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Mathematical Finance: Theory, Modeling, Implementation

520 pages, 94 figures, 9 tables.

Christian P. Fries

Version 1.5.12 = First Edition, First Printing.
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Das Buch entstand aus dem Vorlesungsskript meiner Vorlesungen an der Universität Mainz und Universität Frankfurt. Es versucht eine ausbalancierte Darstellung der theoretischen Grundlagen, der State-of-the-Art Modelle, die derzeit in der Praxis Verwendung finden, und ihrer Implementierung.

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Contents

1 Introduction 1
1.1 Theory, Modeling, and Implementation 1
1.2 Interest Rate Models and Interest Rate Derivatives 1
1.3 About This Book 3
1.3.1 How to Read This Book 3
1.3.2 Abridged Versions 3
1.3.2.1 Abridged version "Monte Carlo Pricing" 3
1.3.2.2 Abridged version "LIBOR Market Model" 3
1.3.2.3 Abridged version "Markov Functional Model" 3
1.3.3 Special Sections 4
1.3.4 Notation 4
1.3.5 Feedback 5
1.3.6 Resources 5
I Foundations 7
2 Foundations 9
2.1 Probability Theory 9
2.2 Stochastic Processes 18
2.3 Filtration 20
2.4 Brownian Motion 22
2.5 Wiener Measure, Canonical Setup 24
2.6 Ito Calculus 25
2.6.1 Ito Integral 28
2.6.2 Ito Process 30
2.6.3 Ito Lemma and Product Rule 32
2.7 Brownian Motion with Instantaneous Correlation 36
2.8 Martingales 38
2.8.1 Martingale Representation Theorem 38
2.9 Change of Measure 39
2.10 Stochastic Integration 44
2.11 Partial Differential Equations (PDEs) 46
2.11.1 Feynman-Kac Theorem 46
2.12 List of Symbols 48
3 Replication 49
3.1 Replication Strategies 49
3.1.1 Introduction 49
3.1.2 Replication in a Discrete Model 53
3.1.2.1 Example: Two Times, Two States, Two Assets 53
3.2 Foundations: Equivalent Martingale Measure 58
3.2.1 Challenge and Solution Outline 58
3.2.2 Steps toward the Universal Pricing Theorem 61
3.2.2.1 Self-Financing Trading Strategy 62
3.2.2.2 Relative Prices 63
3.2.2.3 Equivalent Martingale Measure 66
3.2.2.4 Payoff Replication 67
3.3 Excursus: Relative Prices and Risk-Neutral Measures 70
3.3.1 Why relative prices? 70
3.3.2 Risk-Neutral Measure 72
II First Applications 73
4 Pricing of a European Stock Option under the Black-Scholes Model 75
5 Excursus: The Density of the Underlying of a European Call Option 81
6 Excursus: Interpolation of European Option Prices 83
6.1 No-Arbitrage Conditions for Interpolated Prices 83
6.2 Arbitrage Violations through Interpolation 85
6.2.1 Example 1: Interpolation of Four Prices 85
6.2.1.1 Linear Interpolation of Prices 85
6.2.1.2 Linear Interpolation of Implied Volatilities 86
6.2.1.3 Spline Interpolation of Prices and Implied Volatilities 87
6.2.2 Example 2: Interpolation of Two Prices 87
6.2.2.1 Linear Interpolation for Decreasing Implied Volatilities 88
Conclusion: 88
6.2.2.2 Linear Interpolation for Increasing Implied Volatilities 88
Conclusion: 89
6.3 Arbitrage-Free Interpolation of European Option Prices 89
7 Hedging in Continuous and Discrete Time and the Greeks 93
7.1 Introduction 93
7.2 Deriving the Replications Strategy from Pricing Theory 94
Conclusion: 95
7.2.1 Deriving the Replication Strategy under the Assumption of a Locally Riskless Product 96
7.2.2 Black-Scholes Differential Equation 97
7.2.3 Derivative V(t) as a Function of Its Underlyings Si(t) 97
7.2.3.1 Path-Dependent Options 98
7.2.4 Example: Replication Portfolio and PDE under a Black-Scholes Model 99
7.2.4.1 Interpretation of V as a Function in (t,S) 101
7.3 Greeks 102
7.3.1 Greeks of a European Call-Option under the Black-Scholes Model 103
7.4 Hedging in Discrete Time: Delta and Delta-Gamma Hedging 103
