You are previewing Financial Mathematics.
O'Reilly logo
Financial Mathematics

Book Description

Versatile for Several Interrelated Courses at the Undergraduate and Graduate Levels

Financial Mathematics: A Comprehensive Treatment provides a unified, self-contained account of the main theory and application of methods behind modern-day financial mathematics. Tested and refined through years of the authors’ teaching experiences, the book encompasses a breadth of topics, from introductory to more advanced ones.

Accessible to undergraduate students in mathematics, finance, actuarial science, economics, and related quantitative areas, much of the text covers essential material for core curriculum courses on financial mathematics. Some of the more advanced topics, such as formal derivative pricing theory, stochastic calculus, Monte Carlo simulation, and numerical methods, can be used in courses at the graduate level. Researchers and practitioners in quantitative finance will also benefit from the combination of analytical and numerical methods for solving various derivative pricing problems.

With an abundance of examples, problems, and fully worked out solutions, the text introduces the financial theory and relevant mathematical methods in a mathematically rigorous yet engaging way. Unlike similar texts in the field, this one presents multiple problem-solving approaches, linking related comprehensive techniques for pricing different types of financial derivatives. The book provides complete coverage of both discrete- and continuous-time financial models that form the cornerstones of financial derivative pricing theory. It also presents a self-contained introduction to stochastic calculus and martingale theory, which are key fundamental elements in quantitative finance.

Table of Contents

  1. Preliminaries
  2. Preface
    1. Objectives and Audience
    2. Guide to Material
    3. Acknowledgements
  3. Part I Introduction to Pricing and Management of Financial Securities
    1. Chapter 1 Mathematics of Compounding
      1. 1.1 Interest and Return
        1. 1.1.1 Amount Function and Return
        2. 1.1.2 Simple Interest
        3. 1.1.3 Periodic Compound Interest
        4. 1.1.4 Continuous Compound Interest
        5. 1.1.5 Equivalent Rates
        6. 1.1.6 Continuously Varying Interest Rates
      2. 1.2 Time Value of Money and Cash Flows
        1. 1.2.1 Equations of Value
        2. 1.2.2 Deterministic Cash Flows
      3. 1.3 Annuities
        1. 1.3.1 Simple Annuities
          1. 1.3.1.1 Ordinary Annuities
          2. 1.3.1.2 Annuities Due
          3. 1.3.1.3 Deferred Annuities
        2. 1.3.2 Determining the Term of an Annuity
        3. 1.3.3 General Annuities
        4. 1.3.4 Perpetuities
        5. 1.3.5 Continuous Annuities
      4. 1.4 Bonds
        1. 1.4.1 Introduction and Terminology
        2. 1.4.2 Zero-Coupon Bonds
        3. 1.4.3 Coupon Bonds
        4. 1.4.4 Serial Bonds, Strip Bonds, and Callable Bonds
      5. 1.5 Yield Rates
        1. 1.5.1 Internal Rate of Return and Evaluation Criteria
        2. 1.5.2 Determining Yield Rates for Bonds
          1. 1.5.2.1 Zero-Coupon Bonds
          2. 1.5.2.2 Coupon Bonds
        3. 1.5.3 Approximation Methods
          1. 1.5.3.1 The Method of Averages
          2. 1.5.3.2 The Method of Interpolation
          3. 1.5.3.3 Numerical Methods
        4. 1.5.4 The Yield Curve
      6. 1.6 Exercises
        1. Figure 1.1
        2. Figure 1.2
        3. Figure 1.4
