Quantitative Finance
Description
Written by accomplished teachers and researchers in the field, this book presents quantitative finance theory through applications to specific practical problems and comes with accompanying coding techniques in R and MATLAB, and some generic pseudo-algorithms to modern finance. It also offers over 300 examples and exercises that are appropriate for the beginning student as well as the practitioner in the field.
The Quantitative Finance book is divided into four parts. Part One begins by providing readers with the theoretical backdrop needed from probability and stochastic processes. We also present some useful finance concepts used throughout the book. In part two of the book we present the classical Black-Scholes-Merton model in a uniquely accessible and understandable way. Implied volatility as well as local volatility surfaces are also discussed. Next, solutions to Partial Differential Equations (PDE), wavelets and Fourier transforms are presented. Several methodologies for pricing options namely, tree methods, finite difference method and Monte Carlo simulation methods are also discussed. We conclude this part with a discussion on stochastic differential equations (SDE's). In the third part of this book, several new and advanced models from current literature such as general Lvy processes, nonlinear PDE's for stochastic volatility models in a transaction fee market, PDE's in a jump-diffusion with stochastic volatility models and factor and copulas models are discussed. In part four of the book, we conclude with a solid presentation of the typical topics in fixed income securities and derivatives. We discuss models for pricing bonds market, marketable securities, credit default swaps (CDS) and securitizations.
* Classroom-tested over a three-year period with the input of students and experienced practitioners
* Emphasizes the volatility of financial analyses and interpretations
* Weaves theory with application throughout the book
* Utilizes R and MATLAB software programs
* Presents pseudo-algorithms for readers who do not have access to any particular programming system
* Supplemented with extensive author-maintained web site that includes helpful teaching hints, data sets, software programs, and additional content
Quantitative Finance is an ideal textbook for upper-undergraduate and beginning graduate students in statistics, financial engineering, quantitative finance, and mathematical finance programs. It will also appeal to practitioners in the same fields.
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Persons
MARIA C. MARIANI, PHD, is Shigeko K. Chan Distinguished Professor and Chair in the Department of Mathematical Sciences at The University of Texas at El Paso. She currently focuses her research on mathematical finance, stochastic and non-linear differential equations, geophysics, and numerical methods. Dr. Mariani is co-organizer of the Conference on Modeling High-Frequency Data in Finance.
IONUT FLORESCU, PHD, is Research Professor in Financial Engineering at Stevens Institute of Technology. He serves as Director of the Hanlon Laboratories as well as Director of the Financial Analytics program. His main research is in probability and stochastic processes and applications to domains such as finance, computer vision, robotics, earthquake studies, weather studies, and many more. Dr. Florescu is lead organizer of the Conference on Modeling High-Frequency Data in Finance.
Content
List of Figures xv
List of Tables xvii
Part I Stochastic Processes and Finance 1
1 Stochastic Processes 3
1.1 Introduction 3
1.2 General Characteristics of Stochastic Processes 4
1.2.1 The Index Set I 4
1.2.2 The State Space S 4
1.2.3 Adaptiveness, Filtration, and Standard Filtration 5
1.2.4 Pathwise Realizations 7
1.2.5 The Finite Dimensional Distribution of Stochastic Processes 8
