Derivatives and Internal Models (eBook)
XXXII, 897 Seiten
Springer International Publishing (Verlag)
978-3-030-22899-6 (ISBN)
Now in its fifth edition, Derivatives and Internal Models provides a comprehensive and thorough introduction to derivative pricing, risk management and portfolio optimization, covering all relevant topics with enough hands-on, depth of detail to enable readers to develop their own pricing and risk tools.
The book provides insight into modern market risk quantification methods such as variance-covariance, historical simulation, Monte Carlo, hedge ratios, etc., including time series analysis and statistical concepts such as GARCH Models or Chi-Square-distributions. It shows how optimal trading decisions can be deduced once risk has been quantified by introducing risk-adjusted performance measures and a complete presentation of modern quantitative portfolio optimization. Furthermore, all the important modern derivatives and their pricing methods are presented; from basic discounted cash flow methods to Black-Scholes, binomial trees, differential equations, finite difference schemes, Monte Carlo methods, Martingales and Numeraires, terms structure models, etc.
The fifth edition of this classic finance book has been comprehensively reviewed. New chapters/content cover multicurve bootstrapping, the valuation and hedging of credit default risk that is inherently incorporated in every derivative-both of which are direct and permanent consequences of the financial crises with a large impact on our understanding of modern derivative valuation.
The book will be accompanied by downloadable Excel spread sheets, which demonstrate how the theoretical concepts explained in the book can be turned into valuable algorithms and applications and will serve as an excellent starting point for the reader's own bespoke solutions for valuation and risk management systems.
Hans-Peter Deutsch is one of the founders of d-fine, a leading financial services consulting firm in Europe. Previously, he was a Partner at Arthur Andersen and head of Andersen's Financial and Commodity Risk Consulting (FCRC) in Germany, which he founded in 1997. He holds a PhD in theoretical physics and is the author of roughly 20 international scientific publications in this field. He is also author of many publications in the field of mathematical finance including books on quantitative portfolio management, derivatives pricing and risk management. For many years Dr. Deutsch has been a guest lecturer and member of the Advisory Board of the Mathematical Finance Programme at the University of Oxford, UK, and also Chairman of the Advisory Board of the MathFinance Institute at Johann Wolfgang Goethe-Universität in Frankfurt, Germany. He was also a member of the supervisory board of GET-Capital AG, a German asset management firm, which manages large asset portfolios for institutional investors using a state of the art quantitative method and software system developed by Dr. Deutsch himself.
Mark W. Beinker serves as Managing Director at d-fine GmbH. Dr. Beinker is responsible for the financial engineering business unit and manages projects in development and implementation of models for valuation and risk sensitivity calculation of structured financial products, approval of valuation methods and tools, risk analysis and re-engineering of financial transactions, integration of valuation libraries into the existing system environment, introduction of innovative financial products, development of hedging strategies and the outsourcing of product valuation services. In addition, Dr. Beinker is responsible for the valuation platform MoCo. His professional career began at Arthur Andersen where he held the position of Manager within in the Financial and Commodity Risk Consulting (FCRC) group in Germany. Though he participated in a broad range of projects throughout is business career, the valuation of financial derivatives has been his focus since 1997. Dr. Beinker earned his PhD at the universities of TU Dresden and Duke University, USA.
