Mathematical Finance: Theory, Modeling, Implementation

Mathematical Finance: Theory, Modeling, Implementation......Page 4
Contents......Page 14
1.2 Interest Rate Models and Interest Rate Derivatives......Page 26
1.3.2 Abridged Versions......Page 28
1.3.4 Notation......Page 29
1.3.6 Resources......Page 30
I Foundations......Page 32
2.1 Probability Theory......Page 34
2.2 Stochastic Processes......Page 43
2.3 Filtration......Page 45
2.4 Brownian Motion......Page 47
2.5 Wiener Measure, Canonical Setup......Page 49
2.6 Itô Calculus......Page 50
2.6.1 Itô Integral......Page 53
2.6.2 Itô Process......Page 55
2.6.3 Itô Lemma and Product Rule......Page 57
2.7 Brownian Motion with Instantaneous Correlation......Page 61
2.8.1 Martingale Representation Theorem......Page 63
2.9 Change of Measure......Page 64
2.10 Stochastic Integration......Page 69
2.11.1 Feynman-Kač Theorem......Page 71
2.12 List of Symbols......Page 73
3.1.1 Introduction......Page 74
3.1.2 Replication in a Discrete Model......Page 78
3.2.1 Challenge and Solution Outline......Page 83
3.2.2 Steps toward the Universal Pricing Theorem......Page 86
3.3.1 Why relative prices?......Page 95
3.3.2 Risk-Neutral Measure......Page 97
II First Applications......Page 98
4 Pricing of a European Stock Option under the Black-Scholes Model......Page 100
5 Excursus: The Density of the Underlying of a European Call Option......Page 106
6.1 No-Arbitrage Conditions for Interpolated Prices......Page 108
6.2.1 Example 1 : Interpolation of Four Prices......Page 110
6.2.2 Example 2: Interpolation of Two Prices......Page 112
6.3 Arbitrage- Free Interpolation of European Option Prices......Page 114
7.1 Introduction......Page 118
7.2 Deriving the Replications Strategy from Pricing Theory......Page 119
7.2.1 Deriving the Replication Strategy under the Assumption of a Locally Riskless Product......Page 121
7.2.3 Derivative V(t) as a Function of Its Underlyings S i(t)......Page 122
7.2.4 Example: Replication Portfolio and PDE under a Black-Scholes Model......Page 124
7.3 Greeks......Page 127
7.4 Hedging in Discrete Time: Delta and Delta-Gamma Hedging......Page 128
7.4.1 Delta Hedging......Page 130
7.4.2 Error Propagation......Page 131
7.4.3 Delta-Gamma Hedging......Page 134
7.5 Hedging in Discrete Time: Minimizing the Residual Error (Bouchaud-Sornette Method)......Page 138
7.5.1 Minimizing the Residual Error at Maturity T......Page 140
7.5.2 Minimizing the Residual Error in Each Time Step......Page 142
III Interest Rate Structures, Interest Rate Products, and Analytic Pricing Formulas......Page 144
Motivation and Overview......Page 146
8.1 Introduction......Page 148
8.2 Definitions......Page 149
8.4 Interpolation of Interest Rate Curves......Page 155
8.5 Implementation......Page 156
9.1.1 Fix, Floating, and Swap......Page 158
9.1.2 Money Market Account......Page 165
9.2.1 Cap, Floor, and Swaption......Page 167
9.2.2 Foreign Caplet, Quanto......Page 169
10 The Black Model for a Caplet......Page 172
11.1 Choice of Numéraire......Page 176
12.1 Prototypical Product Properties......Page 180
12.2 Interest Rate Products Part 3: Exotic Interest Rate Derivatives......Page 182
12.2.1 Structured Bond, Structured Swap, and Zero Structure......Page 183
12.2.2 Bermudan Option......Page 187
12.2.3 Bermudan Callable and Bermudan Cancelable......Page 189
12.2.4 Compound Options......Page 191
12.2.5 Trigger Products......Page 192
12.2.6 Structured Coupons......Page 193
12.2.7 Shout Options......Page 198
12.3 Product Toolbox......Page 199
IV Discretization and Numerical Valuation Methods......Page 202
Motivation and Overview......Page 204
13.1 Discretization of Time: The Euler and the Milstein Schemes......Page 206
13.1.1 Definitions......Page 208
13.1.2 Time Discretization of a Lognormal Process......Page 210
13.2 Discretization of Paths (Monte Carlo Simulation)......Page 211
13.2.2 Weighted Monte Carlo Simulation......Page 212
13.2.3 Implementation......Page 213
13.2.4 Review......Page 218
13.3.1 Definitions......Page 220
13.3.3 Review......Page 222
13.4 Path Simulation through a Lattice: Two Layers......Page 223
14 Numerical Methods for Partial Differential Equations......Page 224
15.1 Introduction......Page 226
15.2 Bermudan Options: Notation......Page 227
15.2.2 Relative Prices......Page 228
15.3.1 Bermudan Option Value as Single (Unconditioned) Expectation: The Optimal Exercise Value......Page 229