1.2.6 Independent Components 9
1.2.7 Stationary Process 9
1.2.8 Stationary and Independent Increments 10
1.3 Variation and Quadratic Variation of Stochastic Processes 11
1.4 Other More Specific Properties 13
1.5 Examples of Stochastic Processes 14
1.5.1 The Bernoulli Process (Simple Random Walk) 14
1.5.2 The Brownian Motion (Wiener Process) 17
1.6 Borel-Cantelli Lemmas 19
1.7 Central Limit Theorem 20
1.8 Stochastic Differential Equation 20
1.9 Stochastic Integral 21
1.9.1 Properties of the Stochastic Integral 22
1.10 Maximization and Parameter Calibration of Stochastic Processes 22
1.10.1 Approximation of the Likelihood Function (Pseudo Maximum Likelihood Estimation) 24
1.10.2 Ozaki Method 24
1.10.3 Shoji-Ozaki Method 25
1.10.4 Kessler Method 25
1.11 Quadrature Methods 26
1.11.1 Rectangle Rule: (n = 1) (Darboux Sums) 27
1.11.2 Midpoint Rule 28
1.11.3 Trapezoid Rule 28
1.11.4 Simpson's Rule 28
1.12 Problems 29
2 Basics of Finance 33
2.1 Introduction 33
2.2 Arbitrage 33
2.3 Options 35
2.3.1 Vanilla Options 35
2.3.2 Put-Call Parity 36
2.4 Hedging 39
2.5 Modeling Return of Stocks 40
2.6 Continuous Time Model 41
2.6.1 Itô's Lemma 42
2.7 Problems 45
Part II Quantitative Finance in Practice 47
3 Some Models Used in Quantitative Finance 49
3.1 Introduction 49
3.2 Assumptions for the Black-Scholes-Merton Derivation 49
3.3 The B-S Model 50
3.4 Some Remarks on the B-S Model 58
3.4.1 Remark 1 58
3.4.2 Remark 2 58
3.5 Heston Model 60
3.5.1 Heston PDE Derivation 61
3.6 The Cox-Ingersoll-Ross (CIR) Model 63
3.7 Stochastic ¿¿¿¿, ¿¿¿¿, ¿¿¿¿ (SABR) Model 64
3.7.1 SABR Implied Volatility 64
3.8 Methods for Finding Roots of Functions: Implied Volatility 65
3.8.1 Introduction 65
3.8.2 The Bisection Method 65
3.8.3 The Newton's Method 66
3.8.4 Secant Method 67
3.8.5 Computation of Implied Volatility Using the Newton's Method 68
3.9 Some Remarks of Implied Volatility (Put-Call Parity) 69
3.10 Hedging Using Volatility 70
3.11 Functional Approximation Methods 73
3.11.1 Local Volatility Model 74
3.11.2 Dupire's Equation 74
3.11.3 Spline Approximation 77
3.11.4 Numerical Solution Techniques 78
3.11.5 Pricing Surface 79
3.12 Problems 79
4 Solving Partial Differential Equations 83
4.1 Introduction 83
4.2 Useful Definitions and Types of PDEs 83
4.2.1 Types of PDEs (2-D) 83
4.2.2 Boundary Conditions (BC) for PDEs 84
4.3 Functional Spaces Useful for PDEs 85
4.4 Separation of Variables 88
4.5 Moment-Generating Laplace Transform 91
4.5.1 Numeric Inversion for Laplace Transform 92
4.5.2 Fourier Series Approximation Method 93
4.6 Application of the Laplace Transform to the Black-Scholes PDE 96
4.7 Problems 99
5 Wavelets and Fourier Transforms 101
5.1 Introduction 101
5.2 Dynamic Fourier Analysis 101
5.2.1 Tapering 102
5.2.2 Estimation of Spectral Density with Daniell Kernel 103
5.2.3 Discrete Fourier Transform 104
5.2.4 The Fast Fourier Transform (FFT) Method 106
5.3 Wavelets Theory 109
5.3.1 Definition 109
5.3.2 Wavelets and Time Series 110
5.4 Examples of Discrete Wavelets Transforms (DWT) 112
5.4.1 Haar Wavelets 112
5.4.2 Daubechies Wavelets 115
5.5 Application of Wavelets Transform 116
5.5.1 Finance 116
5.5.2 Modeling and Forecasting 117
5.5.3 Image Compression 117
5.5.4 Seismic Signals 117
5.5.5 Damage Detection in Frame Structures 118
5.6 Problems 118
6 Tree Methods 121
6.1 Introduction 121
6.2 Tree Methods: the Binomial Tree 122
6.2.1 One-Step Binomial Tree 122
6.2.2 Using the Tree to Price a European Option 125
6.2.3 Using the Tree to Price an American Option 126
6.2.4 Using the Tree to Price Any Path-Dependent Option 127
6.2.5 Using the Tree for Computing Hedge Sensitivities: the Greeks 128
6.2.6 Further Discussion on the American Option Pricing 128
6.2.7 A Parenthesis: the Brownian Motion as a Limit of Simple Random Walk 132
6.3 Tree Methods for Dividend-Paying Assets 135
