Table of Contents
ACKNOWLEDGEMENTS ....................................................................... xix
PART I: INTRODUCTION. OVERVIEW. AND EXERCISE ....................... 1
1 . Introduction and Outline ...................................................................... 3
Who/ Howmat, “Tech . Index”, Messages, Personal Note ................. 3
Summary Outline: Book Contents ....................................................... 5
Overview (Tech . Index 1/10) ............................................................... 7
Objectives of Quantitative Finance and Risk Management .................. 7
Tools of Quantitative Finance and Risk Management .......................... 9
The Traditional Areas of Risk Management ....................................... 11
When Will We Ever See Real-Time Color Movies of Risk? .............. 13
Quants in Quantitative Finance and Risk Management ...................... 15
References .......................................................................................... 17
2 .
Many People Participate in Risk Management ................................... 13
3 . An Exercise (Tech . Index 1/10) ......................................................... 19
Part #1: Data. Statistics. and Reporting Using a Spreadsheet ............. 19
Part #2: Repeat Part #1 Using Programming ...................................... 22
Part #3: A Few Quick and Tricky Hypothetical Questions ................. 23
Messages and Advice ......................................................................... 24
References .......................................................................................... 24
PART II: RISK LAB (NUTS AND BOLTS OF RISK MANAGEMENT) ..... 25
Equity Options (Tech . Index 3/10) ..................................................... 27 4 .
Pricing and Hedging One Option ....................................................... 27
American Options .............................................................................. 30
Basket Options and Index Options ..................................................... 31
Other Types of Equity Options; Exotics ............................................. 33
Portfolio Risk (Introduction) .............................................................. 33
Scenario Analysis (Introduction) ........................................................ 33
References .......................................................................................... 34
vii
viii Quantitative Finance and Risk Management
5 . FX Options (Tech . Index 4/10) .......................................................... 35
FX Forwards and Options .................................................................. 35
Some Practical Details for FX Options .............................................. 38
Hedging FX Options with Greeks: Details and Ambiguities .............. 39
FX Volatility Skew and/or Smile ....................................................... 41
Pricing Barrier Options with Skew ..................................................... 45
Double Barrier Option: Practical Example ......................................... 47
The “Two-Country Paradox” ............................................................. 48
Quanto Options and Correlations ....................................................... 50
FX Options in the presence of Stochastic Interest Rates .................... 51
Numerical Codes, Closed Form Sanity Checks, and Intuition ............ 51
References .......................................................................................... 52
6 . Equity Volatility Skew (Tech . Index 6/10) ......................................... 53
Put-Call Parity: Theory and Violations .............................................. 54
The Volatility Surface ........................................................................ 55
Dealing with Skew ............................................................................. 55
Perturbative Skew and Barrier Options .............................................. 56
Static Replication ............................................................................... 58
Stochastic Volatility ........................................................................... 60
Local Volatility and Skew .................................................................. 62
The Skew-Implied Probability Distribution ....................................... 63
Local vs . Implied Volatility Skew; Derman’s Rules of Thumb .......... 63
Intuitive Models and Different Volatility Regimes ............................ 68
Jump Diffusion Processes .................................................................. 69
Appendix A: Algorithm for “Perturbative Skew” Approach .............. 69
Option Replication with Gadgets ....................................................... 65
The Macro-Micro Model and Equity Volatility Regimes ................... 69
Appendix B: A Technical Issue for Stochastic Volatility ................... 71
References .......................................................................................... 72
7 . Forward Curves (Tech . Index 4/10) ................................................... 73
Market Input Rates ............................................................................. 73
Construction of the Forward-Rate Curve ........................................... 76
References .......................................................................................... 83
8 . Interest-Rate Swaps (Tech . Index 3/10) ............................................. 85
Swaps: Pricing and Risk ..................................................................... 85
Interest Rate Swaps: Pricing and Risk Details .................................... 91
Counterparty Credit Risk and Swaps ............................................... 107
References ........................................................................................ 109
9 . Bonds: An Overview (Tech . Index 2/10) ......................................... 111
Types of Bonds ................................................................................ 111