7.4.1 Delta Hedging 105
7.4.2 Error Propagation 106
7.4.2.1 Example: Time-Discrete Delta Hedge under a Black-Scholes Model 107
7.4.3 Delta-Gamma Hedging 109
7.4.3.1 Example: Time-Discrete Delta-Gamma Hedge under a Black-Scholes Model 111
7.4.4 Vega Hedging 113
7.5 Hedging in Discrete Time: Minimizing the Residual Error (Bouchaud-Sornette Method) 113
7.5.1 Minimizing the Residual Error at Maturity T 115
7.5.2 Minimizing the Residual Error in Each Time Step 117
III Interest Rate Structures, Interest Rate Products, and Analytic Pricing Formulas 119
Motivation and Overview 121
8 Interest Rate Structures 123
8.1 Introduction 123
8.1.1 Fixing Times and Tenor Times 124
8.2 Definitions 124
8.3 Interest Rate Curve Bootstrapping 130
Induction Start (T0): 130
Induction Step (Ti-1 to Ti): 130
8.4 Interpolation of Interest Rate Curves 130
8.5 Implementation 131
9 Simple Interest Rate Products 133
9.1 Interest Rate Products Part 1: Products without Optionality 133
9.1.1 Fix, Floating, and Swap 133
9.1.2 Money Market Account 140
9.2 Interest Rate Products Part 2: Simple Options 142
9.2.1 Cap, Floor, and Swaption 142
9.2.1.1 Example: Option on a Coupon Bond 144
9.2.2 Foreign Caplet, Quanto 144
10 The Black Model for a Caplet 147
11 Pricing of a Quanto Caplet (Modeling the FFX) 151
11.1 Choice of Numeraire 151
12 Exotic Derivatives 155
12.1 Prototypical Product Properties 155
12.2 Interest Rate Products Part 3: Exotic Interest Rate Derivatives 157
12.2.1 Structured Bond, Structured Swap, and Zero Structure 158
12.2.2 Bermudan Option 162
12.2.3 Bermudan Callable and Bermudan Cancelable 164
12.2.4 Compound Options 166
12.2.5 Trigger Products 167
12.2.5.1 Target Redemption Note 167
12.2.6 Structured Coupons 168
12.2.6.1 Capped, Floored, Inverse, Spread, CMS 169
12.2.6.2 Range Accruals 170
12.2.6.3 Path-Dependent Coupons 170
Example: 171
12.2.6.4 Flexi-Cap 171
12.2.7 Shout Options 173
12.3 Product Toolbox 174
IV Discretization and Numerical Valuation Methods 177
Motivation and Overview 179
13 Discretization of Time and State Space 181
13.1 Discretization of Time: The Euler and the Milstein Schemes 181
13.1.1 Definitions 183
13.1.2 Time Discretization of a Lognormal Process 185
13.1.2.1 Discretization via Euler Scheme 185
13.1.2.2 Discretization via Milstein scheme 185
13.1.2.3 Discretization of the Log Process 186
13.1.2.4 Exact Discretization 186
13.2 Discretization of Paths (Monte Carlo Simulation) 186
13.2.1 Monte Carlo Simulation 187
13.2.2 Weighted Monte Carlo Simulation 187
13.2.3 Implementation 188
13.2.3.1 Example: Valuation of a Stock Option under the Black-Scholes Model Using Monte Carlo Simulation 188
13.2.3.2 Separation of Product and Model 189
13.2.3.3 Model-Product Communication Protocol 191
Core Object: Random Variable 192
Aggregation 1: Vector of Random Variables of Same Simulation Time 192
Aggregation 2: Time-Discrete Stochastic Process 192
Storage, Access, and Processing 193
13.2.4 Review 193
13.3 Discretization of State Space 195
13.3.1 Definitions 195
13.3.2 Backward Algorithm 197
13.3.3 Review 197
13.3.3.1 Path Dependencies 197
13.3.3.2 Course of Dimension 198
13.4 Path Simulation through a Lattice: Two Layers 198
14 Numerical Methods for Partial Differential Equations 199
15 Pricing Bermudan Options in a Monte Carlo Simulation 201
15.1 Introduction 201
15.2 Bermudan Options: Notation 202
15.2.1 Bermudan Callable 203
15.2.2 Relative Prices 203
15.3 Bermudan Option as Optimal Exercise Problem 204
15.3.1 Bermudan Option Value as Single (Unconditioned) Expectation: The Optimal Exercise Value 204
15.4 Bermudan Option Pricing---The Backward Algorithm 205
Induction start 205
Induction step 206