        4. Figure 1.5
        5. Figure 1.7
        6. Figure 1.8
        7. Figure 1.9
        8. Figure 1.12
        9. Figure 1.13
        10. Figure 1.14
        11. Figure 1.15
        12. Figure 1.17
        1. Table 1.3
        2. Table 1.6
        3. Table 1.10
        4. Table 1.11
        5. Table 1.16
    2. Chapter 2 Primer on Pricing Risky Securities
      1. 2.1 Stocks and Stock Price Models
        1. 2.1.1 Underlying Assets and Derivative Securities
        2. 2.1.2 Basic Assumptions for Asset Price Models
      2. 2.2 Basic Price Models
        1. 2.2.1 A Single-Period Binomial Model
        2. 2.2.2 A Discrete-Time Model with a Finite Number of States
          1. 2.2.2.1 Asset Returns
        3. 2.2.3 Introducing the Binomial Tree Model
        4. 2.2.4 Self-Financing Investment Strategies in the Binomial Model
        5. 2.2.5 Log-Normal Pricing Model
      3. 2.3 Arbitrage and Risk-Neutral Pricing
        1. 2.3.1 The Law of One Price
        2. 2.3.2 A First Look at Arbitrage in the Single-Period Binomial Model
        3. 2.3.3 Arbitrage in the Binomial Tree Model
        4. 2.3.4 Risk-Neutral Probabilities
        5. 2.3.5 Martingale Property
        6. 2.3.6 Risk-Neutral Log-Normal Model
      4. 2.4 Value at Risk
      5. 2.5 Dividend Paying Stock
      6. 2.6 Exercises
        1. Figure 2.1
        2. Figure 2.2
        3. Figure 2.3
        4. Figure 2.4
        5. Figure 2.5
    3. Chapter 3 Portfolio Management
      1. 3.1 Expected Utility Functions
        1. 3.1.1 Utility Functions
          1. 3.1.1.1 Risk Aversion
        2. 3.1.2 Mean-Variance Criterion
      2. 3.2 Portfolio Optimization for Two Assets
        1. 3.2.1 Portfolio of Two Assets
        2. 3.2.2 Portfolio Lines
          1. 3.2.2.1 Case with |ρ<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cSubscript">12</span>| = 1| = 1
          2. 3.2.2.2 Case with |ρ<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cSubscript">12</span>| &lt; 1| < 1
          3. 3.2.2.3 Case with a Risk-Free Asset
        3. 3.2.3 The Minimum Variance Portfolio
          1. 3.2.3.1 Case without Short Selling
        4. 3.2.4 Selection of Optimal Portfolios
          1. 3.2.4.1 Minimum Variance Portfolio
          2. 3.2.4.2 Maximum Expected Utility Portfolio
          3. 3.2.4.3 Minimum Loss-Probability Portfolio
      3. 3.3 Portfolio Optimization for <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cItalic">N</span> Assets Assets
        1. 3.3.1 Portfolios of Several Assets
        2. 3.3.2 The Minimum Variance Portfolio
        3. 3.3.3 The Minimum Variance Portfolio Line
        4. 3.3.4 Case without Short Selling
        5. 3.3.5 Efficient Frontier and Capital Market Line
      4. 3.4 The Capital Asset Pricing Model
      5. 3.5 Exercises
        1. Figure 3.1
        2. Figure 3.2
        3. Figure 3.3
        4. Figure 3.4
        5. Figure 3.5
        6. Figure 3.6
        7. Figure 3.7
        8. Figure 3.8
        9. Figure 3.9
        10. Figure 3.10
        11. Figure 3.11
        12. Figure 3.12
        13. Figure 3.13
    4. Chapter 4 Primer on Derivative Securities
      1. 4.1 Forward Contracts
        1. 4.1.1 No-Arbitrage Evaluation of Forward Contracts
          1. 4.1.1.1 Forward Price for a Stock Paying No Dividends
          2. 4.1.1.2 Forward Price for a Stock Paying Dividends
        2. 4.1.2 Value of a Forward Contract
      2. 4.2 Basic Options Theory
        1. 4.2.1 Concept of an Option Contract
        2. 4.2.2 Put-Call Parities
