Stochastic Processes with Applications to Finance

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