Preface to Fifth Edition 6
Preface to Fourth Edition 8
Contents 10
List of Figures 25
List of Tables 29
Part I Fundamentals 33
1 Introduction 34
2 Fundamental Risk Factors of Financial Markets 38
2.1 Interest Rates 38
2.1.1 Day Count Conventions 39
2.1.2 Business Day Conventions 43
2.1.3 Discount Factors 44
2.1.4 Compounding Methods 45
Simple Compounding 46
Discrete Compounding 47
Continuous Compounding 48
Linear Compounding 48
Convention-Dependent Interest Rates 49
2.1.5 Spot Rates 51
2.1.6 Forward Rates 51
2.2 Market Prices 55
2.3 An Intuitive Model for Financial Risk Factors 56
2.3.1 Random Walks as the Basis for Pricing and Risk Models 56
2.3.2 Risk Factors as Random Walks 59
2.4 Ito Processes and Stochastic Analysis 67
2.4.1 General Diffusion Processes 67
2.4.2 Ito's Lemma 68
The Process for the Risk Factor Itself 70
The Process for the Risk Factor over a Finite Time Interval 71
The Drift and the Expected Return 73
2.4.3 Transition Probabilities, Forward and Backward Equation 75
The Forward Equation 75
The Backward Equation 76
A Derivation of the Forward and Backward Equations 77
2.4.4 Forward and Backward Equation in the Black-Scholes World 83
3 Financial Instruments: A System of Derivatives and Underlyings 85
3.1 Issuer and Counterparties 85
3.2 Spot Transactions 86
3.3 Forward Transactions 87
3.3.1 Forward Rate Agreements and Forwards 87
3.3.2 Financial Futures 88
3.3.3 Swaps 90
3.4 Options 91
3.5 Classification of Fixed Income Securities 94
3.5.1 Money Market Securities 95
Discount Papers 96
Interest Bearing Securities 96
Time and Notice Deposits 97
Trading Conventions for Money Market Instruments 98
3.5.2 Capital Market Securities 99
Part II Methods 102
4 Overview of the Assumptions 103
5 Present Value Methods, Yields and Traditional Risk Measures 106
5.1 Present Value and Yield to Maturity 106
5.2 Internal Rate of Return and Net Present Value 108
5.3 Accrued Interest, Residual Debt and Par Rates 112
5.4 Traditional Sensitivities of Interest Rate Instruments 116
5.4.1 Average Lifetime and Macaulay Duration 116
5.4.2 Modified Duration and Convexity 117
5.4.3 Summation of Traditional Sensitivities 122
6 Arbitrage 124
6.1 Forward and Futures Contracts 124
6.1.1 Forward Price and Cash and Carry Arbitrage 124
6.1.2 The Stochastic Process for the Forward Price 127
6.1.3 Forward Positions 128
6.1.4 Futures Positions and Basis Risk 128
6.2 Options 129
6.2.1 Upper and Lower Bounds for Option Prices 129
6.2.2 Early Exercise of American Options 131
6.2.3 Relationships Between Puts and Calls 132
7 The Black-Scholes Differential Equation 134
7.1 Derivation of the Black-Scholes Equation from Arbitrage Arguments 135
7.1.1 The Black-Scholes Equation for European Options 136
7.1.2 The Black-Scholes Inequality for American Options 138
7.1.3 A First Contact with the Risk-Neutral World 141
7.2 The Black-Scholes Equation and the Backward Equation 142
7.2.1 A Second Contact with the Risk-Neutral World 145
7.3 The Relationship to the Heat Equation 146
8 Integral Forms and Analytic Solutions in the Black-Scholes World 150
8.1 Option Prices as Solutions of the Heat Equation 151
8.2 Option Prices and Transition Probabilities 153
8.3 Compilation of Black-Scholes Option Prices for Different Underlyings 157
8.3.1 Options on the Spot Price 157
8.3.2 Options on the Forward Price 158
Options on Futures 158
Options on Forwards 160
8.3.3 Options on Interest Rates 161
Forward Volatilities 161