        3. 4.2.3 Properties of European Options
        4. 4.2.4 Early Exercise and American Options
          1. 4.2.4.1 Relation to European Option Prices
          2. 4.2.4.2 Put-Call Parity Estimate
          3. 4.2.4.3 Monotonicity Properties of American Option Prices
        5. 4.2.5 Nonstandard European Options
      3. 4.3 Basics of Option Pricing
        1. 4.3.1 Pricing of European-Style Derivatives in the Binomial Tree Model
          1. 4.3.1.1 Replication and Pricing of Options in the Single-Period Binomial Model
          2. 4.3.1.2 Pricing in the Binomial Tree Model
        2. 4.3.2 Pricing of American Options in the Binomial Tree Model
        3. 4.3.3 Option Pricing in the Log-Normal Model: The Black–Scholes– Merton Formula
        4. 4.3.4 Greeks and Hedging of Options
          1. 4.3.4.1 Delta of a Derivative in the Binomial Model
          2. 4.3.4.2 Delta of a Derivative in the Log-Normal Model
          3. 4.3.4.3 Delta Hedging
          4. 4.3.4.4 Greeks
        5. 4.3.5 Black–Scholes Equation
      4. 4.4 Exercises
        1. Figure 4.1
        2. Figure 4.2
        3. Figure 4.3
        4. Figure 4.4
        5. Figure 4.5
        6. Figure 4.6
        7. Figure 4.7
        8. Figure 4.8
        9. Figure 4.9
        10. Figure 4.10
        11. Figure 4.11
        12. Figure 4.12
        13. Figure 4.13
        14. Figure 4.14
        15. Figure 4.15
        16. Figure 4.16
        17. Figure 4.17
        18. Figure 4.18
  4. Part II Discrete-Time Modelling
    1. Chapter 5 Single-Period Arrow–Debreu Models
      1. 5.1 Specification of the Model
        1. 5.1.1 Finite-State Economy. Vector Space of Payoffs. Securities
        2. 5.1.2 Initial Price Vector and Payoff Matrix
        3. 5.1.3 Portfolios of Base Securities
      2. 5.2 Analysis of the Arrow–Debreu Model
        1. 5.2.1 Redundant Assets and Attainable Securities
        2. 5.2.2 Completeness of the Model
      3. 5.3 No-Arbitrage Asset Pricing
        1. 5.3.1 The Law of One Price
        2. 5.3.2 Arbitrage
        3. 5.3.3 The First Fundamental Theorem of Asset Pricing
          1. 5.3.3.1 The First FTAP: Sufficiency Part
          2. 5.3.3.2 The First FTAP: Necessity Part
          3. 5.3.3.3 A Geometric Interpretation of the First FTAP
        4. 5.3.4 Risk-Neutral Probabilities
        5. 5.3.5 The Second Fundamental Theorem of Asset Pricing
        6. 5.3.6 Investment Portfolio Optimization
      4. 5.4 Pricing in an Incomplete Market
        1. 5.4.1 A Trinomial Model of an Incomplete Market
        2. 5.4.2 Pricing Nonattainable Payoffs: The Bid-Ask Spread
      5. 5.5 Change of Numéraire
        1. 5.5.1 The Concept of a Numéraire Asset
        2. 5.5.2 Change of Numéraire in a Binomial Model
        3. 5.5.3 Change of Numéraire in a Multinomial Model
      6. 5.6 Exercises
        1. Figure 5.1
        2. Figure 5.2
        3. Figure 5.3
    2. Chapter 6 Introduction to Discrete-Time Stochastic Calculus
      1. 6.1 A Multi-Period Binomial Probability Model
        1. 6.1.1 The Binomial Probability Space
          1. 6.1.1.1 A Sample Space
          2. 6.1.1.2 Random Variables
          3. 6.1.1.3 Probability Measure
        2. 6.1.2 Random Processes
          1. 6.1.2.1 Binomial Price Process and Path Probabilities
          2. 6.1.2.2 Random Walk
      2. 6.2 Information Flow
        1. 6.2.1 Partitions and Their Refinements
          1. 6.2.1.1 Partition Generated by a Random Variable
          2. 6.2.1.2 Refinements of Partitions
