Quantitative Finance with Case Studies in Python.pdf
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Section  1 ■ Setting the Stage: Quant Landscape 3
1.1 INTRODUCTION 3
1.2 QUANT FINANCE INSTITUTIONS 4
1.2.1 Sell-Side: Dealers & Market Makers 5
1.2.2 Buy-Side: Asset Managers & Hedge Funds 5
1.2.3 Financial Technology Firms 6
1.3 MOST COMMON QUANT CAREER PATHS 6
1.3.1 Buy Side 7
1.3.2 Sell Side 8
1.3.3 Financial Technology 9
1.3.4 What’s Common between Roles? 9
1.4 STAGES OF A QUANT PROJECT 9
1.4.1 Data Collection 9
1.4.2 Data Cleaning 10
1.4.3 Model Implementation 10
1.4.4 Model Validation 10
1.5 TRENDS: WHAT HAS CHANGED IN THE LAST FEW YEARS, AND
WHERE IS QUANT FINANCE GOING? 11
1.5.1 Change in Inflation Regime 11
1.5.2 Automation 11
1.5.3 Rapid Increase of Available Data 11
vii
viii ■ Contents
1.5.4 Commoditization of Factor Premia 12
1.5.5 Increased Popularity of Multi-Strategy Hedge Funds 12
1.5.6 Proliferation of Artificial Intelligence, Machine Learning and
Large Language Models 13
1.5.7 Increasing Prevalence of Required Quant/Technical Skills 13
1.5.8 Increased Regulatory Complexity and the Potential for
Democratization 14
Section  2 ■ Setting the Stage: Landscape of Financial Instruments 15
2.1 EQUITY INSTRUMENTS 15
2.1.1 Overview & Mechanics 15
2.1.2 Public Equity 16
2.1.3 Modeling Equity via a Discounted Dividend Growth Model 16
2.1.4 Private Equity 17
2.1.5 Private Equity vs. Public Equity Returns in Practice 17
2.1.6 Preferred Stock 18
2.2 DEBT INSTRUMENTS 18
2.2.1 Overview & Mechanics 18
2.2.2 Sovereign Bonds 19
2.2.3 Corporate Bonds 20
2.2.4 Municipal Bonds 20
2.2.5 Municipal Bond Valuation in Practice 20
2.2.6 Inflation Linked Bonds 21
2.2.7 Convertible Bonds 21
2.2.8 Green Bonds 22
2.3 FORWARDS & FUTURES 22
2.4 OPTIONS 24
2.4.1 Mechanics 24
2.4.2 Option Straddles in Practice 25
2.4.3 Put-Call Parity 26
2.5 SWAPS 27
2.5.1 Mechanics 27
2.5.2 Equity Index Total Return Swaps in Practice 28
2.5.3 Over-The-Counter vs. Exchange Traded Products 29
Contents ■ ix
Section  3 ■ Theoretical Underpinnings of Quant Modeling: Modeling
the Risk Neutral Measure 31
3.1 INTRODUCTION 31
3.2 RISK NEUTRAL PRICING & NO ARBITRAGE 32
3.2.1 Risk Neutral vs. Actual Probabilities 32
3.2.2 Theory of No Arbitrage 33
3.2.3 Complete Markets 33
3.2.4 Risk Neutral Valuation Equation 34
3.2.5 Risk Neutral Discounting, Risk Premia & Stochastic
Discount Factors 34
3.3 BINOMIAL TREES 35
3.3.1 Discrete vs. Continuous Time Models 35
3.3.2 Scaled Random Walk 35
3.3.3 Discrete Binomial Tree Model 37
3.3.4 Limiting Distribution of Binomial Tree Model 39
3.4 BUILDING BLOCKS OF STOCHASTIC CALCULUS 40
3.4.1 Deterministic vs. Stochastic Calculus 40
3.4.2 Stochastic Processes 41
3.4.3 Martingales 41
3.4.4 Brownian Motion 41
3.4.5 Properties of Brownian Motion 42
3.5 STOCHASTIC DIFFERENTIAL EQUATIONS 45
3.5.1 Generic SDE Formulation 45
3.5.2 Bachelier SDE 45
3.5.3 Black-Scholes SDE 46
3.5.4 Stochastic Models in Practice 46
3.6 ITO’S LEMMA 47
3.6.1 General Formulation & Theory 47
3.6.2 Ito in Practice: Risk-Free Bond 48
3.6.3 Ito in Practice: Black-Scholes Dynamics 49
3.7 CONNECTION BETWEEN SDEs AND PDEs 51
3.7.1 PDEs & Stochastic Processes 51
3.7.2 Deriving the Black-Scholes PDE 51
3.7.3 General Formulation: Feynman-Kac Formula 53
3.7.4 Working with PDEs in Practice 54
x ■ Contents
3.8 GIRSANOV’S THEOREM 55
3.8.1 Change of Measure via Girsanov’s Theorem 55
3.8.2 Applications of Girsanov’s Theorem 56
Section  4 ■ Theoretical Underpinnings of Quant Modeling: Modeling
the Physical Measure 57
4.1 INTRODUCTION: FORECASTING VS. REPLICATION 57
4.2 MARKET EFFICIENCY AND RISK PREMIA 58
4.2.1 Efficient Market Hypothesis 58
4.2.2 Market Anomalies, Behavioral Finance & Risk Premia 59
4.2.3 Risk Premia Example: Selling Insurance 60
4.3 LINEAR REGRESSION MODELS 61
4.3.1 Introduction & Terminology 61
4.3.2 Univariate Linear Regression 62
4.3.3 Multivariate Linear Regression 64
4.3.4 Standard Errors & Significance Tests 65
4.3.5 T-Statistics, P-Values & Statistical Significance 67
4.3.6 Assumptions of Linear Regression 68
4.3.7 How Are Regression Models Used in Practice? 69
4.3.8 Regression Models in Practice: Calculating High-Yield Betas
to Stocks and Bonds 70
4.4 TIME SERIES MODELS 70
4.4.1 Time Series Data 70
4.4.2 Stationary vs. Non-Stationary Series & Differencing 71
4.4.3 White Noise & Random Walks 72
