This book gives a good general overview of financial engineering but only for those who have had a lot of prior exposure to the subject, at least from a theoretical or academic point of view, but have yet to get their feet wet in actual practice. For physicists with a background in quantum field theory, stochastic dynamical systems, or statistical mechanics, the mathematics in this book will be straightforward, and physicists will be intrigued that some of their ideas are being applied to finance. It is not a book for beginners though, as it will demand a lot of attention to details, as well as a considerable amount of outside reading. Space does not permit a detailed review of such a large book, and so only selected chapters will be reviewed.
In chapter 4, the author analyzes plain-vanilla equity options and discusses in particular the case of American options. The calculation of the probabilities of exercise at different future times involves the determination of the critical path followed by a Monte Carlo simulation to determine to the fraction of paths crossing the critical path in each interval of time. The hedges are then distributed in time as the delta times these probabilities of exercise. The author unfortunately does not give the details of how to obtain the critical path in this chapter, but these details can be found in later chapters on path integrals.
In chapter 5, foreign exchange options are discussed including how to hedge with the Greeks. The author shows how to price FX forwards and FX European options. He mentions that the Garman-Kohlhagen model is used to price the FX options, but he does not elaborate in any detail on the model. This model, which is the standard pricing convention in the FX market, is the analog of the Black-Scholes model, but where a foreign riskless interest rate is used as the payout on the underlying asset. Particularly interesting in this chapter is the author's discussion on the "two-country paradox". This paradox arises because the change of variables in foreign exchange instruments forces one to do a separate normalization of the drift of each variable, and does not arise for ordinary options. The drift after the change of variable is not consistent with interest-rate parity. Also discussed are the `volatility smiles' that are empirically observed in FX. As the author illustrates in a diagram, the smile corresponds to an upward-facing parabola, and he explains its occurrence by a "fear factor" (sometimes called "crash-o-phobia" in the equity option literature), which causes the implied volatilities of OTM puts to be bid up, thus putting a premium on this volatility relative to the ATM volume.
There are five chapters in the book that discuss the use of path integrals in finance, and these chapters include the formalism and how to calculate them numerically. The writing in these chapters is very lucid, and this no doubt reflects the author's background in physics and his consequent bias toward the use of functional integration in financial modeling. The discussion of the Black-Scholes in the context of functional integration is good motivation for later developments, and should convince readers as to the viability of this approach in finance. In addition, the author gives examples where the path integral approach does not merely reproduce the standard results in finance, one of these examples being the inclusion of dividends in options valuation. Including dividends can be done via the use of an "effective drift function", as the author shows in detail. He also shows that jumps in stock price can be studied in the same way as dividends in the context of path integration. Discrete-schedule Bermuda options are also tackled using path integral methods, as well as American options, and the author shows the reader how to calculate the critical path for these scenarios, following up on a promise in an earlier chapter. The chapter on numerical methods for the calculation of path integrals is interesting because it introduces some techniques and concepts that are no doubt new to many readers, such as "geometric volatility", which corresponds to an approximate volatility that would lead to a particular set of paths.
Perhaps the most interesting and "exotic" of the discussions in the book is included in chapter 46, and regards the application of `Reggeon field theory' (RFT) to financial engineering. Even for physicists working in quantum field theory, this type of field theory may be unknown to them, but the author does give a very brief review. He assumes background in scattering theory, the renormalization group, dimensional regularization, and other topics in field theory and high-energy physics, in order to read this chapter. RFT is presented as a theory to describe high-energy diffractive scattering, as a field theory for a particle called the `Pomeron'. The author's interest for the application of RFT to finance concern its ability to model nonlinearities and non-linear diffusion. He writes down the Lagrangian for RFT, which involves the nonlinear product of three fields, and when the interaction is switched off reduces to an ordinary diffusive model in imaginary time. One could apply ordinary perturbation theory to the case of weak interactions, but the author instead is interested in the non-perturbative region for the theory. This he tackles with the renormalization group, the object of which is to find the critical dimension, in order to test for the occurrence of a phase transition. Therefore the Gell-Mann Low beta function is to be calculated (using perturbation theory) and its zeros found. The author summarizes what is known for RFT from the research in the literature. The applications to finance consist of the ability of the RFT model to describe deviations from "square-root time", the latter of which arises from the standard Brownian motion assumption in financial theory. The RFT model reduces to the standard financial model when the interactions vanish. The nonlinear interactions are expected to produce interesting "fat-tail" jump events, but the author does not elaborate on this in any detail.