        2. 6.2.2 Sigma-Algebras
          1. 6.2.2.1 Construction of a Sigma-Algebra from a Partition
          2. 6.2.2.2 Sigma-Algebra Generated by a Random Variable
        3. 6.2.3 Filtration
          1. 6.2.3.1 Construction of a Filtration from an Information Structure
          2. 6.2.3.2 Construction of a Filtration from a Stochastic Process: Natural Filtration
        4. 6.2.4 Filtered Probability Space
      3. 6.3 Conditional Expectation and Martingales
        1. 6.3.1 Measurability of Random Variables and Processes
        2. 6.3.2 Conditional Expectations
          1. 6.3.2.1 Conditioning on an Event
          2. 6.3.2.2 Conditioning on a Sigma-Algebra
          3. 6.3.2.3 Conditioning on a Random Variable
        3. 6.3.3 Properties of Conditional Expectations
          1. 6.3.3.1 Linearity
          2. 6.3.3.2 Independence
          3. 6.3.3.3 Taking out What Is Known
          4. 6.3.3.4 Tower Property (Iterated Conditioning)
        4. 6.3.4 Conditioning in the Binomial Model
        5. 6.3.5 Sub-, Super-, and True Martingales
          1. 6.3.5.1 Examples
        6. 6.3.6 Classification of Stochastic Processes
        7. 6.3.7 Stopping Times
      4. 6.4 Exercises
        1. Figure 6.1
        2. Figure 6.2
        3. Figure 6.3
        4. Figure 6.4
    3. Chapter 7 Replication and Pricing in the Binomial Tree Model
      1. 7.1 The Standard Binomial Tree Model
      2. 7.2 Self-Financing Strategies and Their Value Processes
        1. 7.2.1 Equivalent Martingale Measures for the Binomial Model
      3. 7.3 Dynamic Replication in the Binomial Tree Model
        1. 7.3.1 Dynamic Replication of Payoffs
        2. 7.3.2 Replication and Valuation of Random Cash Flows
      4. 7.4 Pricing and Hedging Non-Path-Dependent Derivatives
      5. 7.5 Pricing Formulae for Standard European Options
      6. 7.6 Pricing and Hedging Path-Dependent Derivatives
        1. 7.6.1 Average Asset Prices and Asian Options
        2. 7.6.2 Extreme Asset Prices and Lookback Options
        3. 7.6.3 Recursive Evaluation of Path-Dependent Options
          1. 7.6.3.1 Pricing Lookback Options on a Two-Dimensional Lattice
      7. 7.7 American Options
        1. 7.7.1 Writer's Perspective: Pricing and Hedging
        2. 7.7.2 Buyer's Perspective: Optimal Exercise
        3. 7.7.3 Early-Exercise Boundary
        4. 7.7.4 Pricing American Options: The Case with Dividends
      8. 7.8 Exercises
        1. Figure 7.1
        2. Figure 7.2
        3. Figure 7.3
        4. Figure 7.4
        5. Figure 7.5
        6. Figure 7.6
        7. Figure 7.7
        8. Figure 7.8
        9. Figure 7.9
        10. Figure 7.10
        11. Figure 7.13
        12. Figure 7.14
        13. Figure 7.15
        14. Figure 7.16
        1. Table 7.11
        2. Table 7.12
    4. Chapter 8 General Multi-Asset Multi-Period Model
      1. 8.1 Main Elements of the Model
      2. 8.2 Assets, Portfolios, and Strategies
        1. 8.2.1 Payoffs and Assets
        2. 8.2.2 Static and Dynamic Portfolios
        3. 8.2.3 Self-Financing Strategies
        4. 8.2.4 Replication of Payoffs
      3. 8.3 Fundamental Theorems of Asset Pricing
        1. 8.3.1 Arbitrage Strategies
        2. 8.3.2 Enhancing the Law of One Price
        3. 8.3.3 Equivalent Martingale Measures
        4. 8.3.4 Calculation of Martingale Measures
        5. 8.3.5 The First and Second FTAPs
        6. 8.3.6 Pricing and Hedging Derivatives
        7. 8.3.7 Radon–Nikodym Derivative Process and Change of Numéraire
