The book arose from my lecture notes for the lectures on mathematical finance held at University of Mainz and University of Frankfurt. It tries to give a balanced representation of the theoretic foundations, state of the art models, which are actually used in practice and their implementation.
In practice, none of the three aspects "theory", "modeling" and "implementation" may be considered alone. Knowledge of the theory is worthless if it isn't applied. Theory gives the tools for a consistent modeling. A model without implementation is essentially worthless. A good implementation requires a deep understanding of the model and the underlying theory.
With this in mind, the book tries to build a bridge from academia to practice and from theory to object oriented implementation.
P435, 15.4MB.
Contents
1 Introduction 19
1.1 How to Read this Book 20
1.1.1 Abridged Versions 20
1.1.1.1 Abridged version ``Monte-Carlo pricing'' 20
1.1.1.2 Abridged version ``LIBOR Market Model'' 20
1.1.1.3 Abridged version ``Markov Functional Model'' 20
1.1.2 Special Sections 21
I Foundations 23
2 Foundations 25
2.1 Probability Theory 25
2.2 Stochastic Processes 33
2.3 Filtration 34
2.4 Brownian Motion 35
2.5 Wiener Measure, Canonical Setup 37
2.6 Ito Calculus 38
2.6.1 Ito Integral 41
2.6.2 Ito Process 43
2.6.3 Ito Lemma and Product Rule 45
2.7 Brownian Motion with Instantaneous Correlation 48
2.8 Martingales 50
2.8.1 Martingale Representation Theorem 50
2.9 Change of Measure (Girsanov, Cameron, Martin) 51
2.10 Stochastic Integration 55
2.11 List of Symbols 57
3 Replication 59
3.1 Replication Strategies 59
3.1.1 Introduction 59
3.1.2 Replication in a discrete Model 63
3.1.2.1 Example: two times ($T_{0},T_{1}$), two states ($\omega _{1},\omega _{2}$), two assets ($S$,$N$) 63
3.2 Foundations: Equivalent Martingale Measure 67
3.2.1 Challenge and Solution Outline 67
3.2.2 Steps towards the Universal Pricing Theorem 69
3.2.2.1 Self-financing Trading Strategy 70
3.2.2.2 Equivalent Martingale Measure 74
3.2.2.3 Payoff Replication 75
3.3 Basic Assumptions 76
3.4 Excursion: Relative Prices and Risk Neutral Measures 77
3.4.1 Why relative Prices? 77
3.4.2 Risk Neutral Measure 79
II First Applications 81
4 Pricing of an European Stock Option under the Black-Scholes Model 83
5 Excursion: The Density of the Underlying of an European Call Option 87
6 Excursion: Interpolation of European Option Prices 89
6.1 No-Arbitrage Conditions for Interpolated Prices 89
6.2 Arbitrage Violations through Interpolation 90
6.2.1 Example (1): Interpolation of four Prices 90
6.2.1.1 Linear Interpolation of Prices 91
6.2.1.2 Linear Interpolation of Implied Volatilities 92
6.2.1.3 Spline Interpolation of Prices respective Implied Volatilities 92
6.2.2 Example (2): Interpolation of two Prices 92
6.2.2.1 Lineare Interpolation for decreasing Implied Volatilities 93
Conclusion: 93
6.2.2.2 Lineare Interpolation for increasing Implied Volatilities 93
Conclusion: 94
6.3 Arbitrage Free Interpolation of European Option Prices 95
7 Hedging in Continuous and Discrete Time and the Greeks 97
7.1 Introduction 97
7.2 Deriving the Replications Strategy from Pricing Theory 98
Conclusion: 99
7.2.1 Deriving the Replication Strategy under the Assumption of a Locally Riskless Product 99
7.2.2 The Black-Scholes Differential Equation 100
7.2.3 The Derivative $V(t)$ as a Function of its Underlyings $S_{i}(t)$ 101
7.2.3.1 Path dependent options 102
7.2.4 Example: Replication Portfolio and PDE under a Black-Scholes Model 102
7.2.4.1 Interpretation of $V$ as a Function in $(t,S)$ 104
7.3 Greeks 105
7.3.1 Greeks of a european Call-Option under the Black-Scholes model 106
7.4 Hedging in Discrete Time: Delta- and Delta-Gamma-Hedging 107
7.4.1 Delta Hedging 108
7.4.2 Error Propagation 108
