Financial modelling: theory, implementation and practice (with Matlab source)

Financial Modelling - Theory, Implementation and Practice with MATLAB Source is a unique combination of quantitative techniques, the application to financial problems and programming using Matlab. The book enables the reader to model, design and implement a wide range of financial models for derivatives pricing and asset allocation, providing practitioners with complete financial modelling workflow, from model choice, deriving prices and Greeks using (semi-) analytic and simulation techniques, and calibration even for exotic options.

The book is split into three parts. The first part considers financial markets in general and looks at the complex models needed to handle observed structures, reviewing models based on diffusions including stochastic-local volatility models and (pure) jump processes. It shows the possible risk-neutral densities, implied volatility surfaces, option pricing and typical paths for a variety of models including SABR, Heston, Bates, Bates-Hull-White, Displaced-Heston, or stochastic volatility versions of Variance Gamma, respectively Normal Inverse Gaussian models and finally, multi-dimensional models. The stochastic-local-volatility Libor market model with time-dependent parameters is considered and as an application how to price and risk-manage CMS spread products is demonstrated.

The second part of the book deals with numerical methods which enables the reader to use the models of the first part for pricing and risk management, covering methods based on direct integration and Fourier transforms, and detailing the implementation of the COS, CONV, Carr-Madan method or Fourier-Space-Time Stepping. This is applied to pricing of European, Bermudan and exotic options as well as the calculation of the Greeks. The Monte Carlo simulation technique is outlined and bridge sampling is discussed in a Gaussian setting and for Lévy processes. Computation of Greeks is covered using likelihood ratio methods and adjoint techniques. A chapter on state-of-the-art optimization algorithms rounds up the toolkit for applying advanced mathematical models to financial problems and the last chapter in this section of the book also serves as an introduction to model risk.

The third part is devoted to the usage of Matlab, introducing the software package by describing the basic functions applied for financial engineering. The programming is approached from an object-oriented perspective with examples to propose a framework for calibration, hedging and the adjoint method for calculating Greeks in a Libor market model.

Source code used for producing the results and analysing the models is provided on the author's dedicated website, http://www.mathworks.de/matlabcentral/fileexchange/authors/246981.

Author Notes

About the authors

JÖRG KIENITZ is the head of Quantitative Analytics at Deutsche Postbank AG. He is primarily involved in developing and implementing models for pricing complex derivatives structures and for asset allocation. He also lectures at university level on advanced financial modelling and implementation including the University of Oxford's part-time Masters of Finance course. Jörg works as an independent consultant for model development and validation as well as giving seminars for finance professionals. He is a speaker at the major financial conferences including Global Derivatives, WBS Fixed Income and RISK. Jörg is a member of the editorial board of International Review of Applied Financial Issues and Economics and holds a Ph.D. in stochastic analysis from the University of Bielefeld.

DANIEL WETTERAU is a specialist in the Quantitative Analytics team of Deutsche Postbank AG. He is responsible for the implementation of term structure models, advanced numerical methods, optimization algorithms and methods for advanced quantitative asset allocation. Further to his work he teaches finance courses for market professionals. Daniel received a Masters in financial mathematics from the University of Wuppertal and was awarded the Barmenia mathematics award for his thesis.