Bond Issuance .................................................................................. 115
Table of Contents ix
Bond Trading ................................................................................... 116
Bond Math ....................................................................................... 118
References ........................................................................................ 121
10 . Interest-Rate Caps (Tech . Index 4/10) .............................................. 123
Introduction to Caps ......................................................................... 123
The Black Caplet Formula ............................................................... 125
Non-USD Caps ................................................................................ 127
Relations between Caps. Floors. and Swaps ..................................... 127
Hedging Delta and Gamma for Libor Caps ...................................... 128
Hedging Volatility and Vega Ladders .............................................. 129
Prime Caps and a Vega Trap ............................................................ 131
References ........................................................................................ 136
Matrices of Cap Prices ..................................................................... 131
CMT Rates and Volatility Dependence of CMT Products ............... 132
11 . Interest-Rate Swaptions (Tech . Index 5/10) ..................................... 137
European Swaptions ........................................................ ~ ................. 137
Delta and Vega Risk: Move Inputs or Forwards? ............................. 143
Swaptions and Corporate Liability Risk Management ..................... 144
Practical Example: A Deal Involving a Swaption ............................ 146
BermuddAmerican Swaption Pricing .............................................. 141
Miscellaneous Swaption Topics ....................................................... 148
References ........................................................................................ 150
12 . Portfolios and Scenarios (Tech . Index 3/10) .................................... 151
Introduction to Portfolio Risk Using Scenario Analysis ................... 1.51
Definitions of Portfolios ................................................................... 151
Definitions of Scenarios ................................................................... 153
Many Portfolios and Scenarios ......................................................... 155
A Scenario Simulator ....................................................................... 157
Risk Analyses and Presentations ...................................................... 157
PART Ill: EXOTICS. DEALS. AND CASE STUDIES ............................ 159
13 . A Complex CVR Option (Tech . Index 5/ 10) .................................... 161
CVR Starting Point: A Put Spread ................................................... 162
A Simplified CVR: Two Put Spreads with Extension Logic ............ 165
The M&A Scenario .......................................................................... 161
CVR Extension Options and Other Complications ........................... 162
The Arbs and the Mispricing of the CVR Option ............................. 164
Non- Academic Corporate Decision for Option Extension ............... 167
The CVR Option Pricing .................................................................. 169
Analytic CVR Pricing Methodology ................................................ 173
X Quantitative Finance and Risk Management
Some Practical Aspects of CVR Pricing and Hedging ..................... 176
The CVR Buyback ........................................................................... 180
A Legal Event Related to the CVR .................................................. 180
References ........................................................................................ 180
14 . Two More Case Studies (Tech . Index 5/10) ..................................... 183
D123 : The Complex DEC Synthetic Convertible .............................. 188
Case Study: DECS and Synthetic Convertibles ................................ 183
Case Study: Equity Call with Variable Strike and Expiration .......... 193
References ........................................................................................ 199
15 . More Exotics and Risk (Tech . Index 5/10) ....................................... 201
Contingent Caps ............................................................................... 201
Digital Options: Pricing and Hedging .............................................. 205
Historical Simulations and Hedging ................................................. 207
Yield-Curve Shape and Principle-Component Options .................... 209
Principal-Component Risk Measures (Tilt Delta etc.) ...................... 210
Reload Options ................................................................................ 214
References ........................................................................................ 217
Hybrid 2-Dimensional Barrier Options-Examples ......................... 211
16 . A Pot Pourri of Deals (Tech . Index 5/10) ......................................... 219
TIPS (Treasury Inflation Protected Securities) ................................. 219
Municipal Derivatives. Muni Issuance. Derivative Hedging ............ 221
Resettable Options: Cliquets ............................................................ 226
Power Options .................................................................................. 230
Path-Dependent Options and Monte Carlo Simulation ..................... 231
Periodic Caps ................................................................................... 231
ARM Caps ....................................................................................... 231
Index-Amortizing Swaps ................................................................. 232