15.5 Resimulation 207
15.6 Perfect Foresight 207
15.7 Conditional Expectation as Functional Dependence 209
Example: 210
15.8 Binning 210
15.8.1 Binning as a Least-Square Regression 212
15.9 Foresight Bias 214
15.10 Regression Methods---Least-Square Monte Carlo 215
15.10.1 Least-Square Approximation of the Conditional Expectation 215
15.10.2 Example: Evaluation of a Bermudan Option on a Stock (Backward Algorithm with Conditional Expectation Estimator) 216
Induction Start 216
Induction Step 217
15.10.3 Example: Evaluation of a Bermudan Callable 217
Induction Start 217
Induction Step 221
15.10.4 Implementation 222
15.10.5 Binning as Linear Least-Square Regression 223
15.11 Optimization Methods 224
15.11.1 Andersen Algorithm for Bermudan Swaptions 224
Induction Start 225
Induction Step 225
15.11.2 Review of the Threshold Optimization Method 225
15.11.2.1 Fitting the Exercise Strategy to the Product 225
15.11.2.2 Disturbance of the Optimizer through Discontinuities and Local Minima 227
15.11.3 Optimization of Exercise Strategy: A More General Formulation 228
15.11.4 Comparison of Optimization Method and Regression Method 228
15.12 Duality Method: Upper Bound for Bermudan Option Prices 230
15.12.1 Foundations 230
15.12.2 American Option Evaluation as Optimal Stopping Problem 232
15.13 Primal-Dual Method: Upper and Lower Bound 235
16 Pricing Path-Dependent Options in a Backward Algorithm 237
16.1 State Space Extension 237
16.2 Implementation 238
16.3 Path-Dependent Bermudan Options 239
16.4 Examples 240
16.4.1 Evaluation of a Snowball in a Backward Algorithm 240
16.4.2 Evaluation of a Autocap in a Backward Algorithm 240
17 Sensitivities (Partial Derivatives) of Monte Carlo Prices 243
17.1 Introduction 243
17.2 Problem Description 244
17.2.1 Pricing using Monte-Carlo Simulation 244
17.2.2 Sensitivities from Monte Carlo Pricing 245
17.2.3 Example: The Linear and the Discontinuous Payout 245
17.2.3.1 Linear Payout 246
17.2.3.2 Discontinuous Payout 246
17.2.4 Example: Trigger Products 247
17.3 Generic Sensitivities: Bumping the Model 249
17.4 Sensitivities by Finite Differences 251
17.4.1 Example: Finite Differences Applied to Smooth and Discontinuous Payout 252
Simplified Example: 253
17.5 Sensitivities by Pathwise Differentiation 254
17.5.1 Example: Delta of a European Option under a Black-Scholes Model 254
17.5.2 Pathwise Differentiation for Discontinuous Payouts 255
17.6 Sensitivities by Likelihood Ratio Weighting 256
17.6.1 Example: Delta of a European Option under a Black-Scholes Model Using Pathwise Derivative 257
17.6.2 Example: Variance Increase of the Sensitivity when using Likelihood Ratio Method for Smooth Payouts 257
17.7 Sensitivities by Malliavin Weighting 258
17.8 Proxy Simulation Scheme 259
18 Proxy Simulation Schemes for Monte Carlo Sensitivities and Importance Sampling 261
18.1 Full Proxy Simulation Scheme 261
18.1.1 Pricing under a Proxy Simulation Scheme 262
18.1.1.1 Basic Properties of a Proxy Simulation Scheme 262
18.1.2 Calculation of Monte Carlo Weights 262
18.1.3 Sensitivities by Finite Differences on a Proxy Simulation Scheme 263
18.1.4 Localization 264
18.1.5 Object-Oriented Design 265
18.1.6 Importance Sampling 265
18.1.6.1 Example 265
18.2 Partial Proxy Simulation Schemes 268
18.2.1 Linear Proxy Constraint 268
18.2.2 Comparison to Full Proxy Scheme Method 269
18.2.3 Nonlinear Proxy Constraint 269
18.2.3.1 Linearization of the Proxy Constraint 270
18.2.3.2 Finite Difference Approximation of the Nonlinear Proxy Constraint 270
Example: 271
18.2.4 Transition Probability from a Nonlinear Proxy Constraint 271
18.2.4.1 The Proxy Constraint Revisited 271