4.4.4 Autoregressive Processes & Unit Root Tests 73
4.4.5 Moving Average Models 74
4.4.6 ARMA Models 75
4.4.7 State Space Models 76
4.4.8 How Are Time Series Models Used in practice? 76
4.5 PANEL REGRESSION MODELS 77
4.6 CORE PORTFOLIO AND INVESTMENT CONCEPTS 79
4.6.1 Time Value of Money 79
4.6.2 Compounding Returns 80
4.6.3 Portfolio Calculations 81
4.6.4 Portfolio Concepts in Practice: Benefit of Diversification 84
Contents ■ xi
4.7 BOOTSTRAPPING 85
4.7.1 Overview 85
4.8 PRINCIPAL COMPONENT ANALYSIS 87
4.9 CONCLUSIONS: COMPARISON TO RISK NEUTRAL MEASURE
MODELING 89
Section II Fundamentals of Coding and Data Analysis
Section  5 ■ Python Programming Environment 93
5.1 THE PYTHON PROGRAMMING LANGUAGE 93
5.2 ADVANTAGES AND DISADVANTAGES OF PYTHON 93
5.3 PYTHON DEVELOPMENT ENVIRONMENTS 94
5.4 BASIC PROGRAMMING CONCEPTS IN PYTHON 95
5.4.1 Language Syntax 95
5.4.2 Data Types in Python 96
5.4.3 Working with Built-in Functions 96
5.4.4 Conditional Statements 97
5.4.5 Operator Precedence 98
5.4.6 Loops 98
5.4.7 Working with Strings 99
5.4.8 User-Defined Functions 100
5.4.9 Variable Scope 100
5.4.10 Importing Modules 101
5.4.11 Exception Handling 102
5.4.12 Recursive Functions 103
5.4.13 Plotting/Visualizations 103
5.5 WORKING IN A MULTI-PROGRAMMER ENVIRONMENT 104
Section  6 ■ Programming Concepts in Python 106
6.1 INTRODUCTION 106
6.2 NUMPY LIBRARY 106
6.3 PANDAS LIBRARY 107
6.4 DATA STRUCTURES IN PYTHON 108
6.4.1 Tuples 108
6.4.2 Lists 108
6.4.3 Array 109
xii ■ Contents
6.4.4 Differences between Lists and NumPy Arrays 110
6.4.5 Covariance Matrices in Practice 110
6.4.6 Covariance Matrices in Practice: Are Correlations
Stationary? 111
6.4.7 Series 112
6.4.8 DataFrame 112
6.4.9 Dictionary 115
6.5 IMPLEMENTATION OF QUANT TECHNIQUES IN PYTHON 116
6.5.1 Random Number Generation 116
6.5.2 Linear Regression 117
6.5.3 Linear Regression in Practice: Equity Return Decomposition
by Fama-French Factors 118
6.5.4 Autocorrelation Tests 118
6.5.5 ARMA Models in Practice: Testing for Mean-Reversion in
Equity Index Returns 119
6.5.6 Matrix Decompositions 120
6.6 OBJECT-ORIENTED PROGRAMMING IN PYTHON 121
6.6.1 Principles of Object-Oriented Programming 121
6.6.2 Classes in Python 122
6.6.3 Constructors 123
6.6.4 Destructors 124
6.6.5 Class Attributes 125
6.6.6 Class Methods 126
6.6.7 Class Methods vs. Global Functions 126
6.6.8 Operator Overloading 127
6.6.9 Inheritance in Python 129
6.6.10 Polymorphism in Python 130
6.7 DESIGN PATTERNS 130
6.7.1 Types of Design Patterns 130
6.7.2 Abstract Base Classes 131
6.7.3 Factory Pattern 131
6.7.4 Singleton Pattern 131
6.7.5 Template Method 132
6.8 SEARCH ALGORITHMS 132
6.8.1 Binary Search Algorithm 132
6.9 SORT ALGORITHMS 133
6.9.1 Selection Sort 133
Contents ■ xiii
6.9.2 Insertion Sort 133
6.9.3 Bubble Sort 133
6.9.4 Merge Sort 134
Section  7 ■ Working with Financial Datasets 135
7.1 INTRODUCTION 135
7.2 DATA COLLECTION 136
7.2.1 Overview 136
7.2.2 Reading & Writing Files in Python 136
7.2.3 Parsing Data from a Website 138
7.2.4 Interacting with Databases in Python 138
7.3 COMMON FINANCIAL DATASETS 139
7.3.1 Stock Data 140
7.3.2 Currency Data 140
7.3.3 Futures Data 140
7.3.4 Options Data 141
7.3.5 Fixed Income Data 142
7.4 COMMON FINANCIAL DATA SOURCES 142
7.5 CLEANING DIFFERENT TYPES OF FINANCIAL DATA 143
7.5.1 Proper Handling of Corporate Actions 143
7.5.2 Avoiding Survivorship Bias 144
7.5.3 Detecting Arbitrage in the Data 145
7.6 HANDLING MISSING DATA 146
7.6.1 Interpolation & Filling Forward 146
7.6.2 Filling via Regression 147
7.6.3 Filling via Bootstrapping 148
7.6.4 Filling via K-Nearest Neighbor 149
Section  8 ■ Data Science Techniques in Finance 150
8.1 INTRODUCTION: WHAT IS DATA SCIENCE? 150
8.2 EXPLORATORY DATA ANALYSIS 151
8.2.1 Single vs. Multivariate Exploratory Data Analysis 152
8.2.2 Data Visualization 152
8.2.3 Statistical Analysis of Data 153
8.2.4 Factor & Principal Component Analysis 154
xiv ■ Contents
8.2.5 t-distributed Stochastic Neighbor Embedding 155
8.2.6 EDA in Practice 155
8.3 MACHINE LEARNING TECHNIQUES FOR DATA SCIENCE 160
8.3.1 Overview of Machine Learning for Data Science 160
8.3.2 Supervised vs. Unsupervised Learning 160
8.3.3 Structured vs. Unstructured Data 161
8.4 TRANSFORMING DATA USING SQL 161
8.4.1 Modifying Data via Insert, Update and Delete Statements 162
8.4.2 Retrieving Data via Select Statements 163
8.4.3 Retrieving Data from Multiple Tables via JOINs 164