Normally Versus Log Normally Distributed Interest Rates 162
Black-76 Model for Interest Rate Options 163
9 Binomial and Trinomial Trees 165
9.1 General Trees 166
9.1.1 Evolution of the Underlying and the Replicating Portfolio 166
9.1.2 Evolution of the Derivative 167
9.1.3 Forward Contracts 169
9.2 Recombining Trees 170
9.2.1 The Underlying 170
9.2.2 The Binomial Distribution for European Derivatives 171
9.2.3 A Third Contact with the Risk-Neutral World 175
9.3 Random Walk and Binomial Parameters 179
9.4 The Binomial Model with Infinitesimal Steps 182
9.4.1 Components of the Black-Scholes Option Pricing Formula 184
9.5 Trinomial Trees 185
9.5.1 The Trinomial Tree as an Improved Binomial Tree 188
10 Numerical Solutions Using Finite Differences 190
10.1 Discretizing the Black-Scholes Equation 191
10.1.1 The Explicit Method 192
10.1.2 The Implicit Method 193
10.1.3 Combinations of Explicit and Implicit Methods (Crank-Nicolson) 193
10.1.4 Symmetric Finite Differences of the Underlying Price 195
10.2 Difference Schemes 199
10.2.1 Initial Conditions 203
10.2.2 Dirichlet Boundary Conditions 203
10.2.3 Neumann Boundary Condition 209
10.2.4 Generalized Neumann Boundary Conditions 214
10.2.5 Free Boundary Conditions for American Options 216
The Lamberton and Lapeyre Procedure 218
10.3 Convergence Criteria 221
10.3.1 Improving the Convergence Properties 224
10.4 Discrete Dividends 226
10.5 Example 227
10.5.1 Relationship Between Explicit Finite Difference and Tree Methods 231
11 Monte Carlo Simulations 232
Assumptions 233
11.1 A Simple Example: The Area of a Disk 234
11.2 The General Approach to Monte Carlo Simulations 238
11.3 Monte Carlo Simulation of Risk Factors 239
11.3.1 Simulation of the Evolution of a Single Risk Factor 239
11.3.2 Simulation of Several Correlated Risk Factors 243
11.4 Pricing 247
11.5 American Monte Carlo 248
12 Hedging 251
12.1 Hedging Derivatives with Spot Transactions 251
12.1.1 Hedging of Forwards and Futures 254
12.2 Hedging Derivatives with Forward Contracts 256
12.2.1 Hedging with Forwards 256
12.2.2 Hedging with Futures 259
12.3 Hedge-Ratios for Arbitrary Combinations of Financial Instruments 260
12.4 ``Greek'' Risk Management with Sensitivities 263
12.4.1 Sensitivities and a Portfolio's Change in Value 263
12.4.2 Omega and Beta 267
12.4.3 Summation of Sensitivities of Different Underlyings 269
12.5 Computation of the Greek Risk Variables 271
12.5.1 Sensitivities in the Binomial Model 271
12.5.2 Sensitivities in the Black-Scholes Model 273
12.5.3 Sensitivities by Means of Finite Difference Methods 274
12.5.4 Sensitivities by Means of Monte Carlo Simulations 275
13 Martingales and Numeraires 276
13.1 The Martingale Property 276
13.2 The Numeraire 278
13.3 Self-financing Portfolio Strategies 283
13.4 Generalization to Continuous Time 286
13.5 The Drift 297
13.6 The Market Price of Risk 301
13.7 Tradable Underlyings 303
13.8 Applications in the Black-Scholes World 305
14 Interest Rates and Term Structure Models 310
14.1 Instantaneous Spot Rates and Instantaneous Forward Rates 311
14.2 Important Numeraire Instruments 313
14.2.1 The Risk-Neutral Measure 314
14.2.2 The Forward-Neutral Measure 316
14.3 The Special Case of Deterministic Interest Rates 317
14.4 Tradable and Non-tradable Variables 319
14.5 Convexity Adjustments 322
14.5.1 In-Arrears Swaps 325
14.5.2 Money Market Futures 327
Quotation for Money Market Futures 329
14.6 Arbitrage-Free Interest Rate Trees Grid (Tree) Models 330
