Quantitative Finance

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.

• List of Figures xvList of Tables xviiPart I Stochastic Processes and Finance 11 Stochastic Processes 31.1 Introduction 31.2 General Characteristics of Stochastic Processes 41.2.1 The Index Set I 41.2.2 The State Space S 41.2.3 Adaptiveness, Filtration, and Standard Filtration 51.2.4 Pathwise Realizations 71.2.5 The Finite Dimensional Distribution of Stochastic Processes 81.2.6 Independent Components 91.2.7 Stationary Process 91.2.8 Stationary and Independent Increments 101.3 Variation and Quadratic Variation of Stochastic Processes 111.4 Other More Specific Properties 131.5 Examples of Stochastic Processes 141.5.1 The Bernoulli Process (Simple Random Walk) 141.5.2 The Brownian Motion (Wiener Process) 171.6 Borel—Cantelli Lemmas 191.7 Central Limit Theorem 201.8 Stochastic Differential Equation 201.9 Stochastic Integral 211.9.1 Properties of the Stochastic Integral 221.10 Maximization and Parameter Calibration of Stochastic Processes 221.10.1 Approximation of the Likelihood Function (Pseudo Maximum Likelihood Estimation) 241.10.2 Ozaki Method 241.10.3 Shoji-Ozaki Method 251.10.4 Kessler Method 251.11 Quadrature Methods 261.11.1 Rectangle Rule: (n = 1) (Darboux Sums) 271.11.2 Midpoint Rule 281.11.3 Trapezoid Rule 281.11.4 Simpson’s Rule 281.12 Problems 292 Basics of Finance 332.1 Introduction 332.2 Arbitrage 332.3 Options 352.3.1 Vanilla Options 352.3.2 Put–Call Parity 362.4 Hedging 392.5 Modeling Return of Stocks 402.6 Continuous Time Model 412.6.1 Itô’s Lemma 422.7 Problems 45Part II Quantitative Finance in Practice 473 Some Models Used in Quantitative Finance 493.1 Introduction 493.2 Assumptions for the Black–Scholes–Merton Derivation 493.3 The B-S Model 503.4 Some Remarks on the B-S Model 583.4.1 Remark 1 583.4.2 Remark 2 583.5 Heston Model 603.5.1 Heston PDE Derivation 613.6 The Cox–Ingersoll–Ross (CIR) Model 633.7 Stochastic 𝛼, 𝛽, 𝜌 (SABR) Model 643.7.1 SABR Implied Volatility 643.8 Methods for Finding Roots of Functions: Implied Volatility 653.8.1 Introduction 653.8.2 The Bisection Method 653.8.3 The Newton’s Method 663.8.4 Secant Method 673.8.5 Computation of Implied Volatility Using the Newton’s Method 683.9 Some Remarks of Implied Volatility (Put–Call Parity) 693.10 Hedging Using Volatility 703.11 Functional Approximation Methods 733.11.1 Local Volatility Model 743.11.2 Dupire’s Equation 743.11.3 Spline Approximation 773.11.4 Numerical Solution Techniques 783.11.5 Pricing Surface 793.12 Problems 794 Solving Partial Differential Equations 834.1 Introduction 834.2 Useful Definitions and Types of PDEs 834.2.1 Types of PDEs (2-D) 834.2.2 Boundary Conditions (BC) for PDEs 844.3 Functional Spaces Useful for PDEs 854.4 Separation of Variables 884.5 Moment-Generating Laplace Transform 914.5.1 Numeric Inversion for Laplace Transform 924.5.2 Fourier Series Approximation Method 934.6 Application of the Laplace Transform to the Black–Scholes PDE 964.7 Problems 995 Wavelets and Fourier Transforms 1015.1 Introduction 1015.2 Dynamic Fourier Analysis 1015.2.1 Tapering 1025.2.2 Estimation of Spectral Density with Daniell Kernel 1035.2.3 Discrete Fourier Transform 1045.2.4 The Fast Fourier Transform (FFT) Method 1065.3 Wavelets Theory 1095.3.1 Definition 1095.3.2 Wavelets and Time Series 1105.4 Examples of Discrete Wavelets Transforms (DWT) 1125.4.1 Haar Wavelets 1125.4.2 Daubechies Wavelets 1155.5 Application of Wavelets Transform 1165.5.1 Finance 1165.5.2 Modeling and Forecasting 1175.5.3 Image Compression 1175.5.4 Seismic Signals 1175.5.5 Damage Detection in Frame Structures 1185.6 Problems 1186 Tree Methods 1216.1 Introduction 1216.2 Tree Methods: the Binomial Tree 1226.2.1 One-Step Binomial Tree 1226.2.2 Using the Tree to Price a European Option 1256.2.3 Using the Tree to Price an American Option 1266.2.4 Using the Tree to Price Any Path-Dependent Option 1276.2.5 Using the Tree for Computing Hedge Sensitivities: the Greeks 1286.2.6 Further Discussion on the American Option Pricing 1286.2.7 A Parenthesis: the Brownian Motion as a Limit of Simple Random Walk 1326.3 Tree Methods for Dividend-Paying Assets 1356.3.1 Options