7.4.2.1 Example: Time discrete delta hedge under a Black-Scholes model 109
7.4.3 Delta-Gamma Hedging 110
7.4.3.1 Example: Time discrete delta-gamma hedge under a Black-Scholes model 112
7.4.4 Vega Hedging 114
7.5 Hedging in Discrete Time: Minimizing the Residual Error (Bouchaud-Sornette Method) 116
7.5.1 Minimizing the Residual Error at Maturity $T$ 117
7.5.2 Minimizing the Residual Error in each Time Step 118
III Interest Rate Structures, Interest Rate Products and Analytic Pricing Formulas 121
Motivation and Overview 123
8 Interest Rate Structures 125
8.1 Introduction 125
8.1.1 Fixing Times and Tenor Times 126
8.2 Definitions 126
8.3 Interest Rate Curve Bootstrapping 131
Induction start ($T_{0}$): 131
Induction step ($T_{i-1} \rightarrow T_{i}$): 132
8.4 Interpolation of Interest Rate Curves 132
8.5 Implementation 133
9 Simple Interest Rate Products 135
9.1 Interest Rate Products Part 1: Products without Optionality 135
9.1.1 Fix, Floating and Swap 135
9.1.2 Money-Market Account 142
9.2 Interest Rate Products Part 2: Simple Options 143
9.2.1 Cap, Floor, Swaption 143
9.2.2 Foreign Caplet, Quanto 145
10 The Black Model for a Caplet 147
11 Pricing of a Quanto Caplet (Modeling the FFX) 149
11.1 Choice of Num\'eraire 149
12 Exotic Derivatives 153
12.1 Prototypical Product Properties 153
12.2 Interest Rate Products Part 3: Exotic Interest Rate Derivatives 155
12.2.1 Structured Bond, Structured Swap, Zero Structure 155
12.2.2 Bermudan Callables and Cancelable 160
12.2.3 Compound Options 163
12.2.4 Trigger Products 163
12.2.4.1 Target Redemption Note 163
12.2.5 Structured Coupons 165
12.2.5.1 Capped, Floored, Inverse, Spread, CMS 165
12.2.5.2 Range Accruals 166
12.2.5.3 Path Dependent Coupons 167
12.2.5.4 Flexi Cap 167
12.2.6 Shout Options 169
12.3 Product Toolbox 169
IV Discretization and Numerical Valuation Methods 173
Motivation and Overview 175
13 Discretization of time and state space 177
13.1 Discretization of Time: The Euler and the Milstein Scheme 177
13.1.1 Definitions 177
13.1.2 Time-Discretisation of a Lognormal Process 181
13.1.2.1 Discretization via Euler scheme 181
13.1.2.2 Discretization via Milstein scheme 181
13.1.2.3 Discretization of the Log-Process 181
13.1.2.4 Exact Discretization 182
13.2 Discretization of Paths (Monte-Carlo Simulation) 182
13.2.1 Monte-Carlo Simulation 182
13.2.2 Weighted Monte-Carlo Simulation 183
13.2.3 Implementation 183
13.2.3.1 Example: Valuation of a Stock Option under the Black-Scholes Model using Monte-Carlo Simulation 184
13.2.3.2 Separation of Product and Model 186
13.2.4 Review 186
13.3 Discretization of State Space 187
13.3.1 Definitions 187
13.3.2 Backward-Algorithm 189
13.3.3 Review 189
13.3.3.1 Path Dependencies 189
13.3.3.2 Course of Dimension 190
13.4 Path Simulation through a Lattice: Two Layers 190
14 Pricing Options with Early Exercise by Monte Carlo Simulation 191
14.1 Introduction 191
14.2 Bermudan Options: Notation 192
14.2.1 Bermudan Callable 192
14.2.2 Relative Prices 193
14.3 Bermudan Option as Optimal Exercise Problem 194
14.3.1 Bermudan Option Value as single (unconditioned) Expectation: The Optimal Exercise Value 194
14.4 Bermudan Option Pricing - The Backward Algorithm 195
Induction start: 195
Induction step 195
14.5 Re-simulation 196
14.6 Perfect Foresight 197
14.7 Conditional Expectation as Functional Dependence 198
Example: 199
14.8 Binning 199
14.8.1 Binning as a Least-Square Regression 201
14.9 Foresight Bias 203
14.10 Regression Methods - Least Square Monte Carlo 203
14.10.1 Least Square Approximation of the Conditional Expectation 204
14.10.2 Example: Evaluation of a Bermudan Option on a Stock (Backward Algorithm with Conditional Expectation Estimator) 205
Induction start: $t > T_{n}$ 205