A Hypothetical Rep0 + Options Deal ............................................... 236
Convertible Issuance Risk ................................................................ 239
Difference Option on an Equity Index and a Basket of Stocks ......... 224
References ........................................................................................ 241
17 . Single Barrier Options (Tech . Index 6/10) ....................................... 243
Knock-Out Options .......................................................................... 245
The Semi-Group Property including a Barrier ................................. 247
Calculating Barrier Options ............................................................. 248
Knock-In Options ............................................................................. 249
Complicated Barrier Options and Numerical Techniques ................ 252
A Useful Discrete Barrier Approximation ........................................ 252
“Potential Theory” for General Sets of Single Barriers .................... 253
Useful Integrals for Barrier Options ................................................. 251
Table of Contents xi
Barrier Options with Time-Dependent Drifts and Volatilities .......... 255
References ........................................................................................ 256
18 . Double Barrier Options (Tech . Index 7/10) ...................................... 257
Double Barrier Solution with an Infinite Set of Images ................... 258
Double Barrier Option Pricing ......................................................... 260
Rebates for Double Barrier Options ................................................. 262
References ........................................................................................ 263
19 . Hybrid 2-D Barrier Options (Tech . Index 7/10) ............................... 265
Pricing the Barrier 2-Dimension Hybrid Options ............................. 267
Useful Integrals for 2D Barrier Options ........................................... 268
References ........................................................................................ 269
20 . Average-Rate Options (Tech . Index 8/10) ........................................ 271
Arithmetic Average Rate Options in General Gaussian Models ....... 272
Results for Average-Rate Options in the MRG Model ..................... 276
Simple Harmonic Oscillator Derivation for Average Options .......... 277
Thermodynamic Identity Derivation for Average Options .............. 278
Average Options with Log-Normal Rate Dynamics ......................... 278
References ........................................................................................ 280
PART IV: QUANTITATIVE RISK MANAGEMENT ................................ 281
Fat Tail Volatility (Tech . Index 5/10) .............................................. 283
Gaussian Behavior and Deviations from Gaussian ........................... 283
Outliers and Fat Tails ....................................................................... 284
Use of the Equivalent Gaussian Fat-Tail Volatility .......................... 287
Practical Considerations for the Fat-Tail Parameters ....................... 288
References ........................................................................................ 294
Correlation Matrix Formalism; the N-Sphere (Tech . Index 8/10) ... 295
The Importance and Difficulty of Correlation Risk .......................... 295
One Correlation in Two Dimensions ................................................ 296
Two Correlations in Three Dimensions; the Azimuthal Angle ......... 297
Correlations in Four Dimensions ..................................................... 300
Correlations in Five and Higher Dimensions ................................... 301
Spherical Representation of the Cholesky Decomposition ............... 303
Numerical Considerations for the N-Sphere ................................... 304
References ........................................................................................ 305
Stressed Correlations and Random Matrices (Tech . Index 5/10) ...... 307
Correlation Stress Scenarios Using Data .......................................... 307
21
22
23
xii Quantitative Finance and Risk Management
Stressed Random Correlation Matrices ............................................ 313
Random Correlation Matrices Using Historical Data ....................... 313
Stochastic Correlation Matrices Using the W-sphere ....................... 314
24 . Optimally Stressed PD Correlation Matrices (Tech . Index 7/10) ..... 319
Least-Squares Fitting for the Optimal PD Stressed Matrix ............... 321
Numerical Considerations for Optimal PD Stressed Matrix ............. 322
Example of Optimal PD Fit to a NPD Stressed Matrix .................... 323
SVD Algorithm for the Starting PD Correlation Matrix ................... 325
PD Stressed Correlations by Wallung through the Matrix ................ 328
References ........................................................................................ 328
25 . Models for Correlation Dynamics. Uncertainties
(Tech . Index 6/10) ............................................................................ 329
“Just Make the Correlations Zero” Model; Three Versions .............. 329
The Macro-Micro Model for Quasi-Random Correlations ............... 331
Implied. Current. and Historical Correlations for Baskets ................ 338
Plain-Vanilla VAR (Tech . Index 4/10) ............................................. 341
Quadratic Plain-Vanilla VAR and CVARs ...................................... 344
Monte-Carlo VAR ........................................................................... 346
Backtesting ...................................................................................... 347
Monte-Carlo CVARs and the CVAR Volatility ............................... 347
Confidence Levels for Individual Variables in VAR ........................ 350
Correlation Dependence on Volatility .............................................. 335
26 .