6.3.1 Options on Assets Paying a Continuous Dividend 135
6.3.2 Options on Assets Paying a Known Discrete Proportional Dividend 136
6.3.3 Options on Assets Paying a Known Discrete Cash Dividend 136
6.3.4 Tree for Known (Deterministic) Time-Varying Volatility 137
6.4 Pricing Path-Dependent Options: Barrier Options 139
6.5 Trinomial Tree Method and Other Considerations 140
6.6 Markov Process 143
6.6.1 Transition Function 143
6.7 Basic Elements of Operators and Semigroup Theory 146
6.7.1 Infinitesimal Operator of Semigroup 150
6.7.2 Feller Semigroup 151
6.8 General Diffusion Process 152
6.8.1 Example: Derivation of Option Pricing PDE 155
6.9 A General Diffusion Approximation Method 156
6.10 Particle Filter Construction 159
6.11 Quadrinomial Tree Approximation 163
6.11.1 Construction of the One-Period Model 164
6.11.2 Construction of the Multiperiod Model: Option Valuation 170
6.12 Problems 173
7 Approximating PDEs 177
7.1 Introduction 177
7.2 The Explicit Finite Difference Method 179
7.2.1 Stability and Convergence 180
7.3 The Implicit Finite Difference Method 180
7.3.1 Stability and Convergence 182
7.4 The Crank-Nicolson Finite Difference Method 183
7.4.1 Stability and Convergence 183
7.5 A Discussion About the Necessary Number of Nodes in the Schemes 184
7.5.1 Explicit Finite Difference Method 184
7.5.2 Implicit Finite Difference Method 185
7.5.3 Crank-Nicolson Finite Difference Method 185
7.6 Solution of a Tridiagonal System 186
7.6.1 Inverting the Tridiagonal Matrix 186
7.6.2 Algorithm for Solving a Tridiagonal System 187
7.7 Heston PDE 188
7.7.1 Boundary Conditions 189
7.7.2 Derivative Approximation for Nonuniform Grid 190
7.8 Methods for Free Boundary Problems 191
7.8.1 American Option Valuations 192
7.8.2 Free Boundary Problem 192
7.8.3 Linear Complementarity Problem (LCP) 193
7.8.4 The Obstacle Problem 196
7.9 Methods for Pricing American Options 199
7.10 Problems 201
8 Approximating Stochastic Processes 203
8.1 Introduction 203
8.2 Plain Vanilla Monte Carlo Method 203
8.3 Approximation of Integrals Using the Monte Carlo Method 205
8.4 Variance Reduction 205
8.4.1 Antithetic Variates 205
8.4.2 Control Variates 206
8.5 American Option Pricing with Monte Carlo Simulation 208
8.5.1 Introduction 209
8.5.2 Martingale Optimization 210
8.5.3 Least Squares Monte Carlo (LSM) 210
8.6 Nonstandard Monte Carlo Methods 216
8.6.1 Sequential Monte Carlo (SMC) Method 216
8.6.2 Markov Chain Monte Carlo (MCMC) Method 217
8.7 Generating One-Dimensional Random Variables by Inverting the cdf 218
8.8 Generating One-Dimensional Normal Random Variables 220
8.8.1 The Box-Muller Method 221
8.8.2 The Polar Rejection Method 222
8.9 Generating Random Variables: Rejection Sampling Method 224
8.9.1 Marsaglia's Ziggurat Method 226
8.10 Generating Random Variables: Importance Sampling 236
8.10.1 Sampling Importance Resampling 240
8.10.2 Adaptive Importance Sampling 241
8.11 Problems 242
9 Stochastic Differential Equations 245
9.1 Introduction 245
9.2 The Construction of the Stochastic Integral 246
9.2.1 Itô Integral Construction 249
9.2.2 An Illustrative Example 251
9.3 Properties of the Stochastic Integral 253
9.4 Itô Lemma 254
9.5 Stochastic Differential Equations (SDEs) 257
9.5.1 Solution Methods for SDEs 259
9.6 Examples of Stochastic Differential Equations 260
9.6.1 An Analysis of Cox-Ingersoll-Ross (CIR)-Type Models 263
9.6.2 Moments Calculation for the CIR Model 265
9.6.3 Interpretation of the Formulas for Moments 267
9.6.4 Parameter Estimation for the CIR Model 267
9.7 Linear Systems of SDEs 268
9.8 Some Relationship Between SDEs and Partial Differential Equations (PDEs) 271
9.9 Euler Method for Approximating SDEs 273
9.10 Random Vectors: Moments and Distributions 277
9.10.1 The Dirichlet Distribution 279
9.10.2 Multivariate Normal Distribution 280
9.11 Generating Multivariate (Gaussian) Distributions with Prescribed Covariance Structure 281