      4. 8.4 Examples of Discrete-Time Models
        1. 8.4.1 Binomial Tree Model with Stochastic Volatility
        2. 8.4.2 Binomial Tree Model for Interest Rates
      5. 8.5 Exercises
        1. Figure 8.1
        2. Figure 8.2
        3. Figure 8.3
        4. Figure 8.4
        5. Figure 8.5
        6. Figure 8.6
  5. Part III Continuous-Time Modelling
    1. Chapter 9 Essentials of General Probability Theory
      1. 9.1 Random Variables and Lebesgue Integration
      2. 9.2 Multidimensional Lebesgue Integration
      3. 9.3 Multiple Random Variables and Joint Distributions
      4. 9.4 Conditioning
      5. 9.5 Changing Probability Measures
    2. Chapter 10 One-Dimensional Brownian Motion and Related Processes
      1. 10.1 Multivariate Normal Distributions
        1. 10.1.1 Multivariate Normal Distribution
        2. 10.1.2 Conditional Normal Distributions
      2. 10.2 Standard Brownian Motion
        1. 10.2.1 One-Dimensional Symmetric Random Walk
        2. 10.2.2 Formal Definition and Basic Properties of Brownian Motion
        3. 10.2.3 Multivariate Distribution of Brownian Motion
        4. 10.2.4 The Markov Property and the Transition PDF
        5. 10.2.5 Quadratic Variation and Nondifferentiability of Paths
      3. 10.3 Some Processes Derived from Brownian Motion
        1. 10.3.1 Drifted Brownian Motion
        2. 10.3.2 Geometric Brownian Motion
        3. 10.3.3 Brownian Bridge
        4. 10.3.4 Gaussian Processes
      4. 10.4 First Hitting Times and Maximum and Minimum of Brownian Motion
        1. 10.4.1 The Reflection Principle: Standard Brownian Motion
        2. 10.4.2 Translated and Scaled Driftless Brownian Motion
        3. 10.4.3 Brownian Motion with Drift
          1. 10.4.3.1 Translated and Scaled Brownian Motion with Drift
      5. 10.5 Exercises
        1. Figure 10.1
        2. Figure 10.2
        3. Figure 10.3
        4. Figure 10.4
        5. Figure 10.5
        6. Figure 10.6
    3. Chapter 11 Introduction to Continuous-Time Stochastic Calculus
      1. 11.1 The Riemann Integral of Brownian Motion
        1. 11.1.1 The Riemann Integral
        2. 11.1.2 The Integral of a Brownian Path
      2. 11.2 The Riemann–Stieltjes Integral of Brownian Motion
        1. 11.2.1 The Riemann–Stieltjes Integral
        2. 11.2.2 Integrals w.r.t. Brownian Motion
      3. 11.3 The Itô Integral and Its Basic Properties
        1. 11.3.1 The Itô Integral for Simple Processes
        2. 11.3.2 Properties of the Itô Integral
      4. 11.4 Itô Processes and Their Properties
        1. 11.4.1 Gaussian Processes Generated by Itô Integrals
        2. 11.4.2 Itô Processes
        3. 11.4.3 Quadratic (Co-) Variation
      5. 11.5 Itô's Formula for Functions of BM and Itô Processes
        1. 11.5.1 Itô's Formula for Functions of BM
        2. 11.5.2 Itô's Formula for Itô Processes
      6. 11.6 Stochastic Differential Equations
        1. 11.6.1 Solutions to Linear SDEs
        2. 11.6.2 Existence and Uniqueness of a Strong Solution of an SDE
      7. 11.7 The Markov Property, Feynman–Kac Formulae, and Transition CDFs and PDFs
        1. 11.7.1 Forward Kolmogorov PDE
        2. 11.7.2 Transition CDF/PDF for Time-Homogeneous Diffusions
      8. 11.8 Radon–Nikodym Derivative Process and Girsanov's Theorem
        1. 11.8.1 Some Applications of Girsanov’s Theorem
      9. 11.9 Brownian Martingale Representation Theorem
      10. 11.10 Stochastic Calculus for Multidimensional BM