14.6.1 Backward Induction 331
14.6.2 Forward Induction and Green's Functions 335
14.7 Market Rates vs. Instantaneous Rates 340
14.7.1 Arrow-Debreu Prices 341
14.7.2 Pricing Caplets Using Arrow-Debreu Prices 345
Practical Implementation of Arrow-Debreu Prices 346
14.8 Explicit Specification of Short Rate Models 347
14.8.1 The Effect of Volatility 348
14.8.2 Normal Models 350
14.8.3 Lognormal Models 353
Exact Reproduction of the Term Structure with the Lognormal Model 356
14.9 The Example Program TermStructureModels.xlsm 357
14.9.1 Construction of Interest Rate Trees and Option Pricing 357
14.9.2 Absolute and Relative Volatilities 360
14.9.3 Calibration of Volatilities 361
14.10 Monte Carlo on the Tree 363
14.11 The Drift in Term Structure Models 365
14.11.1 Heath-Jarrow-Morton Models 365
14.11.2 Short Rate Models 366
14.12 Short Rate Models with Discrete Compounding 371
14.12.1 Normal Models 372
14.12.2 Lognormal Models 373
14.13 Other Interest Rate Models 373
Part III Instruments 375
15 Simple Interest Rate Products 376
15.1 Zero Bonds 377
15.1.1 Cash Flows and Present Value 377
15.1.2 Yield to Maturity and Par Rate 377
15.1.3 Sensitivities 378
15.2 Forward Rate Agreements 378
15.3 Coupon Bonds 381
15.3.1 Cash Flows and Present Value 381
15.3.2 Yield to Maturity 382
15.3.3 Par Rates 384
15.3.4 Sensitivities 386
15.4 Forward Bonds 387
15.4.1 Present Value 387
15.4.2 Forward Par Rate and Forward Yield to Maturity 387
15.5 Interest Rate Futures 388
15.5.1 Futures on Zero Bonds 388
15.5.2 Futures on Coupon Bonds 389
15.6 Floaters 392
15.6.1 Cash Flows and Present Value 393
15.6.2 Yield to Maturity, Par Rate and Sensitivities 396
15.7 Swaps 396
15.7.1 Cash Flows and Present Value 397
15.7.2 Par Swap Rate and Yield to Maturity 399
15.7.3 Sensitivities 402
15.8 Forward Swaps 403
15.8.1 Present Value 404
15.8.2 Forward Par Swap Rates 404
15.8.3 Annuity as Numeraire 406
16 FX Derivatives 409
16.1 FX Forward Rate and Cross Currency Basis 409
16.2 FX Swaps 410
16.2.1 Short Term Financing of Positions in Foreign Currency 412
16.3 FX Forwards and FX Futures 412
16.4 Cross Currency Swaps 413
16.4.1 Plain Vaniall CCY Swap 413
16.4.2 Mark to Market CCY Swaps 413
16.4.3 Cash Flows and Present Value 414
16.5 CCY Basis 416
17 Variants of Fixed Income Instruments 418
17.1 Basis Swaps 418
17.1.1 Present Value and Cash Flows of Basis Swaps 419
17.1.2 Par Basis Swap Spread 420
17.2 Annuity Loans 420
17.2.1 Cash Flows and Residual Debt 420
17.2.2 Present Value 422
17.2.3 Yield to Maturity and Par Rates 426
17.2.4 Sensitivities 428
17.3 Fixing-in-Arrears 429
17.4 Float Rate with Cap/Floor 429
17.5 Call Rights and Break Clause 430
17.6 Reverse Floater 430
17.7 Constant Maturity Swaps 431
18 Plain Vanilla Options 433
18.1 Traditional and General Definition of an Option 433
18.2 Conventions 434
18.3 Options on Spot and Forward Prices 435
18.3.1 European Options 435
18.3.2 American Options 438
18.4 Index Options and Index Futures 440
18.5 Foreign Exchange Options 441
18.5.1 Put-Call-Equivalence for FX Options 441
18.6 Interest Rate Options 442
18.6.1 Options on Bonds 442
18.6.2 Options on Bond Futures 443
18.6.3 Caps and Floors 444
Valuation as Interest Rate Options (Lognormally Distributed Interest Rates) 445
Valuation as Bond Options (Normally Distributed Interest Rates) 447
Collars and Put-Call Parity for Caps and Floors 449
18.6.4 Swaptions 450
Valuation as Interest Rate Options (Lognormally Distributed Interest Rates) 452