8.4.4 Aggregating Data via a Group by Clause 165
8.5 DIMENSIONALITY REDUCTION 166
8.5.1 Principal Component Analysis 166
8.5.2 Autoencoder 167
8.5.3 Dimensionality Reduction in Practice: Reducing the Dimen￾sionality of a Covariance Matrix Using PCA 168
8.6 PYTHON TOOLS FOR BIG DATA 170
8.6.1 Working with Large Datasets 170
8.6.2 Memory & Large Datasets 170
8.6.3 Parallel Computing 170
8.6.4 Spark & Hadoop 171
8.7 OUTLIER DETECTION 171
8.7.1 Single vs. Multi-Variate Outlier Detection 171
8.7.2 Plotting 172
8.7.3 Standard Deviation 172
8.7.4 Density Analysis 172
8.7.5 Distance from K Nearest Neighbor 173
8.7.6 Outlier Detection in Practice: Identifying Anomalies in ETF
Returns 173
Section  9 ■ Model Validation 176
9.1 WHY IS MODEL VALIDATION SO IMPORTANT? 176
9.2 HOW DO WE ENSURE OUR MODELS ARE CORRECT? 177
9.3 COMPONENTS OF A MODEL VALIDATION PROCESS 178
9.3.1 Model Documentation 178
9.3.2 Code Review 178
Contents ■ xv
9.3.3 Unit Tests 179
9.3.4 Production Model Change Process 180
9.4 GOALS OF MODEL VALIDATION 180
9.4.1 Validating Model Implementation 180
9.4.2 Understanding Model Strengths and Weaknesses 181
9.4.3 Identifying Model Assumptions 181
9.5 TRADEOFF BETWEEN REALISTIC ASSUMPTIONS AND
PARSIMONY IN MODELS 182
Section III Options Modeling
Section  10 ■ Stochastic Models 185
10.1 SIMPLE MODELS 185
10.1.1 Black-Scholes Model 185
10.1.2 Black-Scholes Model in Practice: Are Equity Returns
Log-Normally Distributed? 187
10.1.3 Implied Volatility Surfaces in Practice: Equity Options 187
10.1.4 Bachelier Model 189
10.1.5 CEV Model 190
10.1.6 CEV Model in Practice: Impact of Beta 191
10.1.7 Ornstein-Uhlenbeck Process 192
10.1.8 Cox-Ingersol-Ross Model 194
10.1.9 Conclusions 194
10.2 STOCHASTIC VOLATILITY MODELS 195
10.2.1 Introduction 195
10.2.2 Heston Model 195
10.2.3 SABR Model 197
10.2.4 SABR Model in Practice: Relationship between Model
Parameters and Volatility Surface 198
10.2.5 Stochastic Volatility Models: Comments 200
10.3 JUMP DIFFUSION MODELS 201
10.3.1 Introduction 201
10.3.2 Merton’s Jump Diffusion Model 201
10.3.3 SVJ Model 202
10.3.4 Variance Gamma Model 204
10.3.5 VGSA Model 205
10.3.6 Comments on Jump Processes 206
xvi ■ Contents
10.4 LOCAL VOLATILITY MODELS 207
10.4.1 Dupire’s Formula 207
10.4.2 Local Volatility Model in Practice: S&P Option Local
Volatility Surface 208
10.5 STOCHASTIC LOCAL VOLATILITY MODELS 209
10.6 PRACTICALITIES OF USING THESE MODELS 209
10.6.1 Comparison of Stochastic Models 209
10.6.2 Leveraging Stochastic Models in Practice 210
Section  11 ■ Options Pricing Techniques for European Options 212
11.1 MODELS WITH CLOSED FORM SOLUTIONS OR ASYMPTOTIC
APPROXIMATIONS 212
11.2 OPTION PRICING VIA QUADRATURE 213
11.2.1 Overview 213
11.2.2 Quadrature Approximations 213
11.2.3 Approximating a Pricing Integral via Quadrature 215
11.2.4 Quadrature Methods in Practice: Digital Options Prices in
Black-Scholes vs. Bachelier Model 217
11.3 OPTION PRICING VIA FFT 217
11.3.1 Fourier Transforms & Characteristic Functions 217
11.3.2 European Option Pricing via Transform 219
11.3.3 Digital Option Pricing via Transform 223
11.3.4 Calculating Outer Pricing Integral via Quadrature 225
11.3.5 Summary of FFT Algorithm 226
11.3.6 Calculating Outer Pricing Integral via FFT 227
11.3.7 Summary: Option Pricing via FFT 228
11.3.8 Strike Spacing Functions 229
11.3.9 Interpolation of Option Prices 230
11.3.10 Technique Parameters 230
11.3.11 Dependence on Technique Parameters 231
11.3.12 Strengths and Weaknesses 232
11.3.13 Variants of FFT Pricing Technique 233
11.3.14 FFT Pricing in Practice: Sensitivity to Technique
Parameters 233
11.4 ROOT FINDING 234
11.4.1 Setup 234
Contents ■ xvii
11.4.2 Newton’s Method 235
11.4.3 First Calibration: Implied Volatility 236
11.4.4 Implied Volatility in Practice: Volatility Skew for VIX
Options 237
11.5 OPTIMIZATION TECHNIQUES 237
11.5.1 Background & Terminology 238
11.5.2 Global vs. Local Minima & Maxima 239
11.5.3 First- & Second-Order Conditions 240
11.5.4 Unconstrained Optimization 241
11.5.5 Lagrange Multipliers 242
11.5.6 Optimization with Equality Constraints 242
11.5.7 Minimum Variance Portfolios in Practice: Stock & Bond
Minimum Variance Portfolio Weights 243
11.5.8 Convex Functions 244
11.5.9 Optimization Methods in Practice 244
11.6 CALIBRATION OF VOLATILITY SURFACES 245
11.6.1 Optimization Formulation 245