        1. 11.10.1 The Itô Integral and Itô's Formula for Multiple Processes on Multidimensional BM
        2. 11.10.2 Multidimensional SDEs, Feynman–Kac Formulae, and Transition CDFs and PDFs
        3. 11.10.3 Girsanov’s Theorem for Multidimensional BM
        4. 11.10.4 Martingale Representation Theorem for Multidimensional BM
      11. 11.11 Exercises
        1. Figure 11.1
    4. Chapter 12 Risk-Neutral Pricing in the (B, S) Economy: One Underlying Stock
      1. 12.1 Replication (Hedging) and Derivative Pricing in the Simplest Black–Scholes Economy
        1. 12.1.1 Pricing Standard European Calls and Puts
        2. 12.1.2 Hedging Standard European Calls and Puts
        3. 12.1.3 Europeans with Piecewise Linear Payoffs
        4. 12.1.4 Power Options
        5. 12.1.5 Dividend Paying Stock
          1. 12.1.5.1 The Case of Continuous Dividend Paying Stock
          2. 12.1.5.2 The Case of Discrete-Time Dividends
      2. 12.2 Forward Starting and Compound Options
      3. 12.3 Some European-Style Path-Dependent Derivatives
        1. 12.3.1 Risk-Neutral Pricing under GBM
        2. 12.3.2 Pricing Single Barrier Options
        3. 12.3.3 Pricing Lookback Options
      4. 12.4 Exercises
        1. Figure 12.1
        2. Figure 12.2
        3. Figure 12.3
        4. Figure 12.4
        5. Figure 12.5
    5. Chapter 13 Risk-Neutral Pricing in a Multi-Asset Economy
      1. 13.1 General Multi-Asset Market Model: Replication and Risk-Neutral Pricing
      2. 13.2 Black–Scholes PDE and Delta Hedging for Standard Multi-Asset Derivatives within a General Diffusion Model
        1. 13.2.1 Standard European Option Pricing for Multi-Stock GBM
        2. 13.2.2 Explicit Pricing Formulae for the GBM Model
          1. 13.2.2.1 Exchange and Other Related Options
          2. 13.2.2.2 Other Basket Options
        3. 13.2.3 Cross-Currency Option Valuation
      3. 13.3 Equivalent Martingale Measures: Derivative Pricing with General Numéraire Assets
      4. 13.4 Exercises
    6. Chapter 14 American Options
      1. 14.1 Basic Properties of Early-Exercise Options
      2. 14.2 Arbitrage-Free Pricing of American Options
        1. 14.2.1 Optimal Stopping Formulation and Early-Exercise Boundary
        2. 14.2.2 The Smooth Pasting Condition
        3. 14.2.3 Put-Call Symmetry Relation
        4. 14.2.4 Dynamic Programming Approach for Bermudan Options
      3. 14.3 Perpetual American Options
        1. 14.3.1 Pricing a Perpetual Put Option
        2. 14.3.2 Pricing a Perpetual Call Option
      4. 14.4 Finite-Expiration American Options
        1. 14.4.1 The PDE Formulation
        2. 14.4.2 The Integral Equation Formulation
      5. 14.5 Exercises
        1. Figure 14.1
        2. Figure 14.2
        3. Figure 14.3
        4. Figure 14.4
        5. Figure 14.5
    7. Chapter 15 Interest-Rate Modelling and Derivative Pricing
      1. 15.1 Basic Fixed Income Instruments
        1. 15.1.1 Bonds
        2. 15.1.2 Forward Rates
        3. 15.1.3 Arbitrage-Free Pricing
        4. 15.1.4 Fixed Income Derivatives
          1. 15.1.4.1 Options on Bonds
          2. 15.1.4.2 Cap and Caplets
          3. 15.1.4.3 Swap and Swaptions
      2. 15.2 Single-Factor Models
        1. 15.2.1 Diffusion Models for the Short Rate Process
        2. 15.2.2 PDE for the Zero-Coupon Bond Value
        3. 15.2.3 Affine Term Structure Models
        4. 15.2.4 The Ho–Lee Model
        5. 15.2.5 The Vasiček Model
        6. 15.2.6 The Cox–Ingersoll–Ross Model
      3. 15.3 Heath–Jarrow–Morton Formulation