Induction step: $t = T_{i}$, $i = n, n-1, n-2, \ldots 1$ 206
14.10.3 Example: Evaluation of an Bermudan Callable 206
Induction start: $t > T_{n}$ 206
Induction step: $t = T_{i}$, $i = n, n-1, n-2, \ldots 1$ 209
14.10.4 Implementation 210
14.10.5 Binning as linear Least-Square Regression 210
14.11 Optimization Methods 212
14.11.1 Andersen Algorithm for Bermudan Swaptions 212
Induction start: $t > T_{n}$ 212
Induction step: $t = T_{i}$, $i = n, n-1, n-2, \ldots 1$ 213
14.11.2 Review of the Threshold Optimization Method 213
14.11.2.1 Fitting the exercise strategy to the product 213
14.11.2.2 Disturbance of the Optimizer through Discontinuities and local Minima 215
14.11.3 Optimization of Exercise Strategy: A more general Formulation 215
14.11.4 Comparison of Optimization Method and Regression Method 216
14.12 Duality Method: Upper Bound for Bermudan Option Prices - The Method of Rogers 217
14.12.1 Foundations 217
14.12.2 American Option Evaluation as Optimal Stopping Problem 219
14.13 Primal-Dual Method: Upper and Lower Bound 222
15 Sensitivities (Partial Derivatives) of Monte-Carlo Prices 225
15.1 Introduction 225
15.2 Problem Description 225
15.2.1 Pricing using Monte Carlo Simulation 226
15.2.2 Sensitivities from Monte Carlo Pricing 227
15.2.3 Example: The Linear and the Discontinuous Payout 227
15.2.3.1 Linear Payout 227
15.2.3.2 Discontinuous Payout 228
15.2.4 Example: Trigger Products 228
15.3 Generic Sensitivities: Bumping the Model 228
15.4 Sensitivities by Finite Differences 232
15.4.1 Example: Finite Differences applied to Smooth and Discontinuous Payout 233
Simplified Example: 234
15.5 Sensitivities by Pathwise Differentiation 234
15.5.1 Example: Delta of a European Option under a Black-Scholes Model 235
15.5.2 Pathwise Differentiation for Discontinuous Payouts 236
15.6 Sensitivities by Likelihood Ratio Weighting 236
15.6.1 Example: Delta of a European Option under a Black-Scholes Model using Pathwise Derivative 237
15.6.2 Example: Variance Increase of the Sensitivity when using Likelihood Ratio Method for Smooth Payouts 237
15.7 Sensitivities by Malliavin Weighting 238
15.8 Proxy Simulation Scheme 239
16 Proxy Simulation Schemes for Monte-Carlo Sensitivities and Importance Sampling 241
16.1 Full Proxy Simulation Scheme 241
16.1.1 Calculation of Monte-Carlo weights 242
16.2 Sensitivities by Finite Differences on a Proxy Simulation Scheme 244
16.2.1 Localization 244
16.2.2 Object Oriented Design 245
16.3 Importance Sampling 245
16.3.1 Example 247
16.4 Partial Proxy Simulation Schemes 248
V Pricing Models for Interest Rate Derivatives 249
17 Market Models 251
17.1 LIBOR Market Model 252
17.1.1 Derivation of the Drift Term 253
17.1.1.1 Derivation of the Drift Term under the Terminal Measure 253
17.1.1.2 Derivation of the Drift Term under the Spot LIBOR Measure 255
17.1.1.3 Derivation of the Drift Term under the $T_{k}$-Forward Measure 257
17.1.2 The Short Period Bond $P(T_{m(t)+1};t)$ 258
17.1.2.1 Role of the short bond in a LIBOR Market Model 258
17.1.2.2 Link to continuous time tenors 258
17.1.2.3 Drift of the short bond in a LIBOR Market Model 258
17.1.3 Discretization and (Monte Carlo) Simulation 259
17.1.3.1 Generation of the (time-discrete) Forward Rate Process 259
17.1.3.2 Generation of the Sample Paths 260
17.1.3.3 Generation of the Num\'eraire 260
17.1.4 Calibration - Choice of the free Parameters 260
17.1.4.1 Choice of the Initial Conditions 261
Reproduction of Bond Market Prices 261
17.1.4.2 Choice of the Volatilities 262
Reproduction of \hyperref [def:caplet]{Caplet} Market Prices 262
Reproduction of \hyperref [def:swaption]{Swaption} Market Prices 262
Functional Forms 264
17.1.4.3 Choice of the Correlations 265
Factors 265
Functional Forms 265
Factor Reduction 265
Calibration 266