9.11.1 Generating Gaussian Vectors 281
9.12 Problems 283
Part III Advanced Models for Underlying Assets 287
10 Stochastic Volatility Models 289
10.1 Introduction 289
10.2 Stochastic Volatility 289
10.3 Types of Continuous Time SV Models 290
10.3.1 Constant Elasticity of Variance (CEV) Models 291
10.3.2 Hull-White Model 292
10.3.3 The Stochastic Alpha Beta Rho (SABR) Model 293
10.3.4 Scott Model 294
10.3.5 Stein and Stein Model 295
10.3.6 Heston Model 295
10.4 Derivation of Formulae Used: Mean-Reverting Processes 296
10.4.1 Moment Analysis for CIR Type Processes 299
10.5 Problems 301
11 Jump Diffusion Models 303
11.1 Introduction 303
11.2 The Poisson Process (Jumps) 303
11.3 The Compound Poisson Process 304
11.4 The Black-Scholes Models with Jumps 305
11.5 Solutions to Partial-Integral Differential Systems 310
11.5.1 Suitability of the Stochastic Model Postulated 311
11.5.2 Regime-Switching Jump Diffusion Model 312
11.5.3 The Option Pricing Problem 313
11.5.4 The General PIDE System 314
11.6 Problems 322
12 General Lévy Processes 325
12.1 Introduction and Definitions 325
12.2 Lévy Processes 325
12.3 Examples of Lévy Processes 329
12.3.1 The Gamma Process 329
12.3.2 Inverse Gaussian Process 330
12.3.3 Exponential Lévy Models 330
12.4 Subordination of Lévy Processes 331
12.5 Rescaled Range Analysis (Hurst Analysis) and Detrended Fluctuation Analysis (DFA) 332
12.5.1 Rescaled Range Analysis (Hurst Analysis) 332
12.5.2 Detrended Fluctuation Analysis 334
12.5.3 Stationarity and Unit Root Test 335
12.6 Problems 336
13 Generalized Lévy Processes, Long Range Correlations, and Memory Effects 337
13.1 Introduction 337
13.1.1 Stable Distributions 337
13.2 The Lévy Flight Models 339
13.2.1 Background 339
13.2.2 Kurtosis 343
13.2.3 Self-Similarity 345
13.2.4 The H - ¿¿¿¿ Relationship for the Truncated Lévy Flight 346
13.3 Sum of Lévy Stochastic Variables with Different Parameters 347
13.3.1 Sum of Exponential Random Variables with Different Parameters 348
13.3.2 Sum of Lévy Random Variables with Different Parameters 351
13.4 Examples and Applications 352
13.4.1 Truncated Lévy Models Applied to Financial Indices 352
13.4.2 Detrended Fluctuation Analysis (DFA) and Rescaled Range Analysis Applied to Financial Indices 357
13.5 Problems 362
14 Approximating General Derivative Prices 365
14.1 Introduction 365
14.2 Statement of the Problem 368
14.3 A General Parabolic Integro-Differential Problem 370
14.3.1 Schaefer's Fixed Point Theorem 371
14.4 Solutions in Bounded Domains 372
14.5 Construction of the Solution in the Whole Domain 385
14.6 Problems 386
15 Solutions to Complex Models Arising in the Pricing of Financial Options 389
15.1 Introduction 389
15.2 Option Pricing with Transaction Costs and Stochastic Volatility 389
15.3 Option Price Valuation in the Geometric Brownian Motion Case with Transaction Costs 390
15.4 Stochastic Volatility Model with Transaction Costs 392
15.5 The PDE Derivation When the Volatility is a Traded Asset 393
15.5.1 The Nonlinear PDE 395
15.5.2 Derivation of the Option Value PDEs in Arbitrage Free and Complete Markets 397
15.6 Problems 400
16 Factor and Copulas Models 403
16.1 Introduction 403
16.2 Factor Models 403
16.2.1 Cross-Sectional Regression 404
16.2.2 Expected Return 406
16.2.3 Macroeconomic Factor Models 407
16.2.4 Fundamental Factor Models 408
16.2.5 Statistical Factor Models 408
16.3 Copula Models 409
16.3.1 Families of Copulas 411
16.4 Problems 412
Part IV Fixed Income Securities and Derivatives 413
17 Models for the Bond Market 415
17.1 Introduction and Notations 415
17.2 Notations 415
17.3 Caps and Swaps 417
17.4 Valuation of Basic Instruments: Zero Coupon and Vanilla Options on Zero Coupon 419
17.4.1 Black Model 419
17.4.2 Short Rate Models 420
17.5 Term Structure Consistent Models 422
17.6 Inverting the Yield Curve 426