11.6.2 Objective Functions 246
11.6.3 Constraints 247
11.6.4 Regularization 247
11.6.5 Gradient-Based vs. Gradient-Free Optimizers 248
11.6.6 Gradient-Based Methods with Linear Constraints 248
11.6.7 Practicalities of Calibrating Volatility Surfaces 249
11.6.8 Calibration in Practice: BRLJPY Currency Options 249
Section  12 ■ Options Pricing Techniques for Exotic Options 251
12.1 INTRODUCTION 251
12.2 SIMULATION 252
12.2.1 Overview 252
12.2.2 Central Limit Theorem & Law of Large Numbers 254
12.2.3 Random Number Generators 256
12.2.4 Generating Random Variables 257
12.2.5 Transforming Random Numbers 257
12.2.6 Transforming Random Numbers: Inverse Transform
Technique 258
xviii ■ Contents
12.2.7 Transforming Random Numbers: Acceptance Rejection
Method 259
12.2.8 Generating Normal Random Variables 262
12.2.9 Quasi Random Numbers 264
12.2.10 Euler Discretization of SDEs 264
12.2.11 Simulating from Geometric Brownian Motion 266
12.2.12 Simulating from the Heston Model 267
12.2.13 Simulating from the Variance Gamma Model 268
12.2.14 Variance Reduction Techniques 269
12.2.15 Strengths and Weaknesses 273
12.2.16 Simulation in Practice: Impact of Skew on Lookback Options
Values in the Heston Model 274
12.3 NUMERICAL SOLUTIONS TO PDEs 275
12.3.1 Overview 275
12.3.2 PDE Representations of Stochastic Processes 276
12.3.3 Finite Differences 278
12.3.4 Time & Space Grid 280
12.3.5 Boundary Conditions 280
12.3.6 Explicit Scheme 282
12.3.7 Implicit Scheme 284
12.3.8 Crank-Nicolson 286
12.3.9 Stability 287
12.3.10 Multi-Dimension PDEs 287
12.3.11 Partial Integro Differential Equations 288
12.3.12 Strengths & Weaknesses 288
12.3.13 American vs. European Digital Options in Practice 289
12.4 MODELING EXOTIC OPTIONS IN PRACTICE 291
Section  13 ■ Greeks and Options Trading 293
13.1 INTRODUCTION 293
13.2 BLACK-SCHOLES GREEKS 294
13.2.1 Delta 294
13.2.2 Gamma 295
13.2.3 Delta and Gamma in Practice: Delta and Gamma by Strike 296
13.2.4 Theta 298
Contents ■ xix
13.2.5 Theta in Practice: How Does Theta Change by Option
Expiry? 299
13.2.6 Vega 300
13.2.7 Practical Uses of Greeks 300
13.3 THETA VS. GAMMA 301
13.4 MODEL DEPENDENCE OF GREEKS 302
13.5 GREEKS FOR EXOTIC OPTIONS 303
13.6 ESTIMATION OF GREEKS VIA FINITE DIFFERENCES 303
13.7 SMILE ADJUSTED GREEKS 304
13.7.1 Smile Adjusted Greeks in Practice: USDBRL Options 306
13.8 HEDGING IN PRACTICE 306
13.8.1 Re-Balancing Strategies 307
13.8.2 Delta Hedging in Practice 307
13.8.3 Vega Hedging in Practice 308
13.8.4 Validation of Greeks Out-of-Sample 309
13.9 COMMON OPTIONS TRADING STRUCTURES 310
13.9.1 Benefits of Trading Options 310
13.9.2 Covered Calls 310
13.9.3 Call & Put Spreads 311
13.9.4 Straddles & Strangles 312
13.9.5 Butterflies 314
13.9.6 Condors 315
13.9.7 Calendar Spreads 315
13.9.8 Risk Reversals 317
13.9.9 1x2s 318
13.10 VOLATILITY AS AN ASSET CLASS 319
13.11 RISK PREMIA IN THE OPTIONS MARKET: IMPLIED VS. REALIZED
VOLATILITY 320
13.11.1 Delta-Hedged Straddles 320
13.11.2 Implied vs. Realized Volatility 321
13.11.3 Implied Volatility Premium in Practice: S&P 500 322
13.12 CASE STUDY: GAMESTOP REDDIT MANIA 323
Section  14 ■ Extraction of Risk Neutral Densities 325
14.1 MOTIVATION 325
14.2 BREDEN-LITZENBERGER 326
xx ■ Contents
14.2.1 Derivation 326
14.2.2 Breeden-Litzenberger in the Presence of Imprecise Data 327
14.2.3 Strengths and Weaknesses 328
14.2.4 Applying Breden-Litzenberger in Practice 328
14.3 CONNECTION BETWEEN RISK NEUTRAL DISTRIBUTIONS AND
MARKET INSTRUMENTS 329
14.3.1 Butterflies 329
14.3.2 Digital Options 330
14.4 OPTIMIZATION FRAMEWORK FOR NON-PARAMETRIC DENSITY
EXTRACTION 331
14.5 WEIGTHED MONTE CARLO 333
14.5.1 Optimization Directly on Terminal Probabilities 333
14.5.2 Inclusion of a Prior Distribution 334
14.5.3 Weighting Simulated Paths Instead of Probabilities 335
14.5.4 Strengths and Weaknesses 335
14.5.5 Implementation of Weighted Monte Carlo in Practice: S&P
Options 336
14.6 RELATIONSHIP BETWEEN VOLATILITY SKEW AND RISK
NEUTRAL DENSITIES 336
14.7 RISK PREMIA IN THE OPTIONS MARKET: COMPARISON OF RISK
NEUTRAL VS. PHYSICAL MEASURES 338
14.7.1 Comparison of Risk Neutral vs. Physical Measure: Example 339
14.7.2 Connection to Market Implied Risk Premia 340
14.7.3 Taking Advantage of Deviations between the Risk Neutral &
Physical Measure 340
14.8 CONCLUSIONS & ASSESSMENT OF PARAMETRIC VS.
NON-PARAMETRIC METHODS 341