References ........................................................................................ 351
27 . ImprovedEnhancedKtressed VAR (Tech . Index 5/10) .................... 353
Improved Plain-Vanilla VAR (IPV-VAR) ........................................ 353
EnhancedStressed VAR (ES-VAR) .................................................. 357
Other VAR Topics ........................................................................... 365
References ........................................................................................ 368
28 . VAR. CVAR. CVAR Volatility Formalism (Tech . Index 7/10) ....... 369
Set-up and Overview of the Formal VAR Results ............................ 369
Calculation of the Generating Function ............................................ 371
VAR, the CVARs, and the CVAR Volatilities ................................. 374
Effective Number of SD for Underlying Variables .......................... 376
Extension to Multiple Time Steps using Path Integrals .................... 378
29 . VAR and CVAR for Two Variables (Tech . Index 5/10) .................. 381
Geometry for Risk Ellipse. VAR Line. CVAR. CVAR Vol ............. 382
The CVAR Volatility with Two Variables ....................................... 381
Table of Contents xiii
30 . Corporate-Level VAR (Tech . Index 3/10) ........................................ 387
Desk CVARs and Correlations between Desk Risks ........................ 389
Aggregation. Desks. and Business Units .......................................... 387
Aged Inventory and Illiquidity ......................................................... 391
31 . Issuer Credit Risk (Tech . Index 5/10) .............................................. 393
Transition/Default Probability Matrices ........................................... 394
Calculation of Issuer Risk-Generic Case ....................................... 399
Example of Issuer Credit Risk Calculation ...................................... 403
Issuer Credit Risk and Market Risk: Separation via Spreads ............ 406
Separating Market and Credit Risk without Double Counting ......... 407
A Unified Credit + Market Risk Model ............................................ 410
References ........................................................................................ 413
32 . Model Risk Overview (Tech . Index 3/10) ........................................ 415
Summary of Model Risk .................................................................. 415
Model Risk and Risk Management .................................................. 416
Time Scales and Models .................................................................. 416
Long-Term Macro Component with Quasi-Random Behavior ........ 417
Liquidity Model Limitations ............................................................ 417
Which Model Should We Use? ........................................................ 418
Model Risk, Model Reserves, and Bid-Offer Spreads ...................... 418
Model Quality Assurance ................................................................. 419
Models and Parameters .................................................................... 419
References ........................................................................................ 420
33 . Model Quality Assurance (Tech . Index 4/10) .................................. 421
Model Quality Assurance Goals. Activities. and Procedures ........... 421
Model QA: Sample Documentation ................................................. 424
User Section of Model QA Documentation ...................................... 425
Quantitative Section of Model QA Documentation .......................... 425
Systems Section of Model QA Documentation ................................ 428
References ........................................................................................ 430
34 . Systems Issues Overview (Tech . Index 2/10) ................................... 431
Advice and a Message to Non-Technical Managers ......................... 431
What are Some Systems Traps and Risks? ....................................... 432
.
What are the “Three-Fives Systems Criteria”? ................................. 431
What is the Fundamental Theorem of Systems? ............................... 432
The Birth and Development of a System .......................................... 433
Systems in Mergers and Startups ..................................................... 435
Vendor Systems ............................................................................... 436
New Paradigms in Systems and Parallel Processing ........................ 437
Languages for Models: Fortran 90, C++, C, and Others .................... 438
What‘s the “Systems Solution”? ....................................................... 440
xiv Quantitative Finance and Risk Management
Are Software Development Problems Unique to Wall Street? ......... 440
References ........................................................................................ 440
35 . Strategic Computing (Tech . Index 3/10) .......................................... 441
Introduction and Background ........................................................... 442
Illustration of Parallel Processing for Finance .................................. 442
Some Aspects of Parallel Processing ............................................... 443
Technology, Strategy and Change .................................................... 446
References ........................................................................................ 447
36 . Qualitative Overview of Data Issues (Tech . Index 2/10) .................. 449
Data Consistency ............................................................................. 449
Data Reliability ................................................................................ 450
Data Vendors ................................................................................... 450
Data Completeness ........................................................................... 450
Historical Data Problems and Data Groups ...................................... 451
Preparation of the Data .................................................................... 451
Bad Data Points and Other Data Traps ............................................. 451
37 . Correlations and Data (Tech . Index 5/10) ........................................ 453
Fluctuations and Uncertainties in Measured Correlations ................ 453
Time Windowing ............................................................................. 454
Correlations, the Number of Data Points, and Variables .................. 456
Intrinsic and Windowing Uncertainties: Example ............................ 458
Two Miscellaneous Aspects of Data and Correlations ..................... 460
References ........................................................................................ 460
38 . Wishart’s Theorem and Fisher’s Transform (Tech . Index 9/10) ....... 461
The Wishart Distribution .................................................................. 464
The Probability Function for One Estimated Correlation ................. 465
Fisher’s Transform and the Correlation Probability Function .......... 466
The Wishart Distribution Using Fourier Transforms ........................ 468
Warm Up: The Distribution for a Volatility Estimate ...................... 462
References ........................................................................................ 473
39 . Economic Capital (Tech . Index 4/10) ............................................... 475
Basic Idea of Economic Capital ....................................................... 475
Exposures for Economic Capital: What Should They Be? ............... 480
Attacks on Economic Capital at High CL ........................................ 480
The Cost of Economic Capital ......................................................... 483
An Economic-Capital Utility Function ............................................. 484
Sharpe Ratios ................................................................................... 484
The Classification of Risk Components of Economic Capital .......... 479
Allocation: Standalone, CVAR, or Other? ....................................... 481
Revisiting Expected Losses; the Importance of Time Scales ........... 485
Table of Contents XV
Cost Cutting and Economic Capital ................................................. 487
Traditional Measures of Capital. Sharpe Ratios. Allocation ............. 487
References ........................................................................................ 488
40 . Unused-Limit Rsk (Tech . Index 6/10) ............................................. 489
General Aspects of Risk Limits ........................................................ 489
The Unused Limit Risk Model: Overview ....................................... 491
Unused Limit Economic Capital for Issuer Credit Risk ................... 497
PART V: PATH INTEGRALS. GREEN FUNCTIONS.