15.4 Bermudan Option Pricing—The Backward Algorithm......Page 230
15.6 Perfect Foresight......Page 232
15.7 Conditional Expectation as Functional Dependence......Page 234
15.8 Binning......Page 235
15.8.1 Binning as a Least-Square Regression......Page 237
15.9 Foresight Bias......Page 239
15.10.1 Least-Square Approximation of the Conditional Expectation......Page 240
15.10.2 Example: Evaluation of a Bermudan Option on a Stock (Backward Algorithm with Conditional Expectation Estimator)......Page 241
15.10.3 Example: Evaluation of a Bermudan Callable......Page 242
15.10.4 Implementation......Page 247
15.10.5 Binning as Linear Least-Square Regression......Page 248
15.11.1 Andersen Algorithm for Bermudan Swaptions......Page 249
15.11.2 Review of the Threshold Optimization Method......Page 250
15.11.4 Comparison of Optimization Method and Regression Method......Page 253
15.12.1 Foundations......Page 255
15.12.2 American Option Evaluation as Optimal Stopping Problem......Page 257
15.13 Primal-Dual Method: Upper and Lower Bound......Page 260
16.1 State Space Extension......Page 262
16.2 Implementation......Page 263
16.3 Path-Dependent Bermudan Options......Page 264
16.4.2 Evaluation of a Autocap in a Backward Algorithm......Page 265
17.1 Introduction......Page 268
17.2.1 Pricing using Monte-Carlo Simulation......Page 269
17.2.3 Example: The Linear and the Discontinuous Payout......Page 270
17.2.4 Example: Trigger Products......Page 272
17.3 Generic Sensitivities: Bumping the Model......Page 274
17.4 Sensitivities by Finite Differences......Page 276
17.4.1 Example: Finite Differences Applied to Smooth and Discontinuous Payout......Page 277
17.5.1 Example: Delta of a European Option under a Black-Scholes Model......Page 279
17.5.2 Pathwise Differentiation for Discontinuous Payouts......Page 280
17.6 Sensitivities by Likelihood Ratio Weighting......Page 281
17.6.2 Example: Variance Increase of the Sensitivity when using Likelihood Ratio Method for Smooth Payouts......Page 282
17.7 Sensitivities by Malliavin Weighting......Page 283
17.8 Proxy Simulation Scheme......Page 284
18.1 Full Proxy Simulation Scheme......Page 286
18.1.2 Calculation of Monte Carlo Weights......Page 287
18.1.3 Sensitivities by Finite Differences on a Proxy Simulation Scheme......Page 288
18.1.4 Localization......Page 289
18.1.6 Importance Sampling......Page 290
18.2.1 Linear Proxy Constraint......Page 293
18.2.3 Nonlinear Proxy Constraint......Page 294
18.2.4 Transition Probability from a Nonlinear Proxy Constraint......Page 296
18.2.6 Example: LIBOR Target Redemption Note......Page 299
18.2.7 Example: CMS Target Redemption Note......Page 301
18.3.1 Problem Description......Page 304
18.3.3 Partial Proxy Simulation Scheme (revisited)......Page 307
18.3.4 Localized Proxy Simulation Scheme......Page 310
18.3.6 Implementation......Page 311
18.3.7 Examples and Numerical Results......Page 312
V Pricing Models for Interest Rate Derivatives......Page 318
Motivation and Overview......Page 320
19 LIBOR Market Model......Page 322
19.1.1 Derivation of the Drift Term under the Terminal Measure......Page 324
19.1.2 Derivation of the Drift Term under the Spot LIBOR Measure......Page 326
19.1.3 Derivation of the Drift Term under the Tk-Forward Measure......Page 328
19.2.3 Drift of the Short Bond in a LIBOR Market Model......Page 329
19.3.1 Generation of the (Time-Discrete) Forward Rate Process......Page 330
19.3.3 Generation of the Numéraire......Page 331
19.4 Calibration—Choice of the Free Parameters......Page 332
19.4.2 Choice of the Volatilities......Page 333
19.4.3 Choice of the Correlations......Page 336
19.4.4 Covariance Structure......Page 338
19.4.5 Analytic Evaluation of Caplets, Swaptions and Swap Rate Covariance......Page 339
19.5.1 Interpolation of the Tenor Structure (Ti)......Page 344
19.6 Object-Oriented Design......Page 348
19.6.3 Abstraction of Model Parameters......Page 349
19.6.4 Abstraction of Calibration......Page 350
20 Swap Rate Market Models......Page 354
20.1 The Swap Measure......Page 355
20.2 Derivation of the Drift Term......Page 356
20.3.2 Choice of the Volatilities......Page 357
21 .I Definitions......Page 360
21.2 Terminal Correlation Examined in a LIBOR Market Model Example......Page 361
21.2.1 Decorrelation in a One-Factor Model......Page 362
21.2.2 Impact of the Time Structure of the Instantaneous Volatility on Caplet and Swaption Prices......Page 364
21.2.3 Swaption Value as a Function of Forward Rates......Page 365