Introduction	p. 1
1 Introduction and Management Summary	p. 1
2 Why We Have Written this Book	p. 2
3 Why You Should Read this Book	p. 3
4 The Audience	p. 3
5 The Structure of this Book	p. 4
6 What this Book Does Not Cover	p. 5
7 Credits	p. 6
8 Code	p. 6
Part I Financial Markets and Popular Models
1 Financial Markets - Data, Basics and Derivatives	p. 9
1.1 Introduction and Objectives	p. 9
1.2 Financial Time-Series, Statistical Properties of Market Data and Invariants	p. 10
1.2.1 Real World Distribution	p. 15
1.3 Implied Volatility Surfaces and Volatility Dynamics	p. 17
1.3.1 Is There More than just a Volatility?	p. 19
1.3.2 Implied Volatility	p. 22
1.3.3 Time-Dependent Volatility	p. 22
1.3.4 Stochastic Volatility	p. 23
1.3.5 Volatility from Jumps	p. 23
1.3.6 Traders' Rule of Thumb	p. 24
1.3.7 The Risk Neutral Density	p. 24
1.4 Applications	p. 26
1.4.1 Asset Allocation	p. 26
1.4.2 Pricing, Hedging and Risk Management	p. 27
1.5 General Remarks on Notation	p. 30
1.6 Summary and Conclusions	p. 31
1.7 Appendix - Quotes	p. 32
2 Diffusion Models	p. 35
2.1 Introduction and Objectives	p. 35
2.2 Local Volatility Models	p. 35
2.2.1 The Bachelier and the Black-Scholes Model	p. 37
2.2.2 The Huh-White Model	p. 40
2.2.3 The Constant Elasticity of Variance Model	p. 46
2.2.4 The Displaced Diffusion Model	p. 50
2.2.5 CEV and DD Models	p. 53
2.3 Stochastic Volatility Models	p. 54
2.3.1 Pricing European Options	p. 55
2.3.2 Risk Neutral Density	p. 56
2.3.3 The Heston Model (and Extensions)	p. 57
2.3.4 The SABR Model	p. 67
2.3.5 SABR - Further Remarks	p. 73
2.4 Stochastic Volatility and Stochastic Rates Models	p. 81
2.4.1 The Heston-Hull-White Model	p. 81
2.5 Summary and Conclusions	p. 90
3 Models with Jumps	p. 93
3.1 Introduction and Objectives	p. 93
3.2 Poisson Processes and Jump Diffusions	p. 94
3.2.1 Poisson Processes	p. 94
3.2.2 The Merton Model	p. 95
3.2.3 The Bates Model	p. 99
3.2.4 The Bates-Hull-White Model	p. 104
3.3 Exponential Lévy Models	p. 105
3.3.1 The Variance Gamma Model	p. 107
3.3.2 The Normal Inverse Gaussian Model	p. 112
3.4 Other Models	p. 118
3.4.1 Exponential Lévy Models with Stochastic Volatility	p. 122
3.4.2 Stochastic Clocks	p. 122
3.5 Martingale Correction	p. 129
3.6 Summary and Conclusions	p. 134
4 Multi-Dimensional Models	p. 137
4.1 Introduction and Objectives	p. 137
4.2 Multi-Dimensional Diffusions	p. 137
4.2.1 GBM Baskets	p. 137
4.2.2 Libor Market Models	p. 139
4.3 Multi-Dimensional Heston and SABR Models	p. 141
4.3.1 Stochastic Volatility Models	p. 141
4.4 Parameter Averaging	p. 143
4.4.1 Applications to CMS Spread Options	p. 144
4.5 Markovian Projection	p. 159
4.5.1 Baskets with Local Volatility	p. 162
4.5.2 Markovian Projection on Local Volatility and Heston Models	p. 162
4.5.3 Markovian Projection onto DD SABR Models	p. 164
4.6 Copulae	p. 172
4.6.1 Measures of Concordance and Dependency	p. 174
4.6.2 Examples	p. 175
4.6.3 Elliptical Copulae	p. 175
4.6.4 Archimedean Copulae	p. 177
4.6.5 Building New Copulae from Given Copulae	p. 179
4.6.6 Asymmetric Copulae	p. 179
4.6.7 Applying Copulae to Option Pricing	p. 180
4.6.8 Applying Copulae to Asset Allocation	p. 180
4.7 Multi-Dimensional Variance Gamma Processes	p. 187