Author(s): Jan W. Dash
Publisher: World Scientific Publishing Company
Year: 2004
Language: English
Pages: 802
Quantitative Finance and Risk Management: A Physicist's Approach......Page 2
Table of Contents......Page 8
ACKNOWLEDGEMENTS......Page 20
PART I: INTRODUCTION, OVERVIEW, AND EXERCISE......Page 22
Who/ How/What, “Tech. Index”, Messages, Personal Note......Page 24
Summary Outline: Book Contents......Page 26
Objectives of Quantitative Finance and Risk Management......Page 28
Tools of Quantitative Finance and Risk Management......Page 30
The Traditional Areas of Risk Management......Page 32
Many People Participate in Risk Management......Page 34
Quants in Quantitative Finance and Risk Management......Page 36
References......Page 38
Part #1: Data, Statistics, and Reporting Using a Spreadsheet......Page 40
Part #2: Repeat Part #1 Using Programming......Page 43
Part #3: A Few Quick and Tricky Hypothetical Questions......Page 44
References......Page 45
PART II: RISK LAB (NUTS AND BOLTS OF RISK MANAGEMENT)......Page 46
Pricing and Hedging One Option......Page 48
American Options......Page 51
Basket Options and Index Options......Page 52
Scenario Analysis (Introduction)......Page 54
References......Page 55
FX Forwards and Options......Page 56
Some Practical Details for FX Options......Page 59
Hedging FX Options with Greeks: Details and Ambiguities......Page 60
FX Volatility Skew and/or Smile......Page 62
Pricing Barrier Options with Skew......Page 66
Double Barrier Option: Practical Example......Page 68
The “Two-Country Paradox”......Page 69
Quanto Options and Correlations......Page 71
Numerical Codes, Closed Form Sanity Checks, and Intuition......Page 72
References......Page 73
6. Equity Volatility Skew (Tech. Index 6/10)......Page 74
Put-Call Parity: Theory and Violations......Page 75
Dealing with Skew......Page 76
Perturbative Skew and Barrier Options......Page 77
Static Replication......Page 79
Stochastic Volatility......Page 81
Local Volatility and Skew......Page 83
Local vs. Implied Volatility Skew; Derman's Rules of Thumb......Page 84
Option Replication with Gadgets......Page 86
Intuitive Models and Different Volatility Regimes......Page 89
Appendix A: Algorithm for “Perturbative Skew” Approach......Page 90
Appendix B: A Technical Issue for Stochastic Volatility......Page 92
References......Page 93
Market Input Rates......Page 94
Construction of the Forward-Rate Curve......Page 97
References......Page 104
Swaps: Pricing and Risk......Page 106
Interest Rate Swaps: Pricing and Risk Details......Page 112
Counterparty Credit Risk and Swaps......Page 128
References......Page 130
Types of Bonds......Page 132
Bond Issuance......Page 136
Bond Trading......Page 137
Bond Math......Page 139
References......Page 142
Introduction to Caps......Page 144
The Black Caplet Formula......Page 146
Relations between Caps, Floors, and Swaps......Page 148
Hedging Delta and Gamma for Libor Caps......Page 149
Hedging Volatility and Vega Ladders......Page 150
Prime Caps and a Vega Trap......Page 152
CMT Rates and Volatility Dependence of CMT Products......Page 153
References......Page 157
European Swaptions......Page 158
Bermuda/American Swaption Pricing......Page 162
Delta and Vega Risk: Move Inputs or Forwards?......Page 164
Swaptions and Corporate Liability Risk Management......Page 165
Practical Example: A Deal Involving a Swaption......Page 167
Miscellaneous Swaption Topics......Page 169
References......Page 171
Definitions of Portfolios......Page 172
Definitions of Scenarios......Page 174
Many Portfolios and Scenarios......Page 176
Risk Analyses and Presentations......Page 178
PART III: EXOTICS, DEALS, AND CASE STUDIES......Page 180
The M&A Scenario......Page 182
CVR Extension Options and Other Complications......Page 183
The Arbs and the Mispricing of the CVR Option......Page 185
A Simplified CVR: Two Put Spreads with Extension Logic......Page 186
Non-Academic Corporate Decision for Option Extension......Page 188
The CVR Option Pricing......Page 190
Analytic CVR Pricing Methodology......Page 194
Some Practical Aspects of CVR Pricing and Hedging......Page 197
References......Page 201
Case Study: DECS and Synthetic Convertibles......Page 204
D123 : The Complex DEC Synthetic Convertible......Page 209
Case Study: Equity Call with Variable Strike and Expiration......Page 214
References......Page 220
Contingent Caps......Page 222