17.6.1 Affine Term Structure 427
17.7 Problems 428
18 Exchange Traded Funds (ETFs), Credit Default Swap (CDS), and Securitization 431
18.1 Introduction 431
18.2 Exchange Traded Funds (ETFs) 431
18.2.1 Index ETFs 432
18.2.2 Stock ETFs 433
18.2.3 Bond ETFs 433
18.2.4 Commodity ETFs 433
18.2.5 Currency ETFs 434
18.2.6 Inverse ETFs 435
18.2.7 Leverage ETFs 435
18.3 Credit Default Swap (CDS) 436
18.3.1 Example of Credit Default Swap 437
18.3.2 Valuation 437
18.3.3 Recovery Rate Estimates 439
18.3.4 Binary Credit Default Swaps 439
18.3.5 Basket Credit Default Swaps 439
18.4 Mortgage Backed Securities (MBS) 440
18.5 Collateralized Debt Obligation (CDO) 441
18.5.1 Collateralized Mortgage Obligations (CMO) 441
18.5.2 Collateralized Loan Obligations (CLO) 442
18.5.3 Collateralized Bond Obligations (CBO) 442
18.6 Problems 443
Bibliography 445
Index 459
1
Stochastic Processes: A Brief Review
1.1 Introduction
In this chapter, we introduce the basic mathematical tools we will use. We assume the reader has a good understanding of probability spaces and random variables. For more details we refer to [67, 70]. This chapter is not meant to be a replacement for a book. To get the fundamentals please consult [70, 117]. In this chapter, we are reviewing fundamental notions for the rest of the book.
So, what is a stochastic process? When asked this question, R.A. Fisher famously replied, "What is a stochastic process? Oh, it's just one darn thing after another." We hope to elaborate on Fisher's reply in this introduction.
We start the study of stochastic processes by presenting some commonly assumed properties and characteristics. Generally, these characteristics simplify analysis of stochastic processes. However, a stochastic process with these properties will have simplified dynamics, and the resulting models may not be complex enough to model real-life behavior. In Section 1.6 of this chapter, we introduce the simplest stochastic processes: the coin toss process (also known as the Bernoulli process) which produces the simple random walk.
We start with the definition of a stochastic process.
Definition 1.1.1
Given a probability space , a stochastic process is any collection of random variables defined on this probability space, where is an index set. The notations and are used interchangeably to denote the value of the stochastic process at index value .
Specifically, for any fixed the resulting is just a random variable. However, what makes this index set special is that it confers the collection of random variables a certain structure. This will be explained next.
1.2 General Characteristics of Stochastic Processes
1.2.1 The Index Set
The set indexes and determines the type of stochastic process. This set can be quite general but here are some examples:
- If or equivalent, we obtain the so-called discrete-time stochastic processes. We shall often write the process as in this case.
- If we obtain the continuous-time stochastic processes. We shall write the process as in this case. Most of the time represents time.
- The index set can be multidimensional. For example, with , we may be describing a discrete random field where at any combination we have a value which may represent some node weights in a two-dimensional graph. If we may be describing the structure of some surface where, for instance, could be the value of some electrical field intensity at position .
1.2.2 The State Space
The state space is the domain space of all the random variables . Since we are discussing about random variables and random vectors, then necessarily or . Again, we have several important examples:
- If , then the process is integer valued or a process with discrete state space.
- If , then is a real-valued process or a process with a continuous state space.
- If , then is a -dimensional vector process.