Section IV Quant Modeling in Different Markets
Section  15 ■ Interest Rate Markets 345
15.1 MARKET SETTING 345
15.2 BOND PRICING CONCEPTS 346
15.2.1 Present Value & Discounting Cashflows 346
15.2.2 Pricing a Zero Coupon Bond 347
15.2.3 Pricing a Coupon Bond 347
15.2.4 Daycount Conventions 348
Contents ■ xxi
15.2.5 Yield to Maturity 349
15.2.6 Duration & Convexity 349
15.2.7 Bond Pricing in Practice: Duration and Convexity vs.
Maturity 350
15.2.8 From Yield to Maturity to a Yield Curve 350
15.3 MAIN COMPONENTS OF A YIELD CURVE 351
15.3.1 Overview 351
15.3.2 FRAs & SOFR Futures 351
15.3.3 Swaps 352
15.3.4 Libor vs. SOFR & The Decommissioning of Libor 354
15.4 MARKET RATES 355
15.5 YIELD CURVE CONSTRUCTION 355
15.5.1 Motivation 355
15.5.2 Bootstrapping 357
15.5.3 Optimization 358
15.5.4 Comparison of Methodologies 359
15.5.5 Bootstrapping in Practice: US Swap Rates 359
15.5.6 Empirical Observations of the Yield Curve 360
15.5.7 Fed Policy and the Yield Curve 361
15.6 INFLATION LINKED ASSETS 362
15.6.1 Importance of Inflation Linked Assets 362
15.6.2 Inflation Linked Bonds 362
15.6.3 Inflation Swaps 363
15.6.4 Breakeven Inflation Rates 364
15.7 MODELING INTEREST RATE DERIVATIVES 364
15.7.1 Linear vs. Non-Linear Payoffs 364
15.7.2 Vanilla vs. Exotic Options 365
15.7.3 Most Common Interest Rate Derivatives 365
15.7.4 Modeling the Curve vs. Modeling a Single Rate 366
15.8 MODELING VOLATILITY FOR A SINGLE RATE: CAPS/FLOORS 367
15.8.1 T-Forward Numeraire 367
15.8.2 Caplets/Floorlets via Black’s Model 368
15.8.3 Stripping Cap/Floor Volatilities 369
15.8.4 Fitting the Volatility Skew 370
15.9 MODELING VOLATILITY FOR A SINGLE RATE: SWAPTIONS 370
15.9.1 Annuity Function & Numeraire 370
xxii ■ Contents
15.9.2 Pricing via the Bachelier Model 371
15.9.3 Fitting the Volatility Skew with the SABR Model 371
15.9.4 Swaption Volatility Cube 372
15.10 MODELING THE TERM STRUCTURE: SHORT RATE MODELS 372
15.10.1 Short Rate Models: Overview 372
15.10.2 Ho-Lee 374
15.10.3 Vasicek 375
15.10.4 Cox Ingersol Ross 376
15.10.5 Hull-White 377
15.10.6 Multi-Factor Short Rate Models 377
15.10.7 Two Factor Gaussian Short Rate Model 378
15.10.8 Two Factor Hull-White Model 379
15.10.9 Short Rate Models: Conclusions 379
15.11 MODELING THE TERM STRUCTURE: FORWARD RATE MODELS 380
15.11.1 Libor Market Models: Introduction 380
15.11.2 Log-Normal Libor Market Model 381
15.11.3 SABR Libor Market Model 381
15.11.4 Valuation of Swaptions in an LMM Framework 382
15.12 EXOTIC OPTIONS 383
15.12.1 Spread Options 383
15.12.2 Bermudan Swaptions 384
15.13 INVESTMENT PERSPECTIVE: TRADED STRUCTURES 385
15.13.1 Hedging Interest Rate Risk in Practice 385
15.13.2 Harvesting Carry in Rates Markets: Swaps 386
15.13.3 Swaps vs. Treasuries Basis Trade 387
15.13.4 Conditional Flattener/Steepeners 388
15.13.5 Triangles: Swaptions vs. Mid-Curves 389
15.13.6 Wedges: Caps vs. Swaptions 390
15.13.7 Berm vs. Most Expensive European 391
15.14 CASE STUDY: INTRODUCTION OF NEGATIVE RATES 392
15.15 CASE STUDY: POST-COVID INFLATION SHOCK 393
Section  16 ■ Credit Markets 396
16.1 MARKET SETTING 396
16.2 MODELING DEFAULT RISK: HAZARD RATE MODELS 398
16.3 RISKY BOND 400
Contents ■ xxiii
16.3.1 Modeling Risky Bonds 400
16.3.2 Bonds in Practice: Comparison of Risky & Risk-Free Bond
Duration 402
16.4 CREDIT DEFAULT SWAPS 402
16.4.1 Overview 402
16.4.2 Valuation of CDS 403
16.4.3 Risk Annuity vs. IR Annuity 405
16.4.4 Credit Triangle 405
16.4.5 Mark to Market of a CDS 406
16.4.6 Market Risks of CDS 407
16.5 CDS VS. CORPORATE BONDS 407
16.5.1 CDS Bond Basis 407
16.5.2 What Drives the CDS-Bond Basis? 408
16.6 BOOTSTRAPPING A SURVIVAL CURVE 409
16.6.1 Term Structure of Hazard Rates 409
16.6.2 CDS Curve: Bootstrapping Procedure 410
16.6.3 Alternate Approach: Optimization 410
16.7 INDICES OF CREDIT DEFAULT SWAPS 411
16.7.1 Credit Indices 411
16.7.2 Valuing Credit Indices 412
16.7.3 Index vs. Single Name Basis 413
16.7.4 Credit Indices in Practice: Extracting IG & HY Index Hazard
Rates 414
16.8 MARKET IMPLIED VS. EMPIRICAL DEFAULT PROBABILITIES 414
16.9 OPTIONS ON CDS & CDX INDICES 415
16.9.1 Options on CDS 416
16.9.2 Options on Indices 418
16.10 MODELING CORRELATION: CDOs 418
16.10.1 CDO Subordination Structure 419
16.10.2 Mechanics of CDOs 420
16.10.3 Default Correlation & the Tranche Loss Distribution 420
16.10.4 A Simple Model for CDOs: One Factor Large Pool
Homogeneous Model 421