Digital Options: Pricing and Hedging......Page 226
Historical Simulations and Hedging......Page 228
Yield-Curve Shape and Principle-Component Options......Page 230
Principal-Component Risk Measures (Tilt Delta etc.)......Page 231
Hybrid 2-Dimensional Barrier Options—Examples......Page 232
Reload Options......Page 235
References......Page 238
TIPS (Treasury Inflation Protected Securities)......Page 240
Municipal Derivatives, Muni Issuance, Derivative Hedging......Page 242
Difference Option on an Equity Index and a Basket of Stocks......Page 245
Resettable Options: Cliquets......Page 247
Power Options......Page 251
ARM Caps......Page 252
Index-Amortizing Swaps......Page 253
A Hypothetical Repo + Options Deal......Page 257
Convertible Issuance Risk......Page 260
References......Page 262
17. Single Barrier Options (Tech. Index 6/10)......Page 264
Knock-Out Options......Page 266
The Semi-Group Property including a Barrier......Page 268
Calculating Barrier Options......Page 269
Knock-In Options......Page 270
Useful Integrals for Barrier Options......Page 272
A Useful Discrete Barrier Approximation......Page 273
“Potential Theory” for General Sets of Single Barriers......Page 274
Barrier Options with Time-Dependent Drifts and Volatilities......Page 276
References......Page 277
18. Double Barrier Options (Tech. Index 7/10)......Page 278
Double Barrier Solution with an Infinite Set of Images......Page 279
Double Barrier Option Pricing......Page 281
Rebates for Double Barrier Options......Page 283
References......Page 284
19. Hybrid 2-D Barrier Options (Tech. Index 7/10)......Page 286
Pricing the Barrier 2-Dimension Hybrid Options......Page 288
Useful Integrals for 2D Barrier Options......Page 289
References......Page 290
20. Average-Rate Options (Tech. Index 8/10)......Page 292
Arithmetic Average Rate Options in General Gaussian Models......Page 293
Results for Average-Rate Options in the MRG Model......Page 297
Simple Harmonic Oscillator Derivation for Average Options......Page 298
Average Options with Log-Normal Rate Dynamics......Page 299
References......Page 301
PART IV: QUANTITATIVE RISK MANAGEMENT......Page 302
Gaussian Behavior and Deviations from Gaussian......Page 304
Outliers and Fat Tails......Page 305
Use of the Equivalent Gaussian Fat-Tail Volatility......Page 308
Practical Considerations for the Fat-Tail Parameters......Page 309
References......Page 315
The Importance and Difficulty of Correlation Risk......Page 316
One Correlation in Two Dimensions......Page 317
Two Correlations in Three Dimensions; the Azimuthal Angle......Page 318
Correlations in Four Dimensions......Page 321
Correlations in Five and Higher Dimensions......Page 322
Spherical Representation of the Cholesky Decomposition......Page 324
Numerical Considerations for the N- Sphere......Page 325
References......Page 326
Correlation Stress Scenarios Using Data......Page 328
Random Correlation Matrices Using Historical Data......Page 334
Stochastic Correlation Matrices Using the N-sphere......Page 335
24. Optimally Stressed PD Correlation Matrices (Tech. Index 7/10)......Page 340
Least-Squares Fitting for the Optimal PD Stressed Matrix......Page 342
Numerical Considerations for Optimal PD Stressed Matrix......Page 343
Example of Optimal PD Fit to a NPD Stressed Matrix......Page 344
SVD Algorithm for the Starting PD Correlation Matrix......Page 346
References......Page 349
“Just Make the Correlations Zero” Model; Three Versions......Page 350
The Macro-Micro Model for Quasi-Random Correlations......Page 352
Correlation Dependence on Volatility......Page 356
Implied, Current, and Historical Correlations for Baskets......Page 359
26. Plain-Vanilla VAR (Tech. Index 4/10)......Page 362
Quadratic Plain-Vanilla VAR and CVARs......Page 365
Monte-Carlo VAR......Page 367
Monte-Carlo CVARs and the CVAR Volatility......Page 368
Confidence Levels for Individual Variables in VAR......Page 371
References......Page 372
Improved Plain-Vanilla VAR (IPV-VAR)......Page 374
Enhanced/Stressed VAR (ES- VAR)......Page 378
Other VAR Topics......Page 386
References......Page 389
Set-up and Overview of the Formal VAR Results......Page 390
Calculation of the Generating Function......Page 392
VAR, the CVARs, and the CVAR Volatilities......Page 395
Effective Number of SD for Underlying Variables......Page 397
Extension to Multiple Time Steps using Path Integrals......Page 399