The state space can be more general (for example, an abstract Lie algebra), in which case the definitions work very similarly except that for each we have measurable functions.
We recall that a real-valued function defined on is called measurable with respect to a sigma algebra in that space if the inverse image of set , defined as is a set in sigma algebra , for all Borel sets of .
A sigma algebra is a collection of sets of satisfying the following conditions:
- .
- If then its complement .
- If is a countable collection of sets in then their union
Suppose we have a random variable defined on a space . The sigma algebra generated by is the smallest sigma algebra in that contains all the pre images of sets in through . That is,
This abstract concept is necessary to make sure that we may calculate any probability related to the random variable .
1.2.3 Adaptiveness, Filtration, and Standard Filtration
In the special case when the index set possesses a total order relationship,1 we can discuss about the information contained in the process at some moment . To quantify this information we generalize the notion of sigma algebras by introducing a sequence of sigma algebras: the filtration.
Definition 1.2.1 (Filtration).
A probability space is a filtered probability space if and only if there exists a sequence of sigma algebras included in such that is an increasing collection i.e.:
The filtration is called complete if its first element contains all the null sets of . If, for example, 0 is the first element of the index set (the usual situation) then This particular notion of a complete filtration is not satisfied and may lead to all sorts of contracdictions and counterexamples. To avoid any such case we shall assume that any filtration defined in this book is complete and all filtered probability spaces are complete.
In the particular case of continuous time (i.e. ), it makes sense to discuss about what happens with the filtration when two consecutive times get close to one another. For some specific time we define the left and right sigma algebras:
The countable intersection of sigma algebras is always a sigma algebra [67], but a union of sigma algebras is not necessarily a sigma algebra. This is why we modified the definition of slightly. The notation used represents the smallest sigma algebra that contains the collection of sets .
Definition 1.2.2 (Right and Left Continuous Filtrations).
A filtration is right continuous if and only if for all , and the filtration is left continuous if and only if for all .
In general we shall assume throughout (if applicable) that any filtration is right continuous.
Definition 1.2.3 (Adapted Stochastic Process).
A stochastic process defined on a filtered probability space is called adapted if and only if is -measurable for any .
This is an important concept since in general, quantifies the flow of information available at any moment . By requiring that the process be adapted, we ensure that we can calculate probabilities related to based solely on the information available at time . Furthermore, since the filtration by definition is increasing, this also says that we can calculate the probabilities at any later moment in time as well.
On the other hand, due to the same increasing property of a filtration, it may not be possible to calculate probabilities related to based only on the information available in for a moment earlier than (i.e. ). This is the reason why the conditional expectation is a crucial concept for stochastic processes. Recall that is -measurable. Suppose we are sitting at time and trying to calculate probabilities related to the random variable at some time in the future. Even though we may not calculate the probabilities related to directly (nobody can since will be in the future), we can still calculate its distribution according to its best guess based on the current information. That is precisely .
Definition 1.2.4 (Standard Filtration).
In some cases, we are only given a standard probability space (without a separate filtration defined on the space). This typically corresponds to the case where we assume that all the information available at time comes from the stochastic process itself. No external sources of information are available. In this case, we will be using the standard filtration generated by the process itself. Let
denote the sigma algebra generated by the random variables up to time . The collection of sigma algebras is increasing and obviously the process is adapted with respect to it.
Notation
In the case when the filtration is not specified, we will always construct the standard filtration and denote it with .
In the special case when , the set of natural numbers, and the filtration is generated by the process, we will sometimes substitute the notation instead of . For example we may write
1.2.4 Pathwise Realizations
Suppose a stochastic process is defined on some probability space . Recall that by definition for every fixed, is a random variable. On the other hand, for every fixed we shall find a particular realization for any time 's, this outcome is typically denoted . Therefore, for each we can find a collection of numbers representing the realization of the stochastic process. That is a path. This realization may be thought of as the function .
This pathwise idea means that we can map each into a function from into . Therefore, the process may be identified as a subset of all the functions from into .
In Figure 1.1 we plot three different paths each corresponding to a different realization , . Due to this pathwise representation, calculating probabilities related to stochastic processes is equivalent with calculating the distribution of these paths in subsets of the two-dimensional space. For example, the probability
is the probability of the paths being in the unit square. However, such a...