18.2.4.2 Transition Probabilities for General Proxy Constraints 272
18.2.4.3 Example 273
18.2.4.4 Approximating an Fti+1-measurable Proxy Constraint by an Fti-measurable Proxy Constraint 274
18.2.5 Sensitivity with Respect to the Diffusion Coefficients---Vega 274
18.2.6 Example: LIBOR Target Redemption Note 274
18.2.7 Example: CMS Target Redemption Note 276
18.2.7.1 Delta and Gamma of a CMS TARN 277
18.2.7.2 Vega of a CMS TARN 278
18.3 Localized Proxy Simulation Schemes 279
18.3.1 Problem Description 279
18.3.2 Solution 282
18.3.3 Partial Proxy Simulation Scheme (revisited) 282
18.3.3.1 Reference Scheme and Target Scheme 282
18.3.3.2 Transition Probabilities 283
18.3.3.3 Proxy Constraint and Proxy Scheme 283
18.3.3.4 Calculating Expectations using a Proxy Simulation Scheme 283
18.3.3.5 Example: Euler Schemes 284
18.3.4 Localized Proxy Simulation Scheme 285
18.3.5 Example: Euler Schemes 286
18.3.6 Implementation 286
18.3.7 Examples and Numerical Results 287
18.3.7.1 Localizers 287
18.3.7.2 Model 287
18.3.7.3 Example: Digital Caplet 287
18.3.7.4 Example: Target Redemption Note (TARN) 288
18.3.7.5 Proxy Constraint and Localizer for the Target Redemption Note 291
V Pricing Models for Interest Rate Derivatives 293
Motivation and Overview 295
19 LIBOR Market Model 297
19.1 Derivation of the Drift Term 299
19.1.1 Derivation of the Drift Term under the Terminal Measure 299
19.1.2 Derivation of the Drift Term under the Spot LIBOR Measure 301
19.1.3 Derivation of the Drift Term under the Tk-Forward Measure 303
19.2 The Short Period Bond P(Tm(t)+1;t) 304
19.2.1 Role of the Short Bond in a LIBOR Market Model 304
19.2.2 Link to Continuous Time Tenors 304
19.2.3 Drift of the Short Bond in a LIBOR Market Model 304
19.3 Discretization and (Monte Carlo) Simulation 305
19.3.1 Generation of the (Time-Discrete) Forward Rate Process 305
19.3.2 Generation of the Sample Paths 306
19.3.3 Generation of the Numeraire 306
19.4 Calibration---Choice of the Free Parameters 307
19.4.1 Choice of the Initial Conditions 308
19.4.1.1 Reproduction of Bond Market Prices 308
19.4.2 Choice of the Volatilities 308
19.4.2.1 Reproduction of Caplet Market Prices 308
19.4.2.2 Reproduction of Swaption Market Prices 309
19.4.2.3 Functional Forms 311
19.4.3 Choice of the Correlations 311
19.4.3.1 Factors 311
19.4.3.2 Functional Forms 312
19.4.3.3 Factor Reduction 312
19.4.3.4 Calibration 313
19.4.4 Covariance Structure 313
19.4.5 Analytic Evaluation of Caplets, Swaptions and Swap Rate Covariance 314
19.4.5.1 Analytic Evaluation of a Caplet in the LIBOR Market Model 314
19.4.5.2 Analytic Evaluation of a Swaption in the LIBOR Market Model 314
19.4.5.3 Analytic Calculation of Swap Rate Covariance in the LIBOR Market Model 318
19.5 Interpolation of Forward Rates in the LIBOR Market Model 319
19.5.1 Interpolation of the Tenor Structure {Ti&rculb; 319
19.5.1.1 Assumption 1: No Stochastic Shortly Before Maturity. 320
19.5.1.2 Assumption 2: Linearity Shortly Before Maturity. 321
19.6 Object-Oriented Design 323
19.6.1 Reuse of Implementation 324
19.6.2 Separation of Product and Model 324
19.6.3 Abstraction of Model Parameters 324
19.6.4 Abstraction of Calibration 325
20 Swap Rate Market Models 329
20.1 The Swap Measure 330
20.2 Derivation of the Drift Term 331
20.3 Calibration---Choice of the Free Parameters 332
20.3.1 Choice of the Initial Conditions 332
20.3.1.1 Reproduction of Bond Market Prices or Swap Market Prices 332
20.3.2 Choice of the Volatilities 332
20.3.2.1 Reproduction of Swaption Market Prices 332
21 Excursus: Instantaneous Correlation and Terminal Correlation 335
21.1 Definitions 335
21.2 Terminal Correlation Examined in a LIBOR Market Model Example 336
