Mathematical Finance
COURSE DESCRIPTION 
The aim of this course is to provide the mathematical foundations of asset pricing valuation in different frameworks. The characteristics of a large class of contracts will be analyzed, and the methodology to provide an arbitragefree value will be presented. In addition, the basic binomial and BlackScholes models will be used during the course to analyze the solution of the optimal investment and consumption problems, motivating the connection with solutions of partial differential equations. Prerequisites A course in Probability and Statistics 
COURSE GOALS 
On completion of the course, students will

COURSE CONTENTS 
1. Fundamental elements of a financial market. (1 week) Introduction to financial markets. Different contracts based on risky assets. Structured products on indexes, definitions and characteristics, rules of the markets. Interest rates. 2. Riskneutral valuation and arbitrage considerations. (1 weeks) Arbitrage and riskneutral probability measures. Valuation of contingent claims. Singleperiod models. Valuation and hedging in complete and incomplete markets. Investment strategies. 3. Multiperiod financial markets. (2 weeks) Conditional expectation and martingales in discrete time. Optimal portfolios for the binomial model. Markov models and incomplete markets. Valuation of European options. Cox Ross and Rubinstein model. 4. American options (2 weeks) Stopping time. Snell envelope and decomposition of supermartingales. Valuation of American options. Perpetual and finite horizon cases. Hedging portfolios in complete markets. 5. Dynamic programming (2 week) Optimal portfolios and dynamic programming. Martingale methods for solving optimal consumption problems. Optimal portfolios with constraints. Bibliography
Grading Homework (30%), midterm exam (30%), final exam (40%) 