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Stochastic Processes with Applications to Finance

By Masaaki Kijima

Published July 29th 2002 by Chapman and Hall/CRC – 288 pages

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Description

In recent years, modeling financial uncertainty using stochastic processes has become increasingly important, but it is commonly perceived as requiring a deep mathematical background. Stochastic Processes with Applications to Finance shows that this is not necessarily so. It presents the theory of discrete stochastic processes and their applications in finance in an accessible treatment that strikes a balance between the abstract and the practical.

Using an approach that views sophisticated stochastic calculus as based on a simple class of discrete processes-"random walks"-the author first provides an elementary introduction to the relevant areas of real analysis and probability. He then uses random walks to explain the change of measure formula, the reflection principle, and the Kolmogorov backward equation. The Black-Scholes formula is derived as a limit of binomial model, and applications to the pricing of derivative securities are presented. Another primary focus of the book is the pricing of corporate bonds and credit derivatives, which the author explains in terms of discrete default models.

By presenting important results in discrete processes and showing how to transfer those results to their continuous counterparts, Stochastic Processes with Applications to Finance imparts an intuitive and practical understanding of the subject. This unique treatment is ideal both as a text for a graduate-level class and as a reference for researchers and practitioners in financial engineering, operations research, and mathematical and statistical finance.

Contents

ELEMENTARY CALCULUS: TOWARDS ITO'S FORMULA

Exponential and Logarithmic Functions

Differentiation

Taylor's Expansion

Ito's Formula

Integration

Exercises

ELEMENTS IN PROBABILITY

The Sample Space and Probability

Discrete Random Variables

Continuous Random Variables

Multivariate Random Variables

Expectation

Conditional Expectation

Moment Generating Functions

Exercises

USEFUL DISTRIBUTIONS IN FINANCE

Binomial Distributions

Other Discrete Distribution

Normal and Log-Normal Distributions

Other Continuous Distributions

Multivariate Normal Distributions

Exercises

DERIVATIVE SECURITIES

The Money-Market Account

Various Interest Rates

Forward and Futures Contracts

Options

Interest-Rate Derivatives

Exercises

A DISCRETE-TIME MODEL FOR SECURITIES MARKET

Price Processes

The Portfolio Value and Stochastic Integral

No-Arbitrage and Replication Portfolios

Martingales and the Asset Pricing Theorem

American Options

Change of Measure

Exercises

RANDOM WALKS

The Mathematical Definition

Transition Probabilities

The Reflection Principle

Change of Measure Revisited

A Binomial Securities Market Model

Exercises

THE BINOMIAL MODEL

The Single-Period Model

The Multi-Period Model

The Binomial Model for American Options

The Trinomial Model

The Binomial Model for Interest-Rate Claims

Exercises

A DISCRETE-TIME MODEL FOR DEFAULTABLE SECURITIES

The Hazard Rate

A Discrete Hazard Model

Pricing of Defaultable Securities

Correlated Defaults

Exercises

MARKOV CHAINS

Markov and Strong Markov Properties

Transition Probabilities

Absorbing Markov Chains

Applications to Finance

Exercises

THE MONTE CARLO SIMULATION

Mathematical Backgrounds

The Idea of Monte Carlo

Generation of Random Numbers

Some Examples for Financial Engineering

Variance Reduction Methods

Exercises

FROM DISCRETE TO CONTINUOUS: TOWARDS THE BLACK-SCHOLES

Brownian Motions

The Central Limit Theorem Revisited

The Black-Scholes Formula

More on Brownian Motions

Poisson Processes

Exercises

BASIC STOCHASTIC PROCESSES IN CONTINUOUS TIME

Diffusion Processes

Sample Paths of Brownian Motions

Martingales

Stochastic Integrals

Stochastic Differential Equations

Ito's Formula Revisited

Exercises

A CONTINUOUS-TIME MODEL FOR SECURITIES MARKET

Self-Financing Portfolio and No-Arbitrage

Price Process Models

The Black-Scholes Model

The Risk-Neutral Method

The Forward-Neutral Method

The Interest-Rate Term Structure

Pricing of Interest-Rate Derivatives

Pricing of Corporate Debts

Exercises

REFERENCES

Name: Stochastic Processes with Applications to Finance (Hardback) – Chapman and Hall/CRC 
Description: By Masaaki Kijima. In recent years, modeling financial uncertainty using stochastic processes has become increasingly important, but it is commonly perceived as requiring a deep mathematical background. Stochastic Processes with Applications to Finance shows that this is...
Categories: Finance, Statistics, Financial Mathematics