        1. 15.3.1 HJM under Risk-Neutral Measure
        2. 15.3.2 Relationship between HJM and Affine Yield Models
          1. 15.3.2.1 The Ho–Lee Model in the HJM Framework
          2. 15.3.2.2 The Vasiček Model in the HJM Framework
      4. 15.4 Multifactor Affine Term Structure Models
        1. 15.4.1 Gaussian Multifactor Models
        2. 15.4.2 Equivalent Classes of Affine Models
      5. 15.5 Pricing Derivatives under Forward Measures
        1. 15.5.1 Forward Measures
        2. 15.5.2 Pricing Stock Options under Stochastic Interest Rates
        3. 15.5.3 Pricing Options on Zero-Coupon Bonds
      6. 15.6 LIBOR Model
        1. 15.6.1 LIBOR Rates
        2. 15.6.2 Brace–Gatarek–Musiela Model of LIBOR Rates
        3. 15.6.3 Pricing Caplets, Caps, and Swaps
      7. 15.7 Exercises
    8. Chapter 16 Alternative Models of Asset Price Dynamics
      1. 16.1 Stochastic Volatility Diffusion Models
        1. 16.1.1 Local Volatility Models
        2. 16.1.2 Constant Elasticity of Variance Model
          1. 16.1.2.1 Definition and Basic Properties
          2. 16.1.2.2 Transition Probability Law
          3. 16.1.2.3 Pricing European Options
        3. 16.1.3 The Heston Model
      2. 16.2 Models with Jumps
        1. 16.2.1 The Poisson Process
        2. 16.2.2 Jump-Diffusion Models with a Compound Poisson Component
        3. 16.2.3 The Variance Gamma Model
      3. 16.3 Exercises
        1. Figure 16.1
        2. Figure 16.2
        3. Figure 16.3
  6. Part IV Computational Techniques
    1. Chapter 17 Introduction to Monte Carlo and Simulation Methods
      1. 17.1 Introduction
        1. 17.1.1 The “Hit-or-Miss” Method
        2. 17.1.2 The Law of Large Numbers
        3. 17.1.3 Approximation Error and Confidence Interval
        4. 17.1.4 Parallel Monte Carlo Methods
        5. 17.1.5 One Monte Carlo Application: Numerical Integration
      2. 17.2 Generation of Uniformly Distributed Random Numbers
        1. 17.2.1 Uniform Probability Distributions
        2. 17.2.2 Linear Congruential Generator
      3. 17.3 Generation of Nonuniformly Distributed Random Numbers
        1. 17.3.1 Transformations of Random Variables
        2. 17.3.2 Inversion Method
          1. 17.3.2.1 Inverse Distribution Function
          2. 17.3.2.2 The Chop-Down Search Method
          3. 17.3.2.3 The Binomial Search Method
        3. 17.3.3 Composition Methods
          1. 17.3.3.1 Mixture of PDFs
          2. 17.3.3.2 Randomized Gamma Distributions
          3. 17.3.3.3 The Alias Method by Walker
        4. 17.3.4 Acceptance-Rejection Methods
        5. 17.3.5 Multivariate Sampling
          1. 17.3.5.1 Sampling by Conditioning
          2. 17.3.5.2 The Box–Müller method
          3. 17.3.5.3 Simulation of Multivariate Normals
      4. 17.4 Simulation of Random Processes
        1. 17.4.1 Simulation of Brownian Processes
          1. 17.4.1.1 Sequential Sampling
          2. 17.4.1.2 Bridge Sampling
        2. 17.4.2 Simulation of Gaussian Processes
        3. 17.4.3 Diffusion Processes: Exact Simulation Methods
          1. 17.4.3.1 The Stochastic Calculus Approach
          2. 17.4.3.2 The PDF Approach
        4. 17.4.4 Diffusion Processes: Approximation Schemes
          1. 17.4.4.1 Types of Convergence
          2. 17.4.4.2 The Euler Scheme
          3. 17.4.4.3 Extrapolation
          4. 17.4.4.4 Error Analysis
        5. 17.4.5 Simulation of Processes with Jumps
          1. 17.4.5.1 Poisson Processes
