Quantitative Finance

Quantitative Finance

by Maria C. Mariani, Ionut Florescu
Released December 2019
Publisher(s): Wiley
ISBN: 9781118629956

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Book description

Presents a multitude of topics relevant to the quantitative finance community by combining the best of the theory with the usefulness of applications

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.

Table of contents

  1. Cover
  2. List of Figures
  3. List of Tables
  4. Part I: Stochastic Processes and Finance
    1. 1 Stochastic Processes
      1. 1.1 Introduction
      2. 1.2 General Characteristics of Stochastic Processes
      3. 1.3 Variation and Quadratic Variation of Stochastic Processes
      4. 1.4 Other More Specific Properties
      5. 1.5 Examples of Stochastic Processes
      6. 1.6 Borel—Cantelli Lemmas
      7. 1.7 Central Limit Theorem
      8. 1.8 Stochastic Differential Equation
      9. 1.9 Stochastic Integral
      10. 1.10 Maximization and Parameter Calibration of Stochastic Processes
      11. 1.11 Quadrature Methods
      12. 1.12 Problems
    2. 2 Basics of Finance
      1. 2.1 Introduction
      2. 2.2 Arbitrage
      3. 2.3 Options
      4. 2.4 Hedging
      5. 2.5 Modeling Return of Stocks
      6. 2.6 Continuous Time Model
      7. 2.7 Problems
  5. Part II: Quantitative Finance in Practice
    1. 3 Some Models Used in Quantitative Finance
      1. 3.1 Introduction
      2. 3.2 Assumptions for the Black–Scholes–Merton Derivation
      3. 3.3 The B‐S Model
      4. 3.4 Some Remarks on the B‐S Model
      5. 3.5 Heston Model
      6. 3.6 The Cox–Ingersoll–Ross (CIR) Model
      7. 3.7 Stochastic (SABR) Model
      8. 3.8 Methods for Finding Roots of Functions: Implied Volatility
      9. 3.9 Some Remarks of Implied Volatility (Put–Call Parity)
      10. 3.10 Hedging Using Volatility
      11. 3.11 Functional Approximation Methods
      12. 3.12 Problems
    2. 4 Solving Partial Differential Equations
      1. 4.1 Introduction
      2. 4.2 Useful Definitions and Types of PDEs
      3. 4.3 Functional Spaces Useful for PDEs
      4. 4.4 Separation of Variables
      5. 4.5 Moment‐Generating Laplace Transform
      6. 4.6 Application of the Laplace Transform to the Black–Scholes PDE
      7. 4.7 Problems
    3. 5 Wavelets and Fourier Transforms
      1. 5.1 Introduction
      2. 5.2 Dynamic Fourier Analysis
      3. 5.3 Wavelets Theory
      4. 5.4 Examples of Discrete Wavelets Transforms (DWT)
      5. 5.5 Application of Wavelets Transform
      6. 5.6 Problems
    4. 6 Tree Methods
      1. 6.1 Introduction
      2. 6.2 Tree Methods: the Binomial Tree
      3. 6.3 Tree Methods for Dividend‐Paying Assets
      4. 6.4 Pricing Path‐Dependent Options: Barrier Options
      5. 6.5 Trinomial Tree Method and Other Considerations
      6. 6.6 Markov Process
      7. 6.7 Basic Elements of Operators and Semigroup Theory
      8. 6.8 General Diffusion Process
      9. 6.9 A General Diffusion Approximation Method
      10. 6.10 Particle Filter Construction
      11. 6.11 Quadrinomial Tree Approximation
      12. 6.12 Problems
    5. 7 Approximating PDEs
      1. 7.1 Introduction
      2. 7.2 The Explicit Finite Difference Method
      3. 7.3 The Implicit Finite Difference Method
      4. 7.4 The Crank–Nicolson Finite Difference Method
      5. 7.5 A Discussion About the Necessary Number of Nodes in the Schemes
      6. 7.6 Solution of a Tridiagonal System
      7. 7.7 Heston PDE
      8. 7.8 Methods for Free Boundary Problems
      9. 7.9 Methods for Pricing American Options
      10. 7.10 Problems
    6. 8 Approximating Stochastic Processes
      1. 8.1 Introduction
      2. 8.2 Plain Vanilla Monte Carlo Method
      3. 8.3 Approximation of Integrals Using the Monte Carlo Method
      4. 8.4 Variance Reduction
      5. 8.5 American Option Pricing with Monte Carlo Simulation
      6. 8.6 Nonstandard Monte Carlo Methods
