Contents |
Course Title
Stochastic Calculus for Finance
Course Code
IAM 522
Credit
(3-0)3
Prerequisites
IAM 541, Probability Theory.
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.
Aims
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 stochastc 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.
Learning 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. Suggested Textbooks: Introduction to Stochastic Calculus Applied to Finance, D. Lamberton and B. Lapeyre Chapman & Hall, 1996 Syllabus:
- Discrete-time models: Martingales and arbitrage opportunities, complete markets and option pricing (2 w).
- Optimal stopping problem and American options : Stopping time, the Snell envelope, decomposition of supermartingales, application to the American option (2 w).
- Continuous-time processes and stochastic differential equations : General comments, Brownian motion, continuous-time martingales, stochastic integral and Itô calculus, stochastic differential equations (3 w).
- The Black-Scholes model : Description of the model, the Girsanov theorem, representation of martingales, pricing and hedging of options in the Black-Scholes model, American options in the Black-Scholes model (2 w).
- Option pricing and partial differential equations : European option pricing and diffusions, solving parabolic equations numerically, American options (2 w).
- Interest rate models : Modeling principles, some classical models (1 w).
- Asset models with jumps : Poisson process, dynamics of risky assets, pricing and hedging of options (1 w).
- Simulation and algorithms for financial models (1 w).
Resources:
- Option Pricing and Portfolio Optimization, Ralf Korn and Elke Korn, American Mathematical Cociety, 2001.
- Financial Derivatives in Theory and Practice, P.J.Hunt and J.E. Kennedy, Wiley, 2000
- Arbitrage Theory in Continuous Time, T. Björk Oxford IUniversity Press, 1998.
- Methods of Mathematical Finance, I. Karatzas and S.E. Shrev Second Edition, Springer, 1999.