AND OPTIONS ...................................................................... 499
41 . Path Integrals and Options: Overview (Tech . Index 4/10) ................ 501
42 . Path Integrals and Options I: Introduction (Tech . Index 7/10) ......... 505
Introduction to Path Integrals ........................................................... 506
Path-Integral Warm-up: The Black Scholes Model .......................... 509
Connection of Path Integral with the Stochastic Equations .............. 521
Dividends and Jumps with Path Integrals ......................................... 523
Discrete Bermuda Options ............................................................... 530
American Options ............................................................................ 537
Appendix A: Girsanov’s Theorem and Path Integrals ...................... 538
Appendix C: Perturbation Theory, Local Volatility, Skew ............... 546
References ........................................................................................ 556
Appendix B: No-Arbitrage, Hedging and Path Integrals .................. 541
Figure Captions for this Chapter ...................................................... 546
43 . Path Integrals and Options 11: Interest-Rates (Tech . Index 8/10) ...... 559
I . Path Integrals: Review .................................................................. 561
I1 . The Green Function; Discretized Gaussian Models ..................... 562
I11 . The Continuous-Time Gaussian Limit ....................................... 566
IV . Mean-Reverting Gaussian Models ............................................. 569
V . The Most General Model with Memory ...................................... 574
VI . Wrap-up for this Chapter .......................................................... 578
Appendix B: Rate-Dependent Volatility Models .............................. 586
Figure Captions for This Chapter ..................................................... 591
References ........................................................................................ 594
Appendix A: MRG Formalism, Stochastic Equations, Etc .............. 579
Appendix C: The General Gaussian Model With Memory ............... 589
44 . Path Integrals and Options 111: Numerical (Tech . Index 6/10) .......... 597
Path Integrals and Common Numerical Methods ............................. 598
Basic Numerical Procedure using Path Integrals .............................. 600
The Castresana-Hogan Path-Integral Discretization ......................... 603
xvi Quantitative Finance and Risk Management
45 .
46 .
