Stochastic processes with applications to finance

Description

Elementary Calculus: Towards Ito's FormulaExponential and Logarithmic Functions Differentiation Taylor's ExpansionIto'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 FinanceBinomial 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 OptionsInterest-Rate DerivativesChange 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 ProbabilitiesThe 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 ChainsApplications to FinanceMonte 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 ProcessesBasic 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 ApproachPricing of Credit DerivativesReferences IndexExercises appear at the end of each chapter.