21.3.1 Dependence of the Terminal Density on the Martingale Measure......Page 367
22 Heath-Jarrow-Morton Framework: Foundations......Page 370
22.1 Short-Rate Process in the HJM Framework......Page 371
22.2 The HJM Drift Condition......Page 372
23.1 Introduction......Page 376
23.2 The Market Price of Risk......Page 377
23.3 Overview: Some Common Models......Page 379
23.4.2 Lattice Implementation of Short-Rate Models......Page 380
24.1 Short-Rate Models in the HJM Framework......Page 382
24.1.1 Example: The Ho-Lee Model in the HJM Framework......Page 383
24.1.2 Example: The Hull-White Model in the HJM Framework......Page 384
24.2.1 HJM Volatility Structure of the LIBOR Market Model......Page 385
24.2.2 LIBOR Market Model Drift under the QB Measure......Page 387
24.2.3 LIBOR Market Model as a Short Rate Model......Page 389
25.1 Model......Page 390
25.2 Interpretation of the Figures......Page 391
25.3 Mean Reversion......Page 392
25.4 Factors......Page 393
25.5 Exponential Volatility Function......Page 394
25.6 Instantaneous Correlation......Page 396
26.1 Introduction......Page 398
26.2 Cheyette Model......Page 399
26.3 Implementation: PDE......Page 400
27.1 Introduction......Page 402
27.1.1 The Markov Functional Assumption (Independent of the Model Considered)......Page 403
27.2.1 Markov Functional Assumption......Page 404
27.2.2 Example: The Black-Scholes Model......Page 405
27.2.3 Numerical Calibration to a Full Two-Dimensional European Option Smile Surface......Page 406
27.2.4 Interest Rates......Page 408
27.2.5 Model Dynamics......Page 409
27.3.1 LIBOR Markov Functional Model in Terminal Measure......Page 415
27.3.2 LIBOR Markov Functional Model in Spot Measure......Page 421
27.3.3 Remark on Implementation......Page 425
27.3.4 Change of Numéraire in a Markov Functional Model......Page 426
27.4 Implementation: Lattice......Page 428
27.4.1 Convolution with the Normal Probability Density......Page 429
27.4.2 State Space Discretization......Page 432
VI Extended Models......Page 434
28.1.2 Credit Spread......Page 436
28.2 Defaultable Bonds......Page 437
28.3 Integrating Deterministic Credit Spread into a Pricing Model......Page 439
28.3.1 Deterministic Credit Spread......Page 440
28.3.2 Implementation......Page 441
28.4 Receiver's and Payer's Credit Spreads......Page 443
28.4.1 Example: Defaultable Forward Starting Coupon Bond......Page 444
28.4.2 Example: Option on a Defaultable Coupon Bond......Page 445
29.1 Cross-Currency LIBOR Market Model......Page 446
29.1.1 Derivation of the Drift Term under Spot Measure......Page 447
29.2.1 Derivation of the Drift Term under Spot-Measure......Page 451
29.3 Equity Hybrid Cross-Currency LIBOR Market Model......Page 453
29.3.1 Dynamic of the Foreign Stock under Spot Measure......Page 454
29.3.2 Summary......Page 455
29.3.3 Implementation......Page 456
VII Implementation......Page 458
30.1 Elements of Object-Oriented Programming: Class and Objects......Page 460
30.1.1 Example: Class of a Binomial Distributed Random Variable......Page 461
30.1.3 Methods: Getter, Setter, and Static Methods......Page 463
30.2.1 Encapsulation and Interfaces......Page 465
30.2.2 Abstraction and Inheritance......Page 469
30.2.3 Polymorphism......Page 472
30.3.1 Root Finder for General Functions......Page 474
30.3.2 Root Finder for Functions with Analytic Derivative: Newton’s Method......Page 476
30.3.3 Root Finder for Functions with Derivative Estimation: Secant Method......Page 477
30.4 Anatomy of a Java™ Class......Page 483
30.5.3 Colt......Page 485
30.6.1 Object-Oriented Design (OOD)/Unified Modeling Language (UML)......Page 486
VIII Appendices......Page 488
A A Small Collection of Common Misconceptions......Page 490
B.1.1 Uniform Distributed Random Variables......Page 492
B.1.4 Poisson Distributed Random Variables......Page 493
B.1.5 Generation of Paths of an n-Dimensional Brownian Motion......Page 494
B.2 Factor Decomposition—Generation of Correlated Brownian Motion......Page 496
B.3 Factor Reduction......Page 497
B.4 Optimization (One-Dimensional): Golden Section Search......Page 500
B.5 Linear Regression......Page 501
B.6 Convolution with Normal Density......Page 502
C Exercises......Page 504
D.1 Java™ Classes for Chapter 30......Page 512
List of Symbols......Page 518
List of Figures......Page 520
List of Tables......Page 524
List of Listings......Page 526
Bibliography......Page 528
Index......Page 536