on Assets Paying a Continuous Dividend 1356.3.2 Options on Assets Paying a Known Discrete Proportional Dividend 1366.3.3 Options on Assets Paying a Known Discrete Cash Dividend 1366.3.4 Tree for Known (Deterministic) Time-Varying Volatility 1376.4 Pricing Path-Dependent Options: Barrier Options 1396.5 Trinomial Tree Method and Other Considerations 1406.6 Markov Process 1436.6.1 Transition Function 1436.7 Basic Elements of Operators and Semigroup Theory 1466.7.1 Infinitesimal Operator of Semigroup 1506.7.2 Feller Semigroup 1516.8 General Diffusion Process 1526.8.1 Example: Derivation of Option Pricing PDE 1556.9 A General Diffusion Approximation Method 1566.10 Particle Filter Construction 1596.11 Quadrinomial Tree Approximation 1636.11.1 Construction of the One-Period Model 1646.11.2 Construction of the Multiperiod Model: Option Valuation 1706.12 Problems 1737 Approximating PDEs 1777.1 Introduction 1777.2 The Explicit Finite Difference Method 1797.2.1 Stability and Convergence 1807.3 The Implicit Finite Difference Method 1807.3.1 Stability and Convergence 1827.4 The Crank–Nicolson Finite Difference Method 1837.4.1 Stability and Convergence 1837.5 A Discussion About the Necessary Number of Nodes in the Schemes 1847.5.1 Explicit Finite Difference Method 1847.5.2 Implicit Finite Difference Method 1857.5.3 Crank–Nicolson Finite Difference Method 1857.6 Solution of a Tridiagonal System 1867.6.1 Inverting the Tridiagonal Matrix 1867.6.2 Algorithm for Solving a Tridiagonal System 1877.7 Heston PDE 1887.7.1 Boundary Conditions 1897.7.2 Derivative Approximation for Nonuniform Grid 1907.8 Methods for Free Boundary Problems 1917.8.1 American Option Valuations 1927.8.2 Free Boundary Problem 1927.8.3 Linear Complementarity Problem (LCP) 1937.8.4 The Obstacle Problem 1967.9 Methods for Pricing American Options 1997.10 Problems 2018 Approximating Stochastic Processes 2038.1 Introduction 2038.2 Plain Vanilla Monte Carlo Method 2038.3 Approximation of Integrals Using the Monte Carlo Method 2058.4 Variance Reduction 2058.4.1 Antithetic Variates 2058.4.2 Control Variates 2068.5 American Option Pricing with Monte Carlo Simulation 2088.5.1 Introduction 2098.5.2 Martingale Optimization 2108.5.3 Least Squares Monte Carlo (LSM) 2108.6 Nonstandard Monte Carlo Methods 2168.6.1 Sequential Monte Carlo (SMC) Method 2168.6.2 Markov Chain Monte Carlo (MCMC) Method 2178.7 Generating One-Dimensional Random Variables by Inverting the cdf 2188.8 Generating One-Dimensional Normal Random Variables 2208.8.1 The Box–Muller Method 2218.8.2 The Polar Rejection Method 2228.9 Generating Random Variables: Rejection Sampling Method 2248.9.1 Marsaglia’s Ziggurat Method 2268.10 Generating Random Variables: Importance Sampling 2368.10.1 Sampling Importance Resampling 2408.10.2 Adaptive Importance Sampling 2418.11 Problems 2429 Stochastic Differential Equations 2459.1 Introduction 2459.2 The Construction of the Stochastic Integral 2469.2.1 Itô Integral Construction 2499.2.2 An Illustrative Example 2519.3 Properties of the Stochastic Integral 2539.4 Itô Lemma 2549.5 Stochastic Differential Equations (SDEs) 2579.5.1 Solution Methods for SDEs 2599.6 Examples of Stochastic Differential Equations 2609.6.1 An Analysis of Cox–Ingersoll–Ross (CIR)-Type Models 2639.6.2 Moments Calculation for the CIR Model 2659.6.3 Interpretation of the Formulas for Moments 2679.6.4 Parameter Estimation for the CIR Model 2679.7 Linear Systems of SDEs 2689.8 Some Relationship Between SDEs and Partial Differential Equations (PDEs) 2719.9 Euler Method for Approximating SDEs 2739.10 Random Vectors: Moments and Distributions 2779.10.1 The Dirichlet Distribution 2799.10.2 Multivariate Normal Distribution 2809.11 Generating Multivariate (Gaussian) Distributions with Prescribed Covariance Structure 2819.11.1 Generating Gaussian Vectors 2819.12 Problems 283Part III Advanced Models for Underlying Assets 28710 Stochastic Volatility Models 28910.1 Introduction 28910.2 Stochastic Volatility 28910.3 Types of Continuous Time SV Models 29010.3.1 Constant Elasticity of Variance (CEV) Models 29110.3.2 Hull–White Model 29210.3.3 The Stochastic Alpha Beta Rho (SABR) Model 29310.3.4 