The CVAR Volatility with Two Variables......Page 402
Geometry for Risk Ellipse, VAR Line, CVAR, CVAR Vol......Page 403
Aggregation, Desks, and Business Units......Page 408
Desk CVARs and Correlations between Desk Risks......Page 410
Aged Inventory and Illiquidity......Page 412
31. Issuer Credit Risk (Tech. Index 5/10)......Page 414
Transition/Default Probability Matrices......Page 415
Calculation of Issuer Risk—Generic Case......Page 420
Example of Issuer Credit Risk Calculation......Page 424
Issuer Credit Risk and Market Risk: Separation via Spreads......Page 427
Separating Market and Credit Risk without Double Counting......Page 428
A Unified Credit + Market Risk Model......Page 431
References......Page 434
Summary of Model Risk......Page 436
Time Scales and Models......Page 437
Liquidity Model Limitations......Page 438
Model Risk, Model Reserves, and Bid-Offer Spreads......Page 439
Models and Parameters......Page 440
References......Page 441
Model Quality Assurance Goals, Activities, and Procedures......Page 442
Model QA: Sample Documentation......Page 445
Quantitative Section of Model QA Documentation......Page 446
Systems Section of Model QA Documentation......Page 449
References......Page 451
What are the “Three-Fives Systems Criteria”?......Page 452
What are Some Systems Traps and Risks?......Page 453
The Birth and Development of a System......Page 454
Systems in Mergers and Startups......Page 456
Vendor Systems......Page 457
New Paradigms in Systems and Parallel Processing......Page 458
Languages for Models: Fortran 90, C++, C, and Others......Page 459
References......Page 461
35. Strategic Computing (Tech. Index 3/10)......Page 462
Illustration of Parallel Processing for Finance......Page 463
Some Aspects of Parallel Processing......Page 464
Technology, Strategy and Change......Page 467
References......Page 468
Data Consistency......Page 470
Data Vendors......Page 471
Bad Data Points and Other Data Traps......Page 472
Fluctuations and Uncertainties in Measured Correlations......Page 474
Time Windowing......Page 475
Correlations, the Number of Data Points, and Variables......Page 477
Intrinsic and Windowing Uncertainties: Example......Page 479
References......Page 481
38. Wishart's Theorem and Fisher's Transform (Tech. Index 9/10)......Page 482
Warm Up: The Distribution for a Volatility Estimate......Page 483
The Wishart Distribution......Page 485
The Probability Function for One Estimated Correlation......Page 486
Fisher’s Transform and the Correlation Probability Function......Page 487
The Wishart Distribution Using Fourier Transforms......Page 489
References......Page 494
Basic Idea of Economic Capital......Page 496
The Classification of Risk Components of Economic Capital......Page 500
Attacks on Economic Capital at High CL......Page 501
Allocation: Standalone, CVAR, or Other?......Page 502
The Cost of Economic Capital......Page 504
Sharpe Ratios......Page 505
Revisiting Expected Losses; the Importance of Time Scales......Page 506
Traditional Measures of Capital, Sharpe Ratios, Allocation......Page 508
References......Page 509
General Aspects of Risk Limits......Page 510
The Unused Limit Risk Model: Overview......Page 512
Unused Limit Economic Capital for Issuer Credit Risk......Page 518
PART V: PATH INTEGRALS, GREEN FUNCTIONS, AND OPTIONS......Page 520
41. Path Integrals and Options: Overview (Tech. Index 4/10)......Page 522
42. Path Integrals and Options I: Introduction (Tech. Index 7/10)......Page 526
Introduction to Path Integrals......Page 527
Path-Integral Warm-up: The Black Scholes Model......Page 530
Connection of Path Integral with the Stochastic Equations......Page 542
Dividends and Jumps with Path Integrals......Page 544
Discrete Bermuda Options......Page 551
American Options......Page 558
Appendix A: Girsanov's Theorem and Path Integrals......Page 559
Appendix B: No-Arbitrage, Hedging and Path Integrals......Page 562
Figure Captions for this Chapter......Page 567
References......Page 577
43. Path Integrals and Options II: Interest-Rates (Tech. Index 8/10)......Page 580
I. Path Integrals: Review......Page 582
II. The Green Function; Discretized Gaussian Models......Page 583
III. The Continuous-Time Gaussian Limit......Page 587
IV. Mean-Reverting Gaussian Models......Page 590
V. The Most General Model with Memory......Page 595
VI. Wrap-up for this Chapter......Page 599