17.1.4.4 Covariance Matrix, Calibration by Parameter Optimization 266
17.1.4.5 Analytic Evaluation of Caplets and Swaptions 266
Analytic Evaluation of a Caplet in the LIBOR Market Model 266
Analytic Evaluation of a Swaption in the LIBOR Market Model 267
17.1.5 Interpolation of Forward Rates in the LIBOR Market Model 267
17.1.5.1 Interpolation of the Tenor Structure $\{ T_{i} \}$ 267
Assumption 1: No stochastic shortly before maturity. 268
Assumption 2: Linearity shortly before maturity. 269
17.2 Object Oriented Design 271
17.2.1 Reuse of Implementation 271
17.2.2 Separation of Product and Model 272
17.2.3 Abstraction of Model Parameters 272
17.2.4 Abstraction of Calibration 274
17.3 Swap Rate Market Models (Jamshidian 1997) 275
17.3.1 The Swap Measure 276
17.3.2 Derivation of the Drift Term 277
17.3.3 Calibration - Choice of the free Parameters 277
17.3.3.1 Choice of the Initial Conditions 277
Reproduction of Bond Market Prices or Swap Market Prices 277
17.3.3.2 Choice of the Volatilities 277
Reproduction of \hyperref [def:swaption]{Swaption} Market Prices 277
18 Excursion: Instantaneous Correlation and Terminal Correlation 281
18.1 Definitions 281
18.2 Terminal Correlation studied on the Example of a LIBOR Market Model 282
18.2.1 De-correlation in a One-Factor-Model 283
18.2.2 Impact of the Time Structure of the Instantaneous Volatility on Caplet and Swaption Prices 284
18.2.3 The Swaption Value as a Function of Forward Rates 285
18.3 Terminal Correlation depends on the Equivalent Martingale Measure 288
18.3.1 Dependence of the Terminal Density on the Martingale Measure 288
19 Heath-Jarrow-Morton Framework: Foundations 291
19.1 Short Rate Process in the HJM Framework 292
19.2 The HJM Drift Condition 292
20 Short Rate Models 297
20.1 Introduction 297
20.2 The Market Price of Risk 298
20.3 Overview: Some Common Models 300
20.4 Implementations 300
20.4.1 Monte-Carlo Implementation of Short-Rate Models 300
20.4.2 Lattice Implementation of Short-Rate Models 301
21 Heath-Jarrow-Morton Framework: Immersion of Short Rate Models and LIBOR Market Model 303
21.1 Short Rate Models in the HJM Framework 303
21.1.1 Example: The Ho-Lee Model in the HJM Framework 303
21.1.2 Example: The Hull-White Model in the HJM Framework 304
21.2 LIBOR Market Model in the HJM Framework 306
21.2.1 HJM Volatility Structure of the LIBOR Market Model 306
21.2.2 LIBOR Market Model Drift under the $\@mathbb {Q}^{B}$ Measure 308
21.2.3 LIBOR Market Model as a Short Rate Model 309
22 Excursion: Shape of the Interest Rate Curve under Mean Reversion and a Multi-Factor Model 311
22.1 Model 311
22.2 Interpretation of the Figures 312
22.3 Mean Reversion 312
22.4 Factors 314
22.5 Exponential Volatility Function 315
22.6 Instantaneouse Correlation 317
23 Markov Functional Models 319
23.1 Introduction 319
23.1.1 The Markov Functional Assumption (independent of the model considered) 320
23.1.2 Outline of this Chapter 321
23.2 Equity Markov Functional Model 321
23.2.1 Markov Functional Assumption 321
23.2.2 Example: The Black-Scholes Model 322
23.2.3 Numerical Calibration to a Full Two Dimensional European Option Smile Surface 323
23.2.3.1 Market Price 323
23.2.3.2 Model Price 324
23.2.3.3 Solving for the Functional 324
23.2.4 Interest Rates 324
23.2.4.1 A Note on Interest Rates and the No-Arbitrage Requirement 324
23.2.4.2 Where are the Interest Rates? 325
23.2.5 Model Dynamics 326
23.2.5.1 Introduction 326
23.2.5.2 Interest Rate Dynamics 327
23.2.5.3 Forward Volatility 329
23.2.6 Implementation 331
23.3 LIBOR Markov Functional Model 331
23.3.1 LIBOR Markov Functional Model in Terminal Measure - Hunt, Kennedy, Pelsser 331
23.3.1.1 Evaluation within the LIBOR Markov Functional Model 333
23.3.1.2 Calibration of the LIBOR Functional 333
