IAMINSTITUTE OF APPLIED MATHEMATICS


IAM522

Code IAM522
Name STOCHASTIC CALCULUS FOR FINANCE
Credits 3(3-0)
ECTS 8.0
Objective The 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.
Content Discrete-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.
Outcomes This 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.

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