Financial Modelling: Theory, Implementation and Practice with MATLAB Source

Financial Modelling......Page 3
Contents......Page 9
1 Introduction and Management Summary......Page 17
2 Why We Have Written this Book......Page 18
4 The Audience......Page 19
5 The Structure of this Book......Page 20
6 What this Book Does Not Cover......Page 21
8 Code......Page 22
PART I FINANCIAL MARKETS AND POPULAR MODELS......Page 23
1.1 Introduction and Objectives......Page 25
1.2 Financial Time-Series, Statistical Properties of Market Data and Invariants......Page 26
1.2.1 Real World Distribution......Page 31
1.3 Implied Volatility Surfaces and Volatility Dynamics......Page 33
1.3.1 Is There More than just a Volatility?......Page 35
1.3.3 Time-Dependent Volatility......Page 38
1.3.5 Volatility from Jumps......Page 39
1.3.7 The Risk Neutral Density......Page 40
1.4.1 Asset Allocation......Page 42
1.4.2 Pricing, Hedging and Risk Management......Page 43
1.5 General Remarks on Notation......Page 46
1.6 Summary and Conclusions......Page 47
1.7 Appendix – Quotes......Page 48
2.2 Local Volatility Models......Page 51
2.2.1 The Bachelier and the Black–Scholes Model......Page 53
2.2.2 The Hull–White Model......Page 56
2.2.3 The Constant Elasticity of Variance Model......Page 62
2.2.4 The Displaced Diffusion Model......Page 66
2.2.5 CEV and DD Models......Page 69
2.3 Stochastic Volatility Models......Page 70
2.3.1 Pricing European Options......Page 71
2.3.2 Risk Neutral Density......Page 72
2.3.3 The Heston Model (and Extensions)......Page 73
2.3.4 The SABR Model......Page 83
2.3.5 SABR – Further Remarks......Page 89
2.4.1 The Heston–Hull–White Model......Page 97
2.5 Summary and Conclusions......Page 106
3.1 Introduction and Objectives......Page 109
3.2.1 Poisson Processes......Page 110
3.2.2 The Merton Model......Page 111
3.2.3 The Bates Model......Page 115
3.2.4 The Bates–Hull–White Model......Page 120
3.3 Exponential Lévy Models......Page 121
3.3.1 The Variance Gamma Model......Page 123
3.3.2 The Normal Inverse Gaussian Model......Page 128
3.4 Other Models......Page 134
3.4.2 Stochastic Clocks......Page 138
3.5 Martingale Correction......Page 145
3.6 Summary and Conclusions......Page 150
4.2.1 GBM Baskets......Page 153
4.2.2 Libor Market Models......Page 155
4.3.1 Stochastic Volatility Models......Page 157
4.4 Parameter Averaging......Page 159
4.4.1 Applications to CMS Spread Options......Page 160
4.5 Markovian Projection......Page 175
4.5.2 Markovian Projection on Local Volatility and Heston Models......Page 178
4.5.3 Markovian Projection onto DD SABR Models......Page 180
4.6 Copulae......Page 188
4.6.1 Measures of Concordance and Dependency......Page 190
4.6.3 Elliptical Copulae......Page 191
4.6.4 Archimedean Copulae......Page 193
4.6.6 Asymmetric Copulae......Page 195
4.6.8 Applying Copulae to Asset Allocation......Page 196
4.7 Multi-Dimensional Variance Gamma Processes......Page 203
4.8 Summary and Conclusions......Page 209
PART II NUMERICAL METHODS AND RECIPES......Page 211
5.2 Fourier Transform......Page 213
5.2.1 Discrete Fourier Transform......Page 215
5.2.2 Fast Fourier Transform......Page 216
5.3 The Carr–Madan Method......Page 218
5.3.1 The Optimal α......Page 223
5.4 The Lewis Method......Page 226
5.4.1 Application to Other Payoffs......Page 230
5.5 The Attari Method......Page 231
5.6 The Convolution Method......Page 232
5.7 The Cosine Method......Page 236
5.8 Comparison, Stability and Performance......Page 244
5.8.1 Other Issues......Page 249
5.9 Extending the Methods to Forward Start Options......Page 251
5.9.1 Forward Characteristic Function for Lévy Processes and CIR Time Change......Page 254
5.9.2 Forward Characteristic Function for Lévy Processes and Gamma-OU Time Change......Page 255
5.9.3 Results......Page 258
5.10 Density Recovery......Page 261
5.11 Summary and Conclusions......Page 266
6.2 Pricing Non-Standard Vanilla Options......Page 269
6.3 Bermudan and American Options......Page 270
6.3.1 The Convolution Method......Page 273
6.3.2 The Cosine Method......Page 274
6.3.3 Numerical Results......Page 282
6.3.4 The Fourier Space Time-Stepping......Page 286
6.4 The Cosine Method and Barrier Options......Page 293
6.5 Greeks......Page 294
6.6 Summary and Conclusions......Page 303
7.2 Sampling Diffusion Processes......Page 305
7.2.3 The Predictor-Corrector Scheme......Page 306
7.2.5 Implementation and Results......Page 307
7.3 Special Purpose Schemes......Page 308
7.3.1 Schemes for the Heston Model......Page 310
7.3.2 Unbiased Scheme for the SABR Model......Page 316
7.4.1 Jump Models – Poisson Processes......Page 329
7.4.3 Stochastic Grid Sampling (SGS)......Page 331
7.4.4 Simulation – Lévy Models......Page 338
7.4.5 Schemes for Lévy Models with Stochastic Volatility......Page 346
7.5 Bridge Sampling......Page 355
7.6 Libor Market Model......Page 362
7.7 Multi-Dimensional Lévy Models......Page 367
7.8 Copulae......Page 368