4.8 Summary and Conclusions	p. 193
Part II Numerical Methods and Recipes
5 Option Pricing by Transform Techniques and Direct Integration	p. 197
5.1 Introduction and Objectives	p. 197
5.2 Fourier Transform	p. 197
5.2.1 Discrete Fourier Transform	p. 199
5.2.2 Fast Fourier Transform	p. 200
5.3 The Carr-Madan Method	p. 202
5.3.1 The Optimal ¿	p. 207
5.4 The Lewis Method	p. 210
5.4.1 Application to Other Payoffs	p. 214
5.5 The Attari Method	p. 215
5.6 The Convolution Method	p. 216
5.7 The Cosine Method	p. 220
5.8 Comparison, Stability and Performance	p. 228
5.8.1 Other Issues	p. 233
5.9 Extending the Methods to Forward Start Options	p. 235
5.9.1 Forward Characteristic Function for Lévy Processes and CIR Time Change	p. 238
5.9.2 Forward Characteristic Function for Lévy Processes and Gamma-OU Time Change	p. 239
5.9.3 Results	p. 242
5.10 Density Recovery	p. 245
5.11 Summary and Conclusions	p. 250
6 Advanced Topics Using Transform Techniques	p. 253
6.1 Introduction and Objectives	p. 253
6.2 Pricing Non-Standard Vanilla Options	p. 253
6.2.1 FFT with Lewis Method	p. 254
6.3 Bermudan and American Options	p. 254
6.3.1 The Convolution Method	p. 257
6.3.2 The Cosine Method	p. 258
6.3.3 Numerical Results	p. 266
6.3.4 The Fourier Space Time-Stepping	p. 270
6.4 The Cosine Method and Barrier Options	p. 277
6.5 Greeks	p. 278
6.6 Summary and Conclusions	p. 287
7 Monte Carlo Simulation and Applications	p. 289
7.1 Introduction and Objectives	p. 289
7.2 Sampling Diffusion Processes	p. 289
7.2.1 The Exact Scheme	p. 290
7.2.2 The Euler Scheme	p. 290
7.2.3 The Predictor-Corrector Scheme	p. 290
7.2.4 The Milstein Scheme	p. 291
7.2.5 Implementation and Results	p. 291
7.3 Special Purpose Schemes	p. 292
7.3.1 Schemes for the Heston Model	p. 294
7.3.2 Unbiased Scheme for the SABR Model	p. 300
7.4 Adding Jumps	p. 313
7.4.1 Jump Models - Poisson Processes	p. 313
7.4.2 Fixed Grid Sampling (FGS)	p. 315
7.4.3 Stochastic Grid Sampling (SGS)	p. 315
7.4.4 Simulation-Lévy Models	p. 322
7.4.5 Schemes for Lévy Models with Stochastic Volatility	p. 330
7.5 Bridge Sampling	p. 339
7.6 Libor Market Model	p. 346
7.7 Multi-Dimensional Lévy Models	p. 351
7.8 Copulae	p. 352
7.8.1 Distributional Sampling Approach (DSA)	p. 353
7.8.2 Conditional Sampling Approach (CSA)	p. 356
7.8.3 Simulation from Other Copulae	p. 358
7.9 Summary and Conclusions	p. 359
8 Monte Carlo Simulation - Advanced Issues	p. 361
8.1 Introduction and Objectives	p. 361
8.2 Monte Carlo and Early Exercise	p. 361
8.2.1 Longstaff-Schwarz Regression	p. 362
8.2.2 Policy Iteration Methods	p. 369
8.2.3 Upper Bounds	p. 374
8.2.4 Problems of the Method	p. 376
8.2.5 Financial Examples and Numerical Results	p. 378
8.3 Greeks with Monte Carlo	p. 382
8.3.1 The Finite Difference Method (EDM)	p. 383
8.3.2 The Pathwise Method	p. 385
8.3.3 The Affine Recursion Problem (ARP)	p. 389
8.3.4 Adjoint Method	p. 391
8.3.5 Bermudan ARPs	p. 393
8.4 Euler Schemes and General Greeks	p. 396
8.4.1 SDE of Diffusions	p. 396
8.4.2 Approximation by Euler Schemes	p. 397
8.4.3 Approximating General Greeks Using ARP	p. 397
8.4.4 Greeks	p. 404
8.5 Application to Trigger Swap	p. 407
8.5.1 Mathematical Modelling	p. 408
8.5.2 Numerical Results	p. 410
