IAM522 - Stochastic Calculus for Finance
Instructor(s): Consent of IAM
Prerequisites: Consent of Instructor(s)
Course Catalogue DescriptionThe objective of this course is an introduction to the probabilistic techniques required for understanding the most widely used financial models. In the last few decades, financial quantitative analysts have used sophisticated mathematical concepts in order to describe the behavior of markets and derive computing methods. The course presents the martingales, the Brownian motion, the rules of stochastic calculus and the stochastic differential equations with their applications to finance. Outline of Topics: Discrete time models, Martingales and arbitrage opportunities, complete markets, European options, option pricing, stopping times, the Snell envelope, American options. Continuous time models: Brownian motion, stochastic integral with respect to the Brownian motion, the Itô Calculus, stochastic differential equations, change of probability, representation of martingales; pricing and hedging in the Black-Scholes model, American options in the Black-Scholes model; option pricing and partial differential equations; interest rate models; asset models with jumps.
Course ObjectivesThe objective of this course is an introduction to the probabilistic techniques required for understanding the most widely used financial models. In the last decades financial quantitative analysts have used sophisticated mathematical concepts, such as martingales and stochastic integration, in order to describe the behaviour of markets or derive computing methods. The course presents the martingales, the Brownian motion, the rules of stochastic calculus and the stochastic differential equations oriented to applications to finance.
Course Learning OutcomesThis course contains the most basic tools of mathematical models for financial markets. Therefore the acquired ability will give the students the necessary skill in financial studies.
Tentative (Weekly) OutlineDiscrete-time models: trading strategies, self-financing strategies, admissible strategies, arbitrage, martingales and viable markets, complete markets and option pricing. Optimal stopping problem and American options : Stopping time, Snell envelope, American options, European options. Brownian motion and stochastic differential equations: Brownian motion, martingales, stochastic integral and Itô calculus, Ornstein-Uhlenbeck process, stochastic differential equations. The Black-Scholes model: the behavior of prices, self-financing strategies, the Girsanov theorem, pricing and hedging of options, hedging of calls and puts, American options, perpetual puts. Option pricing and partial differential equations: European option pricing and diffusions, partial differential equations and computation of expectations, numerical solutions, application to American options. Interest rate models: modelling principles, some classical models. Asset models with jumps: Poisson process, dynamics of risky assets, pricing and hedging of options. Simulation and algorithms for financial models.
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