Scott Model 29410.3.5 Stein and Stein Model 29510.3.6 Heston Model 29510.4 Derivation of Formulae Used: Mean-Reverting Processes 29610.4.1 Moment Analysis for CIR Type Processes 29910.5 Problems 30111 Jump Diffusion Models 30311.1 Introduction 30311.2 The Poisson Process (Jumps) 30311.3 The Compound Poisson Process 30411.4 The Black–Scholes Models with Jumps 30511.5 Solutions to Partial-Integral Differential Systems 31011.5.1 Suitability of the Stochastic Model Postulated 31111.5.2 Regime-Switching Jump Diffusion Model 31211.5.3 The Option Pricing Problem 31311.5.4 The General PIDE System 31411.6 Problems 32212 General Lévy Processes 32512.1 Introduction and Definitions 32512.2 Lévy Processes 32512.3 Examples of Lévy Processes 32912.3.1 The Gamma Process 32912.3.2 Inverse Gaussian Process 33012.3.3 Exponential Lévy Models 33012.4 Subordination of Lévy Processes 33112.5 Rescaled Range Analysis (Hurst Analysis) and Detrended Fluctuation Analysis (DFA) 33212.5.1 Rescaled Range Analysis (Hurst Analysis) 33212.5.2 Detrended Fluctuation Analysis 33412.5.3 Stationarity and Unit Root Test 33512.6 Problems 33613 Generalized Lévy Processes, Long Range Correlations, and Memory Effects 33713.1 Introduction 33713.1.1 Stable Distributions 33713.2 The Lévy Flight Models 33913.2.1 Background 33913.2.2 Kurtosis 34313.2.3 Self-Similarity 34513.2.4 The H - 𝛼 Relationship for the Truncated Lévy Flight 34613.3 Sum of Lévy Stochastic Variables with Different Parameters 34713.3.1 Sum of Exponential Random Variables with Different Parameters 34813.3.2 Sum of Lévy Random Variables with Different Parameters 35113.4 Examples and Applications 35213.4.1 Truncated Lévy Models Applied to Financial Indices 35213.4.2 Detrended Fluctuation Analysis (DFA) and Rescaled Range Analysis Applied to Financial Indices 35713.5 Problems 36214 Approximating General Derivative Prices 36514.1 Introduction 36514.2 Statement of the Problem 36814.3 A General Parabolic Integro-Differential Problem 37014.3.1 Schaefer’s Fixed Point Theorem 37114.4 Solutions in Bounded Domains 37214.5 Construction of the Solution in the Whole Domain 38514.6 Problems 38615 Solutions to Complex Models Arising in the Pricing of Financial Options 38915.1 Introduction 38915.2 Option Pricing with Transaction Costs and Stochastic Volatility 38915.3 Option Price Valuation in the Geometric Brownian Motion Case with Transaction Costs 39015.4 Stochastic Volatility Model with Transaction Costs 39215.5 The PDE Derivation When the Volatility is a Traded Asset 39315.5.1 The Nonlinear PDE 39515.5.2 Derivation of the Option Value PDEs in Arbitrage Free and Complete Markets 39715.6 Problems 40016 Factor and Copulas Models 40316.1 Introduction 40316.2 Factor Models 40316.2.1 Cross-Sectional Regression 40416.2.2 Expected Return 40616.2.3 Macroeconomic Factor Models 40716.2.4 Fundamental Factor Models 40816.2.5 Statistical Factor Models 40816.3 Copula Models 40916.3.1 Families of Copulas 41116.4 Problems 412Part IV Fixed Income Securities and Derivatives 41317 Models for the Bond Market 41517.1 Introduction and Notations 41517.2 Notations 41517.3 Caps and Swaps 41717.4 Valuation of Basic Instruments: Zero Coupon and Vanilla Options on Zero Coupon 41917.4.1 Black Model 41917.4.2 Short Rate Models 42017.5 Term Structure Consistent Models 42217.6 Inverting the Yield Curve 42617.6.1 Affine Term Structure 42717.7 Problems 42818 Exchange Traded Funds (ETFs), Credit Default Swap (CDS), and Securitization 43118.1 Introduction 43118.2 Exchange Traded Funds (ETFs) 43118.2.1 Index ETFs 43218.2.2 Stock ETFs 43318.2.3 Bond ETFs 43318.2.4 Commodity ETFs 43318.2.5 Currency ETFs 43418.2.6 Inverse ETFs 43518.2.7 Leverage ETFs 43518.3 Credit Default Swap (CDS) 43618.3.1 Example of Credit Default Swap 43718.3.2 Valuation 43718.3.3 Recovery Rate Estimates 43918.3.4 Binary Credit Default Swaps 43918.3.5 Basket Credit Default Swaps 43918.4 Mortgage Backed Securities (MBS) 44018.5 Collateralized Debt Obligation (CDO) 44118.5.1 Collateralized Mortgage Obligations (CMO) 44118.5.2 Collateralized Loan Obligations (CLO) 44218.5.3 Collateralized Bond Obligations (CBO) 44218.6 Problems 443Bibliography 445Index 459