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
4 . Equity Options (Tech. Index 3/10) ..................................................... 27
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

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

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

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

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

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