Valuation as Bond Options (Normally Distributed Interest Rates) 455
Put-Call Parity for Swaptions 456
Bermudan Swaptions 456
19 Exotic Options 458
19.1 Payoff Profiles for Selected Exotics 458
19.1.1 Power Options 458
19.1.2 Cliquet and Coupe Options 459
19.1.3 Look-Back Options 460
19.1.4 Asian Options 460
19.1.5 Rainbow and Exchange Options 461
19.1.6 Basket-Options 461
19.1.7 Compound and Bermuda Options 462
19.2 Black-Scholes for Exotics 463
19.2.1 Pay-Later Options 463
19.2.2 Digital Options 465
Cash-or-Nothing 465
Asset-or-Nothing 467
Range Floater 468
19.2.3 Barrier Options 468
Double Barrier Option 473
19.2.4 Ladder Options 474
19.3 Numerical Pricing Methods for Exotics 476
19.3.1 Monte Carlo for European Exotics 478
19.3.2 The Binomial Model for American Exotics 482
Recombinant Trees for Path-Independent Options 482
Binomial Bushy Trees for Path-Dependent Options 484
20 Credit Risk 487
20.1 Expected Positive Exposure, Probability of Default and Loss Given Default 487
20.2 Measures To Reduce Default Risk 491
20.2.1 Collateral Management 491
20.2.2 Central Counterparties 492
20.2.3 Netting Agreements 492
20.2.4 Hedging 492
20.3 Bonds with Default Risk 492
20.4 Credit Spreads 495
20.5 Credit Spread Risk 496
20.6 Credit-Default-Swaps 498
20.6.1 Cashflows and Present Value 500
20.6.2 Approximations 502
20.6.3 Par CDS Rate 503
Part IV Risk 504
21 Fundamentals 505
21.1 Regulatory Requirements 508
21.2 Confidence, Percentile and Risk 509
21.2.1 Other Risk Measures 512
21.3 The Value at Risk of a Single Risk Factor 513
21.4 Approximations in the Distribution of Risk Factors 520
21.5 The Covariance Matrix 522
21.5.1 Logarithmic Changes Versus Returns 525
21.5.2 Covariance Matrices of Data Providers 526
21.5.3 Cholesky Decomposition of the Covariance Matrix 528
Transformation of Uncorrelated Random Variables into Correlated Random Variables 528
Transformation of Correlated Random Variables into Uncorrelated Random Variables 530
The Cholesky Decomposition 532
22 The Variance-Covariance Method 534
22.1 Portfolios vs. Financial Instruments 537
22.2 The Delta-Normal Method 539
22.2.1 The Value at Risk with Respect to a Single Risk Factor 540
22.2.2 The Value at Risk with Respect to Several Risk Factors 541
22.3 The Delta-Gamma Method 545
22.3.1 Decoupling of the Risk Factors 546
22.3.2 Diagonalization of the Gamma Matrix 547
22.3.3 The Distribution of the Portfolio Value Changes 553
22.3.4 Moments of the Portfolio Value Distribution 556
Johnson Transformation 564
Cornish-Fisher Expansion 565
22.3.5 Fourier-Transformation of the Portfolio Value Distribution 566
22.3.6 Monte Carlo Simulation of the Portfolio Value Distribution 569
23 Simulation Methods 571
23.1 Monte Carlo Simulation 571
23.1.1 The Risk Factors as Correlated Random Walks 572
23.1.2 Structured Monte Carlo 573
Simulation 573
Evaluation 574
23.2 Historical Simulation 575
23.3 Crash and Stress Testing: Worst Case Scenarios 577
23.4 Advantages and Disadvantages of the Commonly Used Value at Risk Methods 578
24 Example of a VaR Computation 580
24.1 The Portfolio 580
24.2 Market Data 581
24.3 Calculation of Risk 583
25 Backtesting: Checking the Applied Methods 585
25.1 Profit and Loss Computations 585
25.2 The Traffic Light Approach of the Supervising Authorities 587
25.2.1 Adjusting the Value at Risk (Yellow Zone) 587
25.2.2 Criteria for Rejecting a Model (Red Zone) 589
25.2.3 The Green Zone 591
25.2.4 Multiplication Factor and Add-On 592