      7. 8.7 Generating One‐Dimensional Random Variables by Inverting the cdf
      8. 8.8 Generating One‐Dimensional Normal Random Variables
      9. 8.9 Generating Random Variables: Rejection Sampling Method
      10. 8.10 Generating Random Variables: Importance Sampling
      11. 8.11 Problems
    7. 9 Stochastic Differential Equations
      1. 9.1 Introduction
      2. 9.2 The Construction of the Stochastic Integral
      3. 9.3 Properties of the Stochastic Integral
      4. 9.4 Itô Lemma
      5. 9.5 Stochastic Differential Equations (SDEs)
      6. 9.6 Examples of Stochastic Differential Equations
      7. 9.7 Linear Systems of SDEs
      8. 9.8 Some Relationship Between SDEs and Partial Differential Equations (PDEs)
      9. 9.9 Euler Method for Approximating SDEs
      10. 9.10 Random Vectors: Moments and Distributions
      11. 9.11 Generating Multivariate (Gaussian) Distributions with Prescribed Covariance Structure
      12. 9.12 Problems
  6. Part III: Advanced Models for Underlying Assets
    1. 10 Stochastic Volatility Models
      1. 10.1 Introduction
      2. 10.2 Stochastic Volatility
      3. 10.3 Types of Continuous Time SV Models
      4. 10.4 Derivation of Formulae Used: Mean‐Reverting Processes
      5. 10.5 Problems
    2. 11 Jump Diffusion Models
      1. 11.1 Introduction
      2. 11.2 The Poisson Process (Jumps)
      3. 11.3 The Compound Poisson Process
      4. 11.4 The Black–Scholes Models with Jumps
      5. 11.5 Solutions to Partial‐Integral Differential Systems
      6. 11.6 Problems
    3. 12 General Lévy Processes
      1. 12.1 Introduction and Definitions
      2. 12.2 Lévy Processes
      3. 12.3 Examples of Lévy Processes
      4. 12.4 Subordination of Lvy Processes
      5. 12.5 Rescaled Range Analysis (Hurst Analysis) and Detrended Fluctuation Analysis (DFA)
      6. 12.6 Problems
    4. 13 Generalized Lévy Processes, Long Range Correlations, and Memory Effects
      1. 13.1 Introduction
      2. 13.2 The Lévy Flight Models
      3. 13.3 Sum of Lévy Stochastic Variables with Different Parameters
      4. 13.4 Examples and Applications
      5. 13.5 Problems
    5. 14 Approximating General Derivative Prices
      1. 14.1 Introduction
      2. 14.2 Statement of the Problem
      3. 14.3 A General Parabolic Integro‐Differential Problem
      4. 14.4 Solutions in Bounded Domains
      5. 14.5 Construction of the Solution in the Whole Domain
      6. 14.6 Problems
    6. 15 Solutions to Complex Models Arising in the Pricing of Financial Options
      1. 15.1 Introduction
      2. 15.2 Option Pricing with Transaction Costs and Stochastic Volatility
      3. 15.3 Option Price Valuation in the Geometric Brownian Motion Case with Transaction Costs
      4. 15.4 Stochastic Volatility Model with Transaction Costs
      5. 15.5 The PDE Derivation When the Volatility is a Traded Asset
      6. 15.6 Problems
    7. 16 Factor and Copulas Models
      1. 16.1 Introduction
      2. 16.2 Factor Models
      3. 16.3 Copula Models
      4. 16.4 Problems
  7. Part IV: Fixed Income Securities and Derivatives
    1. 17 Models for the Bond Market
      1. 17.1 Introduction and Notations
      2. 17.2 Notations
      3. 17.3 Caps and Swaps
      4. 17.4 Valuation of Basic Instruments: Zero Coupon and Vanilla Options on Zero Coupon
      5. 17.5 Term Structure Consistent Models
      6. 17.6 Inverting the Yield Curve
      7. 17.7 Problems
    2. 18 Exchange Traded Funds (ETFs), Credit Default Swap (CDS), and Securitization
      1. 18.1 Introduction
      2. 18.2 Exchange Traded Funds (ETFs)
      3. 18.3 Credit Default Swap (CDS)
      4. 18.4 Mortgage Backed Securities (MBS)
      5. 18.5 Collateralized Debt Obligation (CDO)
      6. 18.6 Problems
  8. Bibliography
  9. Index
  10. End User License Agreement

Product information

  • Title: Quantitative Finance
  • Author(s): Maria C. Mariani, Ionut Florescu
  • Release date: December 2019
  • Publisher(s): Wiley
  • ISBN: 9781118629956