## Stochastic Processes with Applications to Finance

Series:
Published:
Author(s):

Hardback
\$115.95
ISBN 9781584882244
Cat# C2247

### Features

• Uses discrete processes to explain important but difficult concepts in stochastic calculus
• Includes an introduction to Monte Carlo simulation that emphasizes its use in financial engineering
• Addresses applications in the pricing of derivative securities, corporate bonds, and credit derivatives
• Describes the dynamics of credit ratings using Markov chains
• Contains numerous exercises that support the concepts
• ### Summary

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.

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