Part V Portfolios 594
26 Classical Portfolio Management 595
26.1 From Risk Management to Portfolio Management 596
26.1.1 Assets and Risk Factors 596
26.1.2 Portfolio Risk and Volatility 599
26.1.3 Risk Contribution and Attribution 604
26.2 Portfolio Optimization 606
26.2.1 The Minimum Risk Portfolio 606
26.2.2 The Efficient Frontier 609
Maximizing the Expected Return 609
Minimizing the Risk 615
26.2.3 The Sharpe Ratio and the Optimal Portfolio 618
A Word of Caution 621
26.2.4 The Capital Market Line 622
Being Not Fully Invested 624
26.3 Alternative Portfolio Management Approaches 626
26.3.1 Value Investing 626
26.3.2 Behavioral Finance 626
26.3.3 Chart Analysis 627
27 Attributes and Their Characteristic Portfolios 628
27.1 General Properties of Characteristic Portfolios 630
27.1.1 Relations Involving Several Characteristic Portfolios 631
27.2 The Leverage 633
27.3 The Excess Return 634
27.4 The Optimal Portfolio 638
27.5 The Efficient Frontier Revisited 641
28 Active Management and Benchmarking 647
28.1 The Capital Asset Pricing Model 647
28.2 Theory of Efficient Markets 649
28.3 Benchmarking Against an Index 650
28.3.1 Active Portfolio Properties 652
28.3.2 Residual Portfolio Properties 654
28.4 Benchmark and Characteristic Portfolios 655
28.4.1 The Fully Invested Minimal Risk Portfolio 657
28.4.2 The Characteristic Portfolio for Beta 658
28.4.3 The Characteristic Portfolio for Alpha 660
28.5 Relations Between Sharpe Ratio and Information Ratio 665
28.5.1 The Market Portfolio 665
28.5.2 The Characteristic Portfolio of the Excess Return 668
Part VI Market Data 670
29 Construction of the Yield Curve Universe 671
29.1 Required Features of Discount Curves 672
29.2 Modeling the Yield Curve 674
29.2.1 Interpolation Methods 674
Linear Interpolation 675
Constant Interpolation 675
Other Interpolation Methods 676
29.2.2 Extrapolation Methods 677
29.3 Parametric Yield Curves 678
Nelson-Siegel-Procedure 679
29.4 Construction of the Curve Hierarchy 680
29.5 Selection of Benchmark Instruments 682
29.5.1 Risk Free Interest Rates 683
29.6 Determination of the Discount Factor Curve 684
29.6.1 Bootstrapping 684
29.6.2 Optimisation 691
29.7 Forward Curves 692
29.8 Cross Currency Curves 695
29.9 Survival Probabilities 696
29.9.1 CDS Spread Curves 697
29.9.2 Synthetic CDS Spreads 697
CDS Spread Component Due to Expected Losses 698
CDS Spread Component Due to Unexpected Losses 700
Liquidity Premium on Top of the CDS Spread 701
Calibrating the Parameters 701
29.10 The Old Yield Curve World 701
29.11 The Future of Yield Curves 702
30 Volatility 704
30.1 Implied Volatilities 704
30.1.1 Smiles and Volatility Indices 704
30.2 Local Volatility Surfaces 707
30.2.1 Implicit Transition Probabilities 708
30.2.2 Implicit Local Volatility Surfaces 711
30.3 Volatility Transformations 716
30.3.1 Transformation Between Relative and Absolute Volatility 716
30.3.2 Summation of Volatilities 718
30.3.3 Transformation Between Yield and Price Volatility 719
30.3.4 Currency Transformation of Volatilities and Correlations 721
The General Transformations for All Risk Factors 721
Transformations for the Exchange Rates and Cross Rates 726
31 Market Parameter from Historical Time Series 733
31.1 Historical Averages as Estimates for Expected Values 734
31.2 Error Estimates 736
31.2.1 Uncorrelated Measurements 737
31.2.2 Error of Autocorrelated Measurements 743
Autocorrelation and Autocovariance 744
Autocorrelation Time and Error Estimates 745
