Stochastic Processes with Applications to Finance: 2nd Edition (Hardback) book cover

Stochastic Processes with Applications to Finance

2nd Edition

By Masaaki Kijima

Chapman and Hall/CRC

343 pages | 27 B/W Illus.

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pub: 2013-04-18
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Description

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.

Table of Contents

Elementary Calculus: Towards Ito’s Formula

Exponential and Logarithmic Functions

Differentiation

Taylor’s Expansion

Ito’s Formula

Integration

Elements in Probability

The Sample Space and Probability

Discrete Random Variables

Continuous Random Variables

Bivariate Random Variables

Expectation

Conditional Expectation

Moment Generating Functions

Copulas

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

Options

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

References

Index

Exercises appear at the end of each chapter.

About the Series

Chapman and Hall/CRC Financial Mathematics Series

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
BUS027000
BUSINESS & ECONOMICS / Finance
MAT000000
MATHEMATICS / General
MAT029000
MATHEMATICS / Probability & Statistics / General