21.2.1 Decorrelation in a One-Factor Model 337
21.2.2 Impact of the Time Structure of the Instantaneous Volatility on Caplet and Swaption Prices 339
21.2.3 Swaption Value as a Function of Forward Rates 340
21.3 Terminal Correlation Is Dependent on the Equivalent Martingale Measure 342
21.3.1 Dependence of the Terminal Density on the Martingale Measure 342
22 Heath-Jarrow-Morton Framework: Foundations 345
22.1 Short-Rate Process in the HJM Framework 346
22.2 The HJM Drift Condition 347
23 Short-Rate Models 351
23.1 Introduction 351
23.2 The Market Price of Risk 352
23.3 Overview: Some Common Models 354
23.4 Implementations 355
23.4.1 Monte Carlo Implementation of Short-Rate Models 355
23.4.2 Lattice Implementation of Short-Rate Models 355
24 Heath-Jarrow-Morton Framework: Immersion of Short-Rate Models and LIBOR Market Model 357
24.1 Short-Rate Models in the HJM Framework 357
24.1.1 Example: The Ho-Lee Model in the HJM Framework 358
24.1.2 Example: The Hull-White Model in the HJM Framework 359
24.2 LIBOR Market Model in the HJM Framework 360
24.2.1 HJM Volatility Structure of the LIBOR Market Model 360
24.2.2 LIBOR Market Model Drift under the QB Measure 362
24.2.3 LIBOR Market Model as a Short Rate Model 364
25 Excursus: Shape of the Interest Rate Curve under Mean Reversion and a Multifactor Model 365
25.1 Model 365
25.2 Interpretation of the Figures 366
25.3 Mean Reversion 367
25.4 Factors 368
25.5 Exponential Volatility Function 369
25.6 Instantaneous Correlation 371
26 Ritchken-Sakarasubramanian Framework: HJM with Low Markov Dimension 373
26.1 Introduction 373
26.2 Cheyette Model 374
26.3 Implementation: PDE 375
27 Markov Functional Models 377
27.1 Introduction 377
27.1.1 The Markov Functional Assumption (Independent of the Model Considered) 378
27.1.2 Outline of This Chapter 379
27.2 Equity Markov Functional Model 379
27.2.1 Markov Functional Assumption 379
27.2.2 Example: The Black-Scholes Model 380
27.2.3 Numerical Calibration to a Full Two-Dimensional European Option Smile Surface 381
27.2.3.1 Market Price 382
27.2.3.2 Model Price 382
27.2.3.3 Solving for the Functional 383
27.2.4 Interest Rates 383
27.2.4.1 A Note on Interest Rates and the No-Arbitrage Requirement 383
27.2.4.2 Where Are the Interest Rates? 383
27.2.5 Model Dynamics 384
27.2.5.1 Introduction 384
27.2.5.2 Interest Rate Dynamics 386
27.2.5.3 Forward Volatility 388
27.2.6 Implementation 390
27.3 LIBOR Markov Functional Model 390
27.3.1 LIBOR Markov Functional Model in Terminal Measure 390
27.3.1.1 Evaluation within the LIBOR Markov Functional Model 392
27.3.1.2 Calibration of the LIBOR Functional 393
Induction start 393
Induction step 393
Induction start 396
Induction step 396
27.3.2 LIBOR Markov Functional Model in Spot Measure 396
27.3.2.1 Calibration of the Markov Functional Model under Spot Measure 397
27.3.2.2 Forward Induction Step 397
27.3.2.3 Dealing With the Path Dependency of the Numeraire 398
27.3.2.4 Efficient Calculation of the LIBOR Functional From Given Market Prices 398
27.3.3 Remark on Implementation 400
27.3.3.1 Fast Calculation of Price Functionals 400
27.3.3.2 Discussion on the Implementation of the Markov Functional Model under Terminal and Spot Measure 401
27.3.4 Change of Numeraire in a Markov Functional Model 401
27.4 Implementation: Lattice 403
27.4.1 Convolution with the Normal Probability Density 404
27.4.1.1 Piecewise Constant Approximation 404
27.4.1.2 Piecewise Polynomial Approximation 405
27.4.2 State Space Discretization 407
27.4.2.1 Equidistant Discretization 407
VI Extended Models 409
28 Credit Spreads 411
28.1 Introduction---Different Types of Spreads 411