7.8.1 Distributional Sampling Approach (DSA)......Page 369
7.8.2 Conditional Sampling Approach (CSA)......Page 372
7.8.3 Simulation from Other Copulae......Page 374
7.9 Summary and Conclusions......Page 375
8.2 Monte Carlo and Early Exercise......Page 377
8.2.1 Longstaff–Schwarz Regression......Page 378
8.2.2 Policy Iteration Methods......Page 385
8.2.3 Upper Bounds......Page 390
8.2.4 Problems of the Method......Page 392
8.2.5 Financial Examples and Numerical Results......Page 394
8.3 Greeks with Monte Carlo......Page 398
8.3.1 The Finite Difference Method (FDM)......Page 399
8.3.2 The Pathwise Method......Page 401
8.3.3 The Affine Recursion Problem (ARP)......Page 405
8.3.4 Adjoint Method......Page 407
8.3.5 Bermudan ARPs......Page 409
8.4.1 SDE of Diffusions......Page 412
8.4.3 Approximating General Greeks Using ARP......Page 413
8.4.4 Greeks......Page 420
8.5 Application to Trigger Swap......Page 423
8.5.1 Mathematical Modelling......Page 424
8.5.2 Numerical Results......Page 426
8.5.3 The Likelihood Ratio Method (LRM)......Page 429
8.5.4 Likelihood Ratio for Finite Differences – Proxy Simulation......Page 432
8.5.5 Numerical Results......Page 435
8.6 Summary and Conclusions......Page 449
8.7 Appendix – Trees......Page 450
9.1 Introduction and Objectives......Page 451
9.2 The Nelder–Mead Method......Page 453
9.2.1 Implementation......Page 458
9.2.2 Calibration Examples......Page 460
9.3 The Levenberg–Marquardt Method......Page 465
9.3.1 Implementation......Page 469
9.3.2 Calibration Examples......Page 471
9.4 The L-BFGS Method......Page 476
9.4.1 Implementation......Page 479
9.4.2 Calibration Examples......Page 480
9.5 The SQP Method......Page 484
9.5.1 The Modified and Globally Convergent SQP Iteration......Page 489
9.5.2 Implementation......Page 491
9.5.3 Calibration Examples......Page 493
9.6 Differential Evolution......Page 498
9.6.1 Implementation......Page 503
9.6.2 Calibration Examples......Page 504
9.7 Simulated Annealing......Page 509
9.7.1 Implementation......Page 513
9.7.2 Calibration Examples......Page 516
9.8 Summary and Conclusions......Page 521
10.1 Introduction and Objectives......Page 523
10.2.1 Similarities – Heston and Bates Models......Page 524
10.2.2 Parameter Stability......Page 527
10.3 Pricing Exotic Options......Page 537
10.4 Hedging......Page 544
10.4.1 Hedging – The Basics......Page 547
10.4.2 Hedging in Incomplete Markets......Page 549
10.4.3 Discrete Time Hedging......Page 557
10.4.4 Numerical Examples......Page 560
10.5 Summary and Conclusions......Page 566
PART III IMPLEMENTATION, SOFTWARE DESIGN AND MATHEMATICS......Page 567
11.2 General Remarks......Page 569
11.3.1 Matrices and Vectors......Page 572
11.3.2 Cell Arrays......Page 578
11.4.1 Functions......Page 580
11.4.2 Function Handles......Page 583
11.5.1 Financial......Page 586
11.5.3 Fixed-Income......Page 587
11.5.4 Optimization......Page 589
11.5.5 Global Optimization......Page 593
11.5.6 Statistics......Page 594
11.5.7 Portfolio Optimization......Page 597
11.6.2 Solving Equations and ODE......Page 605
11.6.3 Useful Functions......Page 607
11.7.1 Two-Dimensional Plots......Page 609
11.7.2 Three-Dimensional Plots – Surfaces......Page 611
11.8 Summary and Conclusions......Page 613
12.2.1 Classes......Page 615
12.2.2 Handling Classes in Matlab......Page 622
12.2.3 Inheritance, Base Classes and Superclasses......Page 623
12.2.4 Handle and Value Classes......Page 625
12.2.5 Overloading......Page 626
12.3 A Model Class Hierarchy......Page 627
12.4 A Pricer Class Hierarchy......Page 629
12.5 An Optimizer Class Hierarchy......Page 634
12.6 Design Patterns......Page 636
12.6.1 The Builder Pattern......Page 637
12.6.2 The Visitor Pattern......Page 640
12.6.3 The Strategy Pattern......Page 642
12.7 Example – Calibration Engine......Page 645
12.7.1 Calibrating a Data Set or a History......Page 647
12.8.1 An Abstract Class for LMM Derivatives......Page 650
12.8.2 A Class for Bermudan Swaptions......Page 653
12.8.3 A Class for Trigger Swaps......Page 655
12.9 Summary and Conclusions......Page 657
13.2 Probability Theory and Stochastic Processes......Page 659
13.2.2 Random Variables......Page 660
13.2.3 Important Results......Page 661
13.2.4 Distributions......Page 665
13.2.5 Stochastic Processes......Page 670
13.2.6 Lévy Processes......Page 671
13.2.7 Stochastic Differential Equations......Page 676
13.3.1 Random Number Generation......Page 681
13.3.2 Methods for Computing Variates......Page 686
13.4.1 Complex Numbers......Page 687
13.4.2 Complex Differentiation and Integration along Paths......Page 688
13.4.3 The Complex Exponential and Logarithm......Page 689
13.4.4 The Residual Theorem......Page 690
13.5 The Characteristic Function and Fourier Transform......Page 691
13.6 Summary and Conclusions......Page 695
List of Figures......Page 697
List of Tables......Page 707
Bibliography......Page 711
Index......Page 721