16.10.5 Correlation Skew 422
16.10.6 CDO Correlation in Practice: Impact of Correlation on
Tranche Valuation 423
16.10.7 Alternative Models for CDOs 424
xxiv ■ Contents
16.11 MODELS CONNECTING EQUITY AND CREDIT 424
16.11.1 Merton’s Model 425
16.11.2 Hirsa-Madan Approach 426
16.12 MORTGAGE BACKED SECURITIES 427
16.12.1 Overview 427
16.12.2 MBS Waterfall Structure 428
16.12.3 Modeling Default & Delinquency Risk 429
16.12.4 Modeling Prepayment Risk 429
16.13 INVESTMENT PERSPECTIVE: TRADED STRUCTURES 429
16.13.1 Hedging Credit Risk 429
16.13.2 Harvesting Carry in Credit Markets 430
16.13.3 CDS Bond Basis 431
16.13.4 Trading Credit Index Calendar Spreads 432
16.13.5 Correlation Trade: Mezzanine vs. Equity Tranches 433
Section  17 ■ Foreign Exchange Markets 435
17.1 MARKET SETTING 435
17.1.1 Overview 435
17.1.2 G10 Major Currencies 436
17.1.3 EM Currencies 436
17.1.4 Major Players 437
17.1.5 Derivatives Market Structure 438
17.2 MODELING IN A CURRENCY SETTING 439
17.2.1 FX Quotations 439
17.2.2 FX Forward Valuations 441
17.2.3 Carry in FX Markets: Do FX Forward Realize? 441
17.2.4 Deliverable vs. Non-Deliverable Forwards 444
17.2.5 FX Triangles 444
17.2.6 Black-Scholes Model in an FX Setting 445
17.2.7 Quoting Conventions in FX Vol. Surfaces 445
17.3 VOLATILITY SMILES IN FOREIGN EXCHANGE MARKETS 448
17.3.1 Persistent Characteristics of FX Volatility Surfaces 448
17.3.2 FX Volatility Surfaces in Practice: Comparison Across
Currency Pairs 449
17.4 EXOTIC OPTIONS IN FOREIGN EXCHANGE MARKETS 450
17.4.1 Digital Options 450
Contents ■ xxv
17.4.2 One Touch Options 450
17.4.3 One-Touches vs. Digis in Practice: Ratio of Prices in
EURJPY 451
17.4.4 Asian Options 452
17.4.5 Barrier Options 453
17.4.6 Volatility & Variance Swaps 454
17.4.7 Dual Digitals 456
17.5 INVESTMENT PERSPECTIVE: TRADED STRUCTURES 457
17.5.1 Hedging Currency Risk 457
17.5.2 Harvesting Carry in FX Markets 459
17.5.3 Trading Dispersion: Currency Triangles 460
17.5.4 Trading Skewness: Digital Options vs. One Touches 461
17.6 CASE STUDY: CHF PEG BREAK IN 2015 462
Section  18 ■ Equity & Commodity Markets 465
18.1 MARKET SETTING 465
18.2 FUTURES CURVES IN EQUITY & COMMODITY MARKETS 466
18.2.1 Determinants of Futures Valuations 466
18.2.2 Futures Curves of Hard to Store Assets 467
18.2.3 Why Are VIX & Commodity Curves Generally in
Contango? 468
18.2.4 Futures Curves in Practice: Excess Contango in Natural Gas
& VIX 468
18.3 VOLATILITY SURFACES IN EQUITY & COMMODITY MARKETS 472
18.3.1 Persistent Characteristics of Equity & Commodity
Volatility Surfaces 472
18.4 EXOTIC OPTIONS IN EQUITY & COMMODITY MARKETS 474
18.4.1 Lookback Options 474
18.4.2 Basket Options 475
18.5 INVESTMENT PERSPECTIVE: TRADED STRUCTURES 476
18.5.1 Hedging Equity Risk 476
18.5.2 Momentum in Single Stocks 477
18.5.3 Harvesting Roll Yield via Commodity Futures Curves 477
18.5.4 Lookback vs. European 479
18.5.5 Dispersion Trading: Index vs. Single Names 480
18.5.6 Leveraged ETF Decay 481
xxvi ■ Contents
18.6 CASE STUDY: NAT. GAS SHORT SQUEEZE 483
18.7 CASE STUDY: VOLATILITY ETP APOCALYPSE OF 2018 486
Section V Portfolio Construction & Risk Management
Section  19 ■ Portfolio Construction & Optimization Techniques 491
19.1 THEORETICAL BACKGROUND 491
19.1.1 Physical vs. Risk-Neutral Measure 491
19.1.2 First- & Second-Order Conditions, Lagrange Multipliers 492
19.1.3 Interpretation of Lagrange Multipliers 493
19.2 MEAN-VARIANCE OPTIMIZATION 495
19.2.1 Investor Utility 495
19.2.2 Unconstrained Mean-Variance Optimization 496
19.2.3 Mean-Variance Efficient Frontier 497
19.2.4 Mean-Variance Fully Invested Efficient Frontier 498
19.2.5 Mean-Variance Optimization in Practice: Efficient Frontier 499
19.2.6 Fully Invested Minimum Variance Portfolio 500
19.2.7 Mean-Variance Optimization with Inequality Constraints 501
19.2.8 Most Common Constraints 501
19.2.9 Mean-Variance Optimization: Market or Factor Exposure
Constraints 502
19.2.10 Mean-Variance Optimization: Turnover Constraint 503
19.2.11 Minimizing Tracking Error to a Benchmark 503
19.2.12 Estimation of Portfolio Optimization Inputs 504
19.3 CHALLENGES ASSOCIATED WITH MEAN-VARIANCE
OPTIMIZATION 505
19.3.1 Estimation Error in Expected Returns 505
19.3.2 Mean-Variance Optimization in Practice: Impact of
Estimation Error 506
19.3.3 Estimation Error of Variance Estimates 508
19.3.4 Singularity of Covariance Matrices 508
