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 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 Buhlmann'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.