8.5.3 The Likelihood Ratio Method (LRM)	p. 413
8.5.4 Likelihood Ratio for Finite Differences - Proxy Simulation	p. 416
8.5.5 Numerical Results	p. 419
8.6 Summary and Conclusions	p. 433
8.7 Appendix-Trees	p. 434
9 Calibration and Optimization	p. 435
9.1 Introduction and Objectives	p. 435
9.2 The Nelder-Mead Method	p. 437
9.2.1 Implementation	p. 442
9.2.2 Calibration Examples	p. 444
9.3 The Levenberg-Marquardt Method	p. 449
9.3.1 Implementation	p. 453
9.3.2 Calibration Examples	p. 455
9.4 The L-BFGS Method	p. 460
9.4.1 Implementation	p. 463
9.4.2 Calibration Examples	p. 464
9.5 The SQP Method	p. 468
9.5.1 The Modified and Globally Convergent SQP Iteration	p. 473
9.5.2 Implementation	p. 475
9.5.3 Calibration Examples	p. 477
9.6 Differential Evolution	p. 482
9.6.1 Implementation	p. 487
9.6.2 Calibration Examples	p. 488
9.7 Simulated Aimealing	p. 493
9.7.1 Implementation	p. 497
9.7.2 Calibration Examples	p. 500
9.8 Summary and Conclusions	p. 505
10 Model Risk - Calibration, Pricing and Hedging	p. 507
10.1 Introduction and Objectives	p. 507
10.2 Calibration	p. 508
10.2.1 Similarities-Heston and Bates Models	p. 508
10.2.2 Parameter Stability	p. 511
10.3 Pricing Exotic Options	p. 521
10.3.1 Exotic Options and Different Models	p. 528
10.4 Hedging	p. 528
10.4.1 Hedging - The Basics	p. 531
10.4.2 Hedging in Incomplete Markets	p. 533
10.4.3 Discrete Time Hedging	p. 541
10.4.4 Numerical Examples	p. 544
10.5 Summary and Conclusions	p. 550
Part III Implementation, Software Design and Mathematics
11 Matlab-Basics	p. 553
11.1 Introduction and Objectives	p. 553
11.2 General Remarks	p. 553
11.3 Matrices, Vectors and Cell Arrays	p. 556
11.3.1 Matrices and Vectors	p. 556
11.3.2 Cell Arrays	p. 562
11.4 Functions and Function Handles	p. 564
11.4.1 Functions	p. 564
11.4.2 Function Handles	p. 567
11.5 Toolboxes	p. 570
11.5.1 Financial	p. 570
11.5.2 Financial Derivatives	p. 571
11.5.3 Fixed-Income	p. 571
11.5.4 Optimization	p. 573
11.5.5 Global Optimization	p. 577
11.5.6 Statistics	p. 578
11.5.7 Portfolio Optimization	p. 581
11.6 Useful Functions and Methods	p. 589
11.6.1 FFT	p. 589
11.6.2 Solving Equations and ODE	p. 589
11.6.3 Useful Functions	p. 591
11.7 Plotting	p. 593
11.7.1 Two-Dimensional Plots	p. 593
11.7.2 Three-Dimensional Plots - Surfaces	p. 595
11.8 Summary and Conclusions	p. 597
12 Matlab - Object Oriented Development	p. 599
12.1 Introduction and Objectives	p. 599
12.2 The Matlab OO Model	p. 599
12.2.1 Classes	p. 599
12.2.2 Handling Classes in Matlab	p. 606
12.2.3 Inheritance, Base Classes and Superclasses	p. 607
12.2.4 Handle and Value Classes	p. 609
12.2.5 Overloading	p. 610
12.3 A Model Class Hierarchy	p. 611
12.4 A Pricer Class Hierarchy	p. 613
12.5 An Optimizer Class Hierarchy	p. 618
12.6 Design Patterns	p. 620
12.6.1 The Builder Pattern	p. 621
12.6.2 The Visitor Pattern	p. 624
12.6.3 The Strategy Pattern	p. 626
12.7 Example - Calibration Engine	p. 629
12.7.1 Calibrating a Data Set or a History	p. 631
12.8 Example - The Libor Market Model and Greeks	p. 634
12.8.1 An Abstract Class for LMM Derivatives	p. 634
12.8.2 A Class for Bermudan Swaptions	p. 637
12.8.3 A Class for Trigger Swaps	p. 639

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