19.3.5 Mean-Variance Optimization in Practice: Analysis of
Covariance Matrices 509
19.3.6 Non-Stationarity of Asset Correlations 511
19.4 CAPITAL ASSET PRICING MODEL 511
19.4.1 Leverage & the Tangency Portfolio 511
19.4.2 CAPM 512
Contents ■ xxvii
19.4.3 Systemic vs. Idiosyncratic Risk 513
19.4.4 CAPM in Practice: Efficient Frontier, Tangency Portfolio
and Leverage 513
19.4.5 Multi-Factor Models 514
19.4.6 Fama-French Factors 515
19.5 BLACK-LITTERMAN 516
19.5.1 Market Implied Equilibrium Expected Returns 516
19.5.2 Bayes’ Rule 517
19.5.3 Incorporating Subjective Views 517
19.5.4 The Black-Litterman Model 518
19.6 RESAMPLING 520
19.6.1 Resampling the Efficient Frontier 520
19.6.2 Resampling in Practice: Comparison to a Mean-Variance
Efficient Frontier 521
19.7 ROBUST PORTFOLIO OPTIMIZATION 522
19.8 DOWNSIDE RISK BASED OPTIMIZATION 522
19.8.1 Value at Risk (VaR) 523
19.8.2 Conditional Value at Risk (CVaR) 524
19.8.3 Mean-VaR Optimal Portfolio 524
19.8.4 Mean-CVaR Optimal Portfolio 525
19.9 RISK PARITY 526
19.9.1 Introduction 526
19.9.2 Inverse Volatility Weighting 527
19.9.3 Marginal Risk Contributions 527
19.9.4 Risk Parity Optimization Formulation 528
19.9.5 Strengths and Weaknesses of Risk Parity 529
19.9.6 Asset Class Risk Parity Portfolio in Practice 529
19.10 COMPARISON OF METHODOLOGIES 530
19.11 CASE STUDY: RISK PARITY AND THE POST-COVID INFLATION
SHOCK 530
Section  20 ■ Modeling Expected Returns and Covariance Matrices 533
20.1 SINGLE & MULTI-FACTOR MODELS FOR EXPECTED RETURNS 533
20.1.1 Building Expected Return Models 533
20.1.2 Employing Regularization Techniques 535
xxviii ■ Contents
20.1.3 Regularization Techniques in Practice: Impact on Expected
Return Model 536
20.1.4 Correcting for Serial Correlation 537
20.1.5 Isolating Signal from Noise 539
20.1.6 Information Coefficient 539
20.1.7 Information Coefficient in Practice: Rolling IC of a Short
Term FX Reversal Signal 540
20.1.8 The Fundamental Law of Active Management: Relationship
between Information Ratio & Information Coefficient 541
20.2 MODELING VOLATILITY 542
20.2.1 Estimating Volatility 542
20.2.2 Rolling & Expanding Windows Volatility Estimates 543
20.2.3 Exponentially Weighted Moving Average Estimates 544
20.2.4 High Frequency & Range Based Volatility Estimators 546
20.2.5 Mean-Reverting Volatility Models: GARCH 547
20.2.6 GARCH in Practice: Estimation of GARCH(1,1) Parameters
to Equity Index Returns 550
20.2.7 Estimation of Covariance Matrices 550
20.2.8 Correcting for Negative Eigenvalues 551
20.2.9 Shrinkage Methods for Covariance Matrices 552
20.2.10 Shrinkage in Practice: Impact on Structure of Principal
Components 553
20.2.11 Random Matrix Theory 553
Section  21 ■ Risk Management 556
21.1 MOTIVATION & SETTING 556
21.1.1 Risk Management in Practice 556
21.1.2 Defined vs. Undefined Risks 557
21.1.3 Types of Risk 558
21.2 COMMON RISK MEASURES 559
21.2.1 Portfolio Value at Risk 559
21.2.2 Marginal VaR Contribution 560
21.2.3 Portfolio Conditional Value at Risk 560
21.2.4 Marginal CVaR Contribution 561
21.2.5 Extreme Loss, Stress Tests & Scenario Analysis 561
21.3 CALCULATION OF PORTFOLIO VaR AND CVaR 562
21.3.1 Overview 562
Contents ■ xxix
21.3.2 Historical Simulation 563
21.3.3 Monte Carlo Simulation 564
21.3.4 Strengths and Weaknesses of Each Approach 565
21.3.5 Validating Our Risk Calculations Out-of-Sample 566
21.3.6 VaR in Practice: Out of Sample Test of Rolling VaR 566
21.4 RISK MANAGEMENT OF NON-LINEAR INSTRUMENTS 568
21.4.1 Non-Linear Risk 568
21.4.2 Hedging Portfolios via Scenarios 570
21.5 RISK MANAGEMENT IN RATES & CREDIT MARKETS 570
21.5.1 Introduction 570
21.5.2 Converting from Change in Yield to Change in Price 571
21.5.3 DV01 and Credit Spread 01: Risk Management via Parallel
Shifts 572
21.5.4 Partial DV01’s: Risk Management via Key Rate Shifts 573
21.5.5 Jump to Default Risk 575
21.5.6 Principal Component Based Shifts 576
21.6 MEASURING COUNTERPARTY RISK 577
21.6.1 Counterparty Risk: Introduction 577
21.6.2 Counterparty Risk: Exchange Traded vs. OTC Contracts 578
21.6.3 Wrong Way Counterparty Risk & CDS 578
21.6.4 Credit Valuation Adjustments: Definitions & Terminology 579
21.6.5 Credit Valuation Adjustments: Modeling 579
21.6.6 Credit Valuation Adjustments & Wrong Way CDS Risk 580
21.6.7 Other Valuation Adjustments 580
21.6.8 Counterparty Risk in Practice: Impact of a Credit Valuation