31.3 Return and Covariance Estimates 747
31.3.1 Return Estimates 747
The Moving Average (MA) Estimator 748
Two Alternatives for the Moving Average 750
The Exponentially Weighted Moving Average (EWMA) 751
31.3.2 Covariance Estimates 752
32 Time Series Modeling 754
32.1 Stationary Time Series and Autoregressive Models 757
32.1.1 AR(p) Processes 758
The Autoregressive Process of First Order 762
32.1.2 Univariate GARCH(p,q) Processes 764
32.1.3 Simulation of GARCH Processes 768
32.2 Calibration of Time Series Models 769
32.2.1 Parameter Estimation for AR(p) Processes 770
32.2.2 Parameter Estimation for GARCH(p,q) Processes 773
32.2.3 Simulated Annealing 775
33 Forecasting with Time Series Models 778
33.1 Forecasting with Autoregressive Models 779
33.2 Volatility Forecasts with GARCH(p,q) Processes 782
33.2.1 Forecast Over Several Time Steps 782
The One-Step Forecast 782
The Two-Step Forecast 783
The Three-Step Forecast 785
The Forecast for h Steps 786
33.2.2 Forecast for the Total Variance 787
33.2.3 Volatility Term Structure 788
33.3 Volatility Forecasts with GARCH (1,1) Processes 789
33.4 Volatility Forecasts with Moving Averages 791
34 Principal Component Analysis 794
34.1 The General Procedure 794
34.2 Principal Component Analysis of the German Term Structure 802
35 Pre-Treatment of Time Series and Assessment of Models 806
35.1 Pre-Treatment of Time Series 806
35.1.1 Differencing 807
35.1.2 Filters 808
35.1.3 Scaling 810
35.2 Measuring the Goodness of Time Series Models 811
35.2.1 Hypothesis Tests 812
Sign Test 812
Inflection Points Test 813
Percentile Test and Kuiper Statistic 813
Estimation of the Autocorrelation Function 814
QQ Plot 815
35.2.2 Goodness of Fit vs. Goodness of Forecast 817
35.2.3 Examples: Goodness of GARCH Models 818
Correction to: Derivatives and Internal Models 824
A Probability and Statistics 825
A.1 Probability, Expectation and Variance 825
A.2 Multivariate Distributions, Covariance, Correlation and Beta 828
A.3 Moments and Characteristic Functions 832
A.3.1 Moment Generating Functions 833
A.3.2 Characteristic Functions 836
A.4 A Collection of Important Distributions 838
A.4.1 The Uniform Distribution 838
A.4.2 The Binomial Distribution and the Bernoulli Experiment 840
A.4.3 The Normal Distribution and the Central Limit Theorem 843
The Multivariate Normal Distribution 850
A.4.4 The Lognormal Distribution 852
A.4.5 The Gamma Distribution 853
A.4.6 The ?2-Distribution 856
The Density of the ?2-Distribution with One Degree of Freedom 861
The Non-central ?2-Distribution 864
A.5 Transformations Between Distributions 866
A.5.1 Summations 866
A.5.2 Box-Muller Transformations 869
A.5.3 Inversion of Cumulative Distribution Functions 869
References 873
Index 882
| Erscheint lt. Verlag | 8.10.2019 |
|---|---|
| Reihe/Serie | Finance and Capital Markets Series | Finance and Capital Markets Series |
| Zusatzinfo | XXXII, 897 p. 39 illus. |
| Sprache | englisch |
| Themenwelt | Wirtschaft ► Betriebswirtschaft / Management ► Finanzierung |
| Schlagworte | Benchmarking • Cash Flow • Derivatives • Financial Instruments • Financial Market • Hedging • Investments and Securities • Management • options • Portfolio • Portfolio Management • Pricing • Risk Management • Statistics • Volatility |
| ISBN-10 | 3-030-22899-1 / 3030228991 |
| ISBN-13 | 978-3-030-22899-6 / 9783030228996 |
| Haben Sie eine Frage zum Produkt? |
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