          2. 17.4.5.2 Subordinated Processes
      5. 17.5 Variance Reduction Methods
        1. 17.5.1 Numerical Integration by a Direct Monte Carlo Method
        2. 17.5.2 Importance Sampling Method
        3. 17.5.3 Change of Probability Measure
        4. 17.5.4 Control Variate Method
        5. 17.5.5 Antithetic Variate
        6. 17.5.6 Conditional Sampling
        7. 17.5.7 Stratified Sampling
      6. 17.6 Exercises
      7. References
        1. Figure 17.2
        2. Figure 17.3
        3. Figure 17.4
        4. Figure 17.5
        5. Figure 17.6
        1. Table 17.1
        2. Table 17.7
    2. Chapter 18 Numerical Applications to Derivative Pricing
      1. 18.1 Overview of Deterministic Numerical Methods
        1. 18.1.1 Quadrature Formulae
          1. 18.1.1.1 Newton–Cotes Quadrature Formulae
          2. 18.1.1.2 Gaussian Quadrature Formulae
          3. 18.1.1.3 Composite Quadrature Formulae
          4. 18.1.1.4 Extrapolation and Romberg Integration
        2. 18.1.2 Finite-Difference Methods
          1. 18.1.2.1 Finite-Difference Approximations for ODEs
          2. 18.1.2.2 Second-Order Linear PDEs
          3. 18.1.2.3 Finite-Difference Approximations for the Heat Equation
            1. The Explicit Method
            2. The Implicit Method
            3. The Crank–Nicolson Method
          4. 18.1.2.4 Stability Analysis
            1. Stability Analysis of the Explicit Method
            2. Stability Analysis of the Implicit Method
            3. Stability Analysis of the Crank–Nicolson Method
      2. 18.2 Pricing European Options
        1. 18.2.1 Pricing European Options by Quadrature Rules
        2. 18.2.2 Pricing European Options by the Monte Carlo Method
        3. 18.2.3 Pricing European Options by Tree Methods
          1. 18.2.3.1 Binomial Model
          2. 18.2.3.2 Multinomial Models
        4. 18.2.4 Pricing European Options by PDEs
          1. 18.2.4.1 Pricing by the Heat Equation
          2. 18.2.4.2 Pricing by the Black–Scholes PDE
            1. The Explicit Method
            2. The Implicit Method
            3. The Crank–Nicolson Method
        5. 18.2.5 Calibration of Asset Price Models to Empirical Data
          1. 18.2.5.1 Least Squares Method
          2. 18.2.5.2 Maximum Likelihood Estimation
      3. 18.3 Pricing Early-Exercise and Path-Dependent Options
        1. 18.3.1 Pricing American and Bermudan Options
          1. 18.3.1.1 Pricing American Options by Tree Methods
          2. 18.3.1.2 Pricing Bermudan Options by the Monte Carlo Method
        2. 18.3.2 Pricing Asian Options
          1. 18.3.2.1 Pricing Discrete-Time Asian Options by the Monte Carlo Method
        1. Figure 18.2
        2. Figure 18.7
        3. Figure 18.9
        4. Figure 18.11
        5. Figure 18.12
        6. Figure 18.14
        7. Figure 18.16
        8. Figure 18.17
        1. Table 18.1
        2. Table 18.3
        3. Table 18.4
        4. Table 18.5
        5. Table 18.6
        6. Table 18.8
        7. Table 18.10
        8. Table 18.13
        9. Table 18.15
        10. Table 18.18
    3. Appendix: Some Useful Integral Identities and Symmetry Properties of Normal Random Variables
    4. Glossary of Symbols and Abbreviations
    5. References
      1. Theory of Probability and Stochastic Processes
      2. Introduction to Mathematics of Finance
      3. Mathematics of Finance (Discrete-Time)
      4. Mathematics of Finance (Continuous-Time)
      5. Computational Methods
      6. Financial Economics