Induction start: 334
Induction step ($T_{i+1} \rightarrow T_{i}$): 334
Induction start: 335
Induction step ($T_{i+1} \rightarrow T_{i}$): 336
23.4 Implementation: Lattice 336
23.4.1 Convolution with the Normal Probability Density 336
23.4.1.1 Piecewise constant Approximation 337
23.4.1.2 Piecewise polynomial Approximation 337
23.4.2 State space discretization 339
23.4.2.1 Equidistant discretization 339
VI Extended Models 341
24 Hybrid Models 343
24.1 Cross Currency LIBOR Market Model 343
24.1.1 Derivation of the Drift Term under Spot-Measure 344
24.1.1.1 Dynamic of the domestic LIBOR under Spot Measure 344
24.1.1.2 Dynamic of the foreign LIBOR under Spot Measure 345
24.1.1.3 Dynamic of the FX Rate under Spot Measure 347
24.1.2 Implementation 348
24.2 Equity Hybrid LIBOR Market Model 348
24.2.1 Derivation of the Drift Term under Spot-Measure 348
24.2.1.1 Dynamic of the Stock Process under Spot Measure 349
24.2.2 Implementation 350
24.3 Equity-Hybrid Cross-Currency LIBOR Market Model 350
24.3.0.1 Dynamic of the Foreign Stock under Spot-Measure 350
24.3.1 Summary 352
24.3.2 Implementation 353
VII Implementation 355
25 Object-Oriented Implementation in Java™ 357
25.1 Elements of Object Oriented Programming: Class and Objects 358
25.1.1 Example: Class of a Binomial Distributed Random Variable 359
25.1.2 Constructor 360
25.1.3 Methods: Getter, Setter, Static Methods 361
25.1.3.1 Aufrufkonvention, Signatur 361
25.1.3.2 Getter, Setter 361
25.1.3.3 Statische Methoden 361
25.2 Principles of Object Oriented Programming 362
25.2.1 Data Hiding and Interfaces 362
25.2.1.1 Kapselung 363
Beispiel für Kapselung: Anbieten alternativer Methoden 364
Beispiel für Kapselung: Geschwindigkeitssteigerung durch Erweitern des internen Datenmodells um einen Cache 364
25.2.1.2 Interfaces 365
25.2.1.3 Beispiel: Diskrete reellwertige Zufallsvariable 366
25.2.2 Abstraction and Inheritance 366
25.2.2.1 Methoden: Überschreiben und überladen 369
25.2.3 Polymorphism 369
25.3 Example: A Class Structure for One Dimensional Root Finders 370
25.3.1 Root Finder for General Functions 370
25.3.1.1 Interface 370
25.3.1.2 Bisection Search 371
25.3.2 Root Finder for Functions with Analytic Derivative: Newton Method 372
25.3.2.1 Interface 372
25.3.2.2 Newtonverfahren 372
25.3.3 Root Finder for Functions with Derivative Estimation: Secant Method 373
25.3.3.1 Vererbung 373
25.3.3.2 Polymorphie 374
25.4 Anatomy of a Java™ Class 376
25.5 Libraries 377
25.5.1 Java™ 2 Platform, Standard Edition (j2se) 377
25.5.2 Java™ 2 Platform, Enterprise Edition (j2ee) 378
25.5.3 Colt 378
25.5.4 Commons-Math: The Jakarta Mathematics Library 378
25.6 Some Final Remarks 378
25.6.1 Object Oriented Design (OOD) / Unified Modeling Language (UML) 378
VIII Appendix 381
A Tools (Selection) 383
A.1 Cholesky Decomposition 383
A.2 Linear Regression 384
A.3 Generation of Random Numbers 385
A.3.1 Uniform Distributed Random Variables 385
A.3.1.1 Mersenne Twister 385
A.3.2 Transformation of the Random Number Distribution via the Inverse Distributionfunction 385
A.3.3 Normal Distributed Random Variables 385
A.3.3.1 Inverse Distribution Function 385
A.3.3.2 Box-Muller Transformation 386
A.3.4 Poisson Distributed Random Variables 386
A.3.4.1 Inverse Distribution Function 386
A.3.5 Generation of Paths of an $n$-dimensional Brownian Motion 386
A.4 Generation of Correlated Brownian Motion 389
A.5 Factor Reduction 390
A.6 Optimization (one dimensional): Golden Section Search 392
A.7 Convolution with the Normal Density 393
B Exercises 395
C List of Symbols 403
D Java™ Source Code (Selection) 405
D.1 Java™ Classes for Chapter \ref {sec:IntroductionToOOWithJava} 405
E Technical Terms (Selection) 409
List of Figures 411
List of Tables 415
Bibliography 419
Index 427 |