Some Numerical Topics Related to Path Integrals ........................... 608
A Few Aspects of Numerical Errors ................................................ 614
Some Miscellaneous Approximation Methods ................................. 618
References ........................................................................................ 624
Path Integrals and Options IV: Multiple Factors
(Tech . Index 9/10) ............................................................................ 625
Calculating Options with Multidimensional Path Integrals .............. 628
Principal-Component Path Integrals ................................................. 629
References ........................................................................................ 630
The Reggeon Field Theory. Fat Tails. Chaos (Tech . Index lO/lO) ... 631
Introduction to the Reggeon Field Theory (RFT) ............................. 631
Summary of the RFT in Physics ....................................................... 632
Aspects of Applications of the RFT to Finance ................................ 637
References ........................................................................................ 638
PART VI: THE MACRO-MICRO MODEL (A RESEARCH TOPIC) ....... 639
47 . The Macro-Micro Model: Overview (Tech . Index 4/10) .................. 641
Explicit Time Scales Separating Dynamical Regions ....................... 641
I . The Macro-Micro Yield-Curve Model .......................................... 642
I1 . Further Developments in the Macro-Micro Model ...................... 646
I11 . A Function Toolkit ..................................................................... 647
References ........................................................................................ 648
48 . A Multivariate Yield-Curve Lognormal Model (Tech . Index 6/10) . 649
Summary of this Chapter ................................................................. 649
The Problem of JQnks in Yield Curves for Models .......................... 650
I . Introduction to this Chapter .......................................................... 650
IIA . Statistical Probes. Data. Quasi-Equilibrium Drift ..................... 653
IIB . Yield-Curve Kinks: BCte Noire of Yield Curve Models ........... 655
I11 . EOF / Principal Component Analysis ......................................... 656
IV . Simpler Lognormal Model with Three Variates ......................... 658
V . Wrap-up and Preview of the Next Chapters ............................... 659
Appendix A: Definitions and Stochastic Equations .......................... 659
Appendix B: EOF or Principal-Component Formalism .................... 662
Figures: Multivariate Lognormal Yield-Curve Model ...................... 667
References ........................................................................................ 680
49 . Strong Mean-Reverting Multifactor YC Model (Tech . Index 7/10) . 681
Summary of this Chapter ................................................................. 681
I . Introduction to this Chapter .......................................................... 682
I1 . Cluster Decomposition Analysis and the SMRG Model .............. 685
Table of Contents xvii
I11 . Other Statistical Tests and the SMRG Model ............................. 691
IV . Principal Components (EOF) and the SMRG Model ................. 694
V . Wrap-up for this Chapter ............................................................ 694
Appendix A: Definitions and Stochastic Equations .......................... 695
Appendix B: The Cluster-Decomposition Analysis (CDA) .............. 697
Figures: Strong Mean-Reverting Multifactor Yield-Curve Model ... 701
References ........................................................................................ 715
50 . The Macro-Micro Yield-Curve Model (Tech . Index 5/10) ............... 717
Summary of this Chapter ................................................................. 717
I . Introduction to this Chapter .......................................................... 718
Prototype: Prime (Macro) and Libor (Macro + Micro) ..................... 720
I1 . Details of the Macro-Micro Yield-Curve Model ......................... 721
I11 . Wrap-up of this Chapter ............................................................ 724
Appendix A . No Arbitrage and Yield-Curve Dynamics ................... 725
References ........................................................................................ 730
Figures: Macro-Micro Model ........................................................... 726
51 . Macro-Micro Model: Further Developments (Tech . Index 6/10) ..... 731
Summary of This Chapter ................................................................ 731
The Macro-Micro Model for the FX and Equity Markets ................ 733
Macro-Micro-Related Models in the Economics Literature ............. 735
Related Models for Interest Rates in the Literature .......................... 735
Related Models for FX in the Literature ........................................... 736
Formal Developments in the Macro-Micro Model ........................... 737
No Arbitrage and the Macro-Micro Model: Formal Aspects ............ 739
Hedging, Forward Prices, No Arbitrage, Options (Equities) ............ 741
Satisfying the Interest-Rate Term-Structure Constraints .................. 744
Other Developments in the Macro-Micro Model ............................. 745
Derman’s Equity Regimes and the Macro-Micro Model .................. 745
Seigel’s Nonequilibrium Dynamics and the MM Model .................. 745
Macroeconomics and Fat Tails (Currency Crises) ............................ 746
Some Remarks on Chaos and the Macro-Micro Model .................... 747
Technical Analysis and the Macro-Micro Model ............................. 749
The Macro-Micro Model and Interest-Rate Data 1950-1996 ........... 750
Data, Models, and Rate Distribution Histograms ............................. 751
Negative Forwards in Multivariate Zero-Rate Simulations .............. 752
References ........................................................................................ 753
52 . A Function Toolkit (Tech . Index 6/10) ............................................ 755
Summary of Desirable Properties of Toolkit Functions ................... 757
Construction of the Toolkit Functions .............................................. 757
Relation of the Function Toolkit to Other Approaches ..................... 762
Example of Standard Micro “Noise” Plus Macro “Signal” .............. 764
Time Thresholds; Time and Frequency; Oscillations ....................... 756
xviii Quantitative Finance and Risk Management
The Total Macro: Quasi-Random Trends + Toolkit Cycles ............. 767
Short-Time Micro Regime. Trading. and the Function Toolkit ........ 768
Appendix: Wavelets. Completeness. and the Function Toolkit ........ 769
References ........................................................................................ 771
Index ............................................................................................................ 773