Adjustment on a CDS Contract 581
Section  22 ■ Quantitative Trading Models 583
22.1 INTRODUCTION TO QUANT TRADING MODELS 583
22.1.1 Quant Strategies 583
22.1.2 What is Alpha Research? 584
22.1.3 Types of Quant Strategies 585
22.2 BACK-TESTING 585
22.2.1 Parameter Estimation 586
22.2.2 Modeling Transactions Costs 587
22.2.3 Evaluating Back-Test Performance 589
xxx ■ Contents
22.2.4 Most Common Quant Traps 589
22.2.5 Common Performance Metrics 590
22.2.6 Back-Tested Sharpe Ratios 595
22.2.7 In-Sample and Out-of-Sample Analysis 596
22.2.8 Out-of-Sample Performance & Slippage 597
22.3 COMMON STAT-ARB STRATEGIES 598
22.3.1 Single Asset Momentum & Mean-Reversion Strategies 599
22.3.2 Cross Asset Autocorrelation Strategies 600
22.3.3 Pairs Trading 600
22.3.4 Pairs Trading in Practice: Gold vs. Gold Miners 602
22.3.5 Factor Models 603
22.3.6 PCA-Based Strategies 605
22.3.7 PCA Decomposition in Practice: How many Principal
Components Explain the S&P 500? 607
22.3.8 Risk Premia Strategies 608
22.3.9 Momentum in Practice: Country ETFs 611
22.3.10 Translating Raw Signals to Positions 612
22.4 SYSTEMATIC OPTIONS BASED STRATEGIES 614
22.4.1 Back-Testing Strategies Using Options 614
22.4.2 Common Options Trading Strategies 615
22.4.3 Options Strategy in Practice: Covered Calls on NASDAQ 622
22.5 COMBINING QUANT STRATEGIES 623
22.6 PRINCIPLES OF DISCRETIONARY VS. SYSTEMATIC INVESTING 629
22.7 CASE STUDY: DO TREND FOLLOWING STRATEGIES PRODUCE
CONVEX RETURNS? 630
Section  23 ■ Artificial Intelligence: Incorporating Machine Learning
Techniques 633
23.1 ARTIFICAL INTELLIGENCE & MACHINE LEARNING: LANDSCAPE 633
23.2 MACHINE LEARNING FRAMEWORK 635
23.2.1 Machine Learning vs. Econometrics 635
23.2.2 Stages of a Machine Learning Project 636
23.2.3 Parameter Tuning & Cross Validation 637
23.2.4 Classes of Machine Learning Algorithms 638
23.2.5 Applications of Machine Learning in Asset Management
& Trading 639
Contents ■ xxxi
23.2.6 Challenges of Using Machine Learning in Finance 640
23.3 SUPERVISED VS. UNSUPERVISED LEARNING METHODS 641
23.3.1 Supervised vs. Unsupervised Learning 641
23.3.2 Supervised Learning Methods 642
23.3.3 Regression vs. Classification Techniques 644
23.3.4 Unsupervised Learning Methods 644
23.4 CLUSTERING 645
23.4.1 What is Clustering? 645
23.4.2 K-Means Clustering 646
23.4.3 Hierarchical Clustering 647
23.4.4 Distance Metrics 648
23.4.5 Optimal Number of Clusters 649
23.4.6 Clustering in Finance 649
23.4.7 Clustering in Practice: Asset Class & Risk-on Risk-off
Clusters 650
23.5 CLASSIFICATION TECHNIQUES 652
23.5.1 What is Classification? 652
23.5.2 K-Nearest Neighbor 653
23.5.3 Probit Regression 654
23.5.4 Logistic Regression 656
23.5.5 Support Vector Machines 658
23.5.6 Confusion Matrices 662
23.5.7 Classification Problems in Finance 663
23.5.8 Classification in Practice: Using Classification Techniques in
an Alpha Signal 664
23.5.9 Other uses of Classification Techniques: Credit Risk Modeling 664
23.6 FEATURE IMPORTANCE & INTERPRETABILITY 665
23.6.1 Feature Importance & Interpretability 665
Section  24 ■ Artificial Intelligence: Incorporating Deep Learning, Large
Language Models and Working with Unstructured Data 667
24.1 OVERVIEW 667
24.1.1 Artificial Intelligence and the Investment Process 668
24.2 DEEP LEARNING 669
24.2.1 Deep Neural Networks 669
24.2.2 Autoencoder 671
xxxii ■ Contents
24.2.3 Autoencoder in Practice: Reducing the Dimensionality of a
Covariance Matrix 672
24.2.4 Convolutional Neural Networks 673
24.2.5 Recurrent Neural Networks 674
24.2.6 Generative Adversarial Networks 675
24.3 NATURAL LANGUAGE PROCESSING 677
24.3.1 Alternative Data Sources 677
24.3.2 Natural Language Processing Algorithms 678
24.3.3 Natural Language Processing in Practice: Parsing News for
Trading Signal 680
24.4 LARGE LANGUAGE MODELS 681
24.4.1 What are LLMs? 681
24.4.2 Transformer Networks 682
24.4.3 Prompt Engineering 682
24.4.4 Hallucinations 683
24.5 APPLICATIONS OF DEEP LEARNING & UNSTRUCTURED DATA 683
24.5.1 Delta Hedging Schemes & Optimal Execution via
Reinforcement Learning 683
24.5.2 Volatility Surface Calibration via Deep Learning 684
24.5.3 Credit Risk Estimation via Deep Learning 684
24.5.4 Leveraging GANs in Risk Modeling 685
24.5.5 Generating Expected Returns via Generative AI & LLMs 685
24.5.6 Challenges & Limitations 686
24.5.7 Responsible Use 687
Bibliography 689
Index 707