28.1.1 Spread on a Coupon 411
28.1.2 Credit Spread 411
28.2 Defaultable Bonds 412
28.3 Integrating Deterministic Credit Spread into a Pricing Model 414
28.3.1 Deterministic Credit Spread 415
28.3.2 Implementation 416
28.4 Receiver's and Payer's Credit Spreads 418
28.4.1 Example: Defaultable Forward Starting Coupon Bond 419
28.4.2 Example: Option on a Defaultable Coupon Bond 420
29 Hybrid Models 421
29.1 Cross-Currency LIBOR Market Model 421
29.1.1 Derivation of the Drift Term under Spot Measure 422
29.1.1.1 Dynamic of the Domestic LIBOR under Spot Measure 422
29.1.1.2 Dynamic of the Foreign LIBOR under Spot Measure 423
29.1.1.3 Dynamic of the FX Rate under Spot Measure 425
29.1.2 Implementation 426
29.2 Equity Hybrid LIBOR Market Model 426
29.2.1 Derivation of the Drift Term under Spot-Measure 426
29.2.1.1 Dynamic of the Stock Process under Spot Measure 427
29.2.2 Implementation 428
29.3 Equity Hybrid Cross-Currency LIBOR Market Model 428
29.3.1 Dynamic of the Foreign Stock under Spot Measure 429
29.3.2 Summary 430
29.3.3 Implementation 431
VII Implementation 433
30 Object-Oriented Implementation in Java™ 435
30.1 Elements of Object-Oriented Programming: Class and Objects 435
30.1.1 Example: Class of a Binomial Distributed Random Variable 436
30.1.2 Constructor 438
30.1.3 Methods: Getter, Setter, and Static Methods 438
30.1.3.1 Calling Convention, Signatures 438
30.1.3.2 Getter, Setter 439
30.1.3.3 Static Methods 439
30.2 Principles of Object Oriented Programming 440
30.2.1 Encapsulation and Interfaces 440
30.2.1.1 Encapsulation 441
Example of Encapsulation: Offering Alternative Methods: 442
Example of Encapsulation: Performance Improvement by Adding a Cache to the Internal Data Modell: 442
30.2.1.2 Interfaces 443
30.2.2 Abstraction and Inheritance 444
30.2.3 Polymorphism 447
30.3 Example: A Class Structure for One-Dimensional Root Finders 449
30.3.1 Root Finder for General Functions 449
30.3.1.1 Interface 449
30.3.1.2 Bisection Search 451
30.3.2 Root Finder for Functions with Analytic Derivative: Newton's Method 451
30.3.2.1 Interface 451
30.3.2.2 Newton Method 451
30.3.3 Root Finder for Functions with Derivative Estimation: Secant Method 452
30.3.3.1 Inheritance 452
30.3.3.2 Polymorphism 454
30.4 Anatomy of a Java™ Class 458
30.5 Libraries 460
30.5.1 Java™ 2 Platform, Standard Edition (j2se) 460
30.5.2 Java™ 2 Platform, Enterprise Edition (j2ee) 460
30.5.3 Colt 460
30.5.4 Commons-Math: The Jakarta Mathematics Library 461
30.6 Some Final Remarks 461
30.6.1 Object-Oriented Design (OOD)/Unified Modeling Language (UML) 461
VIII Appendices 463
A A Small Collection of Common Misconceptions 465
B Tools (Selection) 467
B.1 Generation of Random Numbers 467
B.1.1 Uniform Distributed Random Variables 467
B.1.1.1 Mersenne Twister 467
B.1.2 Transformation of the Random Number Distribution via the Inverse Distribution Function 468
B.1.3 Normal Distributed Random Variables 468
B.1.3.1 Inverse Distribution Function 468
B.1.3.2 Box-Muller Transformation 468
B.1.4 Poisson Distributed Random Variables 468
B.1.4.1 Inverse Distribution Function 468
B.1.5 Generation of Paths of an n-Dimensional Brownian Motion 469
B.2 Factor Decomposition---Generation of Correlated Brownian Motion 471
B.3 Factor Reduction 472
B.4 Optimization (One-Dimensional): Golden Section Search 475
B.5 Linear Regression 476
B.6 Convolution with Normal Density 477
C Exercises 479
D Java™ Source Code (Selection) 487
D.1 Java™ Classes for Chapter 30 487
List of Symbols 493
List of Figures 495
List of Tables 499
List of Listings 500
Bibliography 503
Index 511

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