Appendix A: MRG Formalism, Stochastic Equations, Etc.......Page 600
Appendix B: Rate-Dependent Volatility Models......Page 607
Appendix C: The General Gaussian Model With Memory......Page 610
Figure Captions for This Chapter......Page 612
References......Page 615
44. Path Integrals and Options III: Numerical (Tech. Index 6/10)......Page 618
Path Integrals and Common Numerical Methods......Page 619
Basic Numerical Procedure using Path Integrals......Page 621
The Castresana-Hogan Path-Integral Discretization......Page 624
Some Numerical Topics Related to Path Integrals......Page 629
A Few Aspects of Numerical Errors......Page 635
Some Miscellaneous Approximation Methods......Page 639
References......Page 645
45. Path Integrals and Options IV: Multiple Factors (Tech. Index 9/10)......Page 646
Calculating Options with Multidimensional Path Integrals......Page 649
Principal-Component Path Integrals......Page 650
References......Page 651
Introduction to the Reggeon Field Theory (RFT)......Page 652
Summary of the RFT in Physics......Page 653
Aspects of Applications of the RFT to Finance......Page 658
References......Page 659
PART VI: THE MACRO-MICRO MODEL (A RESEARCH TOPIC)......Page 660
Explicit Time Scales Separating Dynamical Regions......Page 662
I. The Macro-Micro Yield-Curve Model......Page 663
II. Further Developments in the Macro-Micro Model......Page 667
III. A Function Toolkit......Page 668
References......Page 669
Summary of this Chapter......Page 670
I. Introduction to this Chapter......Page 671
IIA. Statistical Probes, Data, Quasi-Equilibrium Drift......Page 674
IIB. Yield-Curve Kinks: Bete Noire of Yield Curve Models......Page 676
III. EOF / Principal Component Analysis......Page 677
IV. Simpler Lognormal Model with Three Variates......Page 679
Appendix A: Definitions and Stochastic Equations......Page 680
Appendix B: EOF or Principal-Component Formalism......Page 683
Figures: Multivariate Lognormal Yield-Curve Model......Page 688
References......Page 701
Summary of this Chapter......Page 702
I. Introduction to this Chapter......Page 703
II. Cluster Decomposition Analysis and the SMRG Model......Page 706
III. Other Statistical Tests and the SMRG Model......Page 712
V. Wrap-up for this Chapter......Page 715
Appendix A: Definitions and Stochastic Equations......Page 716
Appendix B: The Cluster-Decomposition Analysis (CDA)......Page 718
Figures: Strong Mean-Reverting Multifactor Yield-Curve Model......Page 722
References......Page 736
Summary of this Chapter......Page 738
I. Introduction to this Chapter......Page 739
Prototype: Prime (Macro) and Libor (Macro + Micro)......Page 741
II. Details of the Macro-Micro Yield-Curve Model......Page 742
III. Wrap-Up of this Chapter......Page 745
Appendix A. No Arbitrage and Yield-Curve Dynamics......Page 746
Figures: Macro-Micro Model......Page 747
References......Page 751
Summary of This Chapter......Page 752
The Macro-Micro Model for the FX and Equity Markets......Page 754
Related Models for Interest Rates in the Literature......Page 756
Related Models for FX in the Literature......Page 757
Formal Developments in the Macro-Micro Model......Page 758
No Arbitrage and the Macro-Micro Model: Formal Aspects......Page 760
Hedging, Forward Prices, No Arbitrage, Options (Equities)......Page 762
Satisfying the Interest-Rate Term-Structure Constraints......Page 765
Seigel's Nonequilibrium Dynamics and the MM Model......Page 766
Macroeconomics and Fat Tails (Currency Crises)......Page 767
Some Remarks on Chaos and the Macro-Micro Model......Page 768
Technical Analysis and the Macro-Micro Model......Page 770
The Macro-Micro Model and Interest-Rate Data 1950–1996......Page 771
Data, Models, and Rate Distribution Histograms......Page 772
Negative Forwards in Multivariate Zero-Rate Simulations......Page 773
References......Page 775
52. A Function Toolkit (Tech. Index 6/10)......Page 776
Time Thresholds; Time and Frequency; Oscillations......Page 777
Construction of the Toolkit Functions......Page 778
Relation of the Function Toolkit to Other Approaches......Page 783
Example of Standard Micro “Noise” Plus Macro “Signal”......Page 785
The Total Macro: Quasi-Random Trends + Toolkit Cycles......Page 788
Short-Time Micro Regime, Trading, and the Function Toolkit......Page 789
Appendix: Wavelets, Completeness, and the Function Toolkit......Page 790
References......Page 792
Index......Page 794
