Hedging....................................... 57 3.7 Implied Volatilities..................................... 58 3.8 Term Structure of Volatility.............................. 58 3.9 Smiles and Smirks...................................... 60 3.10 Calculations in VBA.................................... 60 Problems... 67 4 Estimating and Modelling Volatility... 71 4.1 Statistics Review....................................... 71 4.2 Estimating a Constant Volatility and Mean................ 73 4.3 Estimating a Changing Volatility......................... 75 4.4 GARCHModels... 77 4.5 Stochastic Volatility Models.............................. 79 4.6 Smiles and Smirks Again................................ 82 4.7 Hedging and Market Completeness........................ 83 Problems... 84 5 Introduction to Monte Carlo and Binomial Models... 87 5.1 Introduction to Monte Carlo............................. 87 5.2 Introduction to Binomial Models......................... 89 5.3 Binomial Models for American Options.................... 91 5.4 Binomial Parameters.................................... 92 5.5 Binomial Greeks........................................ 94 5.6 Monte Carlo Greeks I: Difference Ratios................... 96 5.7 Monte Carlo Greeks II: Pathwise Estimates................ 98 5.8 Calculations in VBA.................................... 101 Problems...107 Part II Advanced Option Pricing 6 Foreign Exchange...111 6.1 Currency Options...................................... 112 6.2 Options on Foreign Assets Struck in Foreign Currency....... 112 6.3 Options on Foreign Assets Struck in Domestic Currency..... 112 6.4 Currency Forwards and Futures.......................... 113 6.5 Quantos...116 6.6 Replicating Quantos.................................... 118 6.7 QuantoForwards...121 6.8 Quanto Options........................................ 121 6.9 ReturnSwaps...123 6.10 Uncovered Interest Parity................................ 125 Problems...125
XIII 7 Forward, Futures, and Exchange Options...129 7.1 Margrabe sformula...130 7.2 Black sformula...132 7.3 Merton sformula...136 7.4 Deferred Exchange Options.............................. 139 7.5 Calculations in VBA.................................... 140 7.6 GreeksandHedging...142 7.7 The Relation of Futures Prices to Forward Prices........... 144 7.8 Futures Options........................................ 145 7.9 Time-Varying Volatility................................. 147 7.10 Hedging with Forwards and Futures....................... 147 7.11 Market Completeness................................... 150 Problems...152 8 Exotic Options...155 8.1 Forward-Start Options.................................. 155 8.2 Compound Options..................................... 158 8.3 American Calls with Discrete Dividends................... 162 8.4 Choosers...166 8.5 Options on the Max or Min.............................. 168 8.6 Barrier Options........................................ 171 8.7 Lookbacks...175 8.8 Basket and Spread Options.............................. 176 8.9 Asian Options.......................................... 178 8.10 Calculations in VBA.................................... 183 Problems...193 9 More on Monte Carlo and Binomial Valuation...197 9.1 Monte Carlo Models for Path-Dependent Options........... 197 9.2 Binomial Valuation of Basket and Spread Options.......... 198 9.3 Monte Carlo Valuation of Basket and Spread Options....... 200 9.4 Antithetic Variates in Monte Carlo....................... 202 9.5 Control Variates in Monte Carlo.......................... 203 9.6 Accelerating Binomial Convergence....................... 205 9.7 Calculations in VBA.................................... 207 Problems...216 10 Finite Difference Methods...219 10.1 Fundamental PDE...................................... 219 10.2 Discretizing the PDE................................... 221 10.3 Explicit and Implicit Methods............................ 222 10.4 Crank-Nicolson......................................... 225 10.5 European Options...................................... 226 10.6 American Options...................................... 227 10.7 Barrier Options........................................ 227
XIV 10.8 Calculations in VBA.................................... 227 Problems...233 Part III Fixed Income 11 Fixed Income Concepts...237 11.1 The Yield Curve....................................... 237 11.2 LIBOR................................................ 239 11.3 Swaps................................................. 240 11.4 Yield to Maturity, Duration, and Convexity................ 242 11.5 Principal Components................................... 245 11.6 Hedging Principal Components........................... 248 Problems...249 12 Introduction to Fixed Income Derivatives...253 12.1 Caps and Floors........................................ 253 12.2 Forward Rates......................................... 254 12.3 Portfolios that Pay Spot Rates........................... 254 12.4 The Market Model for Caps and Floors................... 255 12.5 The Market Model for European Swaptions................ 257 12.6 A Comment on Consistency.............................. 259 12.7 Caplets as Puts on Discount Bonds....................... 260 12.8 Swaptions as Options on Coupon Bonds................... 260 12.9 Calculations in VBA.................................... 261 Problems...262 13 Valuing Derivatives in the Extended Vasicek Model...265 13.1 The Short Rate and Discount Bond Prices................. 265 13.2 The Vasicek Model..................................... 266 13.3 Estimating the Vasicek Model............................ 269 13.4 Hedging in the Vasicek Model............................ 271 13.5 Extensions of the Vasicek Model.......................... 273 13.6 Fitting Discount Bond Prices and Forward Rates........... 275 13.7 Discount Bond Options, Caps and Floors.................. 277 13.8 Coupon Bond Options and Swaptions..................... 280 13.9 Captions and Floortions................................. 283 13.10 Yields and Yield Volatilities.............................. 284 13.11 The General Hull-White Model........................... 285 13.12 Calculations in VBA.................................... 289 Problems...293
XV 14 A Brief Survey of Term Structure Models...295 14.1 Ho-Lee................................................ 295 14.2 Black-Derman-Toy...................................... 300 14.3 Black-Karasinski....................................... 302 14.4 Cox-Ingersoll-Ross...................................... 302 14.5 Longstaff-Schwartz..................................... 307 14.6 Heath-Jarrow-Morton................................... 310 14.7 Market Models Again................................... 312 Problems...316 Appendices A Programming in VBA...319 A.1 VBA Editor and Modules................................ 319 A.2 Subroutines and Functions............................... 320 A.3 Message Box and Input Box............................. 321 A.4 Writing to and Reading from Cells........................ 322 A.5 VariablesandAssignments...323 A.6 Mathematical Operations................................ 324 A.7 RandomNumbers...324 A.8 ForLoops...325 A.9 While Loops and Logical Expressions..................... 326 A.10 If, Else, and ElseIf Statements............................ 326 A.11 Variable Declarations................................... 327 A.12 VariablePassing...328 A.13 Arrays...329 A.14 Debugging............................................. 331 B Miscellaneous Facts about Continuous-Time Models...333 B.1 Girsanov stheorem...333 B.2 The Minimum of a Geometric Brownian Motion............ 336 B.3 BesselSquaredProcessesandtheCIRModel...340 List of Programs...346 List of Symbols...347 References...349 Index...353
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