2nd Edition

Stochastic Processes with Applications to Finance

By Masaaki Kijima Copyright 2013
    344 Pages 27 B/W Illustrations
    by Chapman & Hall

    Financial engineering has been proven to be a useful tool for risk management, but using the theory in practice requires a thorough understanding of the risks and ethical standards involved. Stochastic Processes with Applications to Finance, Second Edition presents the mathematical theory of financial engineering using only basic mathematical tools that are easy to understand even for those with little mathematical expertise. This second edition covers several important developments in the financial industry.

    New to the Second Edition

    • A chapter on the change of measures and pricing of insurance products
    • Many examples of the change of measure technique, including its use in asset pricing theory
    • A section on the use of copulas, especially in the pricing of CDOs
    • Two chapters that offer more coverage of interest rate derivatives and credit derivatives

    Exploring the merge of actuarial science and financial engineering, this edition examines how the pricing of insurance products, such as equity-linked annuities, requires knowledge of asset pricing theory since the equity index can be traded in the market. The book looks at the development of many probability transforms for pricing insurance risks, including the Esscher transform. It also describes how the copula model is used to model the joint distribution of underlying assets.

    By presenting significant results in discrete processes and showing how to transfer the results to their continuous counterparts, this text imparts an accessible, practical understanding of the subject. It helps readers not only grasp the theory of financial engineering, but also implement the theory in business.

    Elementary Calculus: Towards Ito’s Formula
    Exponential and Logarithmic Functions
    Taylor’s Expansion
    Ito’s Formula

    Elements in Probability
    The Sample Space and Probability
    Discrete Random Variables
    Continuous Random Variables
    Bivariate Random Variables
    Conditional Expectation
    Moment Generating Functions

    Useful Distributions in Finance
    Binomial Distributions
    Other Discrete Distributions
    Normal and Log-Normal Distributions
    Other Continuous Distributions
    Multivariate Normal Distributions

    Derivative Securities
    The Money-Market Account
    Various Interest Rates
    Forward and Futures Contracts
    Interest-Rate Derivatives

    Change of Measures and the Pricing of Insurance Products
    Change of Measures Based on Positive Random Variables
    BlackScholes Formula and Esscher Transform
    Premium Principles for Insurance Products
    Bühlmann’s Equilibrium Pricing Model

    A Discrete-Time Model for Securities Market
    Price Processes
    Portfolio Value and Stochastic Integral
    No-Arbitrage and Replicating Portfolios
    Martingales and the Asset Pricing Theorem
    American Options
    Change of Measures Based on Positive Martingales

    Random Walks
    The Mathematical Definition
    Transition Probabilities
    The Reflection Principle
    Change of Measures in Random Walks
    The Binomial Securities Market Model

    The Binomial Model
    The Single-Period Model
    Multi-Period Models
    The Binomial Model for American Options
    The Trinomial Model
    The Binomial Model for Interest-Rate Claims

    A Discrete-Time Model for Defaultable Securities
    The Hazard Rate
    Discrete Cox Processes
    Pricing of Defaultable Securities
    Correlated Defaults

    Markov Chains
    Markov and Strong Markov Properties
    Transition Probabilities
    Absorbing Markov Chains
    Applications to Finance

    Monte Carlo Simulation
    Mathematical Backgrounds
    The Idea of Monte Carlo
    Generation of Random Numbers
    Some Examples from Financial Engineering
    Variance Reduction Methods

    From Discrete to Continuous: Towards the BlackScholes
    Brownian Motions
    The Central Limit Theorem Revisited
    The BlackScholes Formula
    More on Brownian Motions
    Poisson Processes

    Basic Stochastic Processes in Continuous Time
    Diffusion Processes
    Sample Paths of Brownian Motions
    Continuous-Time Martingales
    Stochastic Integrals
    Stochastic Differential Equations
    Ito;s Formula Revisited

    A Continuous-Time Model for Securities Market
    Self-Financing Portfolio and No-Arbitrage
    Price Process Models
    The BlackScholes Model
    The Risk-Neutral Method
    The Forward-Neutral Method

    Term-Structure Models and Interest-Rate Derivatives
    Spot-Rate Models
    The Pricing of Discount Bonds
    Pricing of Interest-Rate Derivatives
    Forward LIBOR and Black’s Formula

    A Continuous-Time Model for Defaultable Securities
    The Structural Approach
    The Reduced-Form Approach
    Pricing of Credit Derivatives



    Exercises appear at the end of each chapter.


    M. Kijima (Author)