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 41.3 Variation and Quadratic Variation of Stochastic Processes 111.4 Other More Specific Properties 131.5 Examples of Stochastic Processes 141.6 Borel—Cantelli Lemmas 191.7 Central Limit Theorem 201.8 Stochastic Differential Equation 201.9 Stochastic Integral 211.10 Maximization and Parameter Calibration of Stochastic Processes 221.11 Quadrature Methods 261.12 Problems 292 Basics of Finance 332.1 Introduction 332.2 Arbitrage 332.3 Options 352.4 Hedging 392.5 Modeling Return of Stocks 402.6 Continuous Time Model 412.7 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 583.5 Heston Model 603.6 The Cox–Ingersoll–Ross (CIR) Model 633.7 Stochastic α, β, ρ (SABR) Model 643.8 Methods for Finding Roots of Functions: Implied Volatility 653.9 Some Remarks of Implied Volatility (Put–Call Parity) 693.10 Hedging Using Volatility 703.11 Functional Approximation Methods 733.12 Problems 794 Solving Partial Differential Equations 834.1 Introduction 834.2 Useful Definitions and Types of PDEs 834.3 Functional Spaces Useful for PDEs 854.4 Separation of Variables 884.5 Moment-Generating Laplace Transform 914.6 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 1015.3 Wavelets Theory 1095.4 Examples of Discrete Wavelets Transforms (DWT) 1125.5 Application of Wavelets Transform 1165.6 Problems 1186 Tree Methods 1216.1 Introduction 1216.2 Tree Methods: the Binomial Tree 1226.3 Tree Methods for Dividend-Paying Assets 1356.4 Pricing Path-Dependent Options: Barrier Options 1396.5 Trinomial Tree Method and Other Considerations 1406.6 Markov Process 1436.7 Basic Elements of Operators and Semigroup Theory 1466.8 General Diffusion Process 1526.9 A General Diffusion Approximation Method 1566.10 Particle Filter Construction 1596.11 Quadrinomial Tree Approximation 1636.12 Problems 1737 Approximating PDEs 1777.1 Introduction 1777.2 The Explicit Finite Difference Method 1797.3 The Implicit Finite Difference Method 1807.4 The Crank–Nicolson Finite Difference Method 1837.5 A Discussion About the Necessary Number of Nodes in the Schemes 1847.6 Solution of a Tridiagonal System 1867.7 Heston PDE 1887.8 Methods for Free Boundary Problems 1917.9 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 2058.5 American Option Pricing with Monte Carlo Simulation 2088.6 Nonstandard Monte Carlo Methods 2168.7 Generating One-Dimensional Random Variables by Inverting the cdf 2188.8 Generating One-Dimensional Normal Random Variables 2208.9 Generating Random Variables: Rejection Sampling Method 2248.10 Generating Random Variables: Importance Sampling 2368.11 Problems 2429 Stochastic Differential Equations 2459.1 Introduction 2459.2 The Construction of the Stochastic Integral 2469.3 Properties of the Stochastic Integral 2539.4 Itô Lemma 2549.5 Stochastic Differential Equations (SDEs) 2579.6 Examples of Stochastic Differential Equations 2609.7 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 2779.11 Generating Multivariate (Gaussian) Distributions with Prescribed Covariance Structure 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 29010.4 Derivation of Formulae Used: Mean-Reverting Processes 29610.5 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 31011.6 Problems 32212 General Lévy Processes 32512.1 Introduction and Definitions 32512.2 Lévy Processes 32512.3 Examples of Lévy Processes 32912.4 Subordination of Lévy Processes 33112.5 Rescaled Range Analysis (Hurst Analysis) and Detrended Fluctuation Analysis (DFA) 33212.6 Problems 33613 Generalized Lévy Processes, Long Range Correlations, and Memory Effects 33713.1 Introduction 33713.2 The Lévy Flight Models 33913.3 Sum of Lévy Stochastic Variables with Different Parameters 34713.4 Examples and Applications 35213.5 Problems 36214 Approximating General Derivative Prices 36514.1 Introduction 36514.2 Statement of the Problem 36814.3 A General Parabolic Integro-Differential Problem 37014.4 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 39315.6 Problems 40016 Factor and Copulas Models 40316.1 Introduction 40316.2 Factor Models 40316.3 Copula Models 40916.4 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 41917.5 Term Structure Consistent Models 42217.6 Inverting the Yield Curve 42617.7 Problems 42818 Exchange Traded Funds (ETFs), Credit Default Swap (CDS), and Securitization 43118.1 Introduction 43118.2 Exchange Traded Funds (ETFs) 43118.3 Credit Default Swap (CDS) 43618.4 Mortgage Backed Securities (MBS) 44018.5 Collateralized Debt Obligation (CDO) 44118.6 Problems 443Bibliography 445Index 459