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 arbitrage-free value will be presented. In addition, the basic binomial and Black-Scholes 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.
On completion of the course, students will
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. Risk-neutral valuation and arbitrage considerations. (1 weeks)
Arbitrage and risk-neutral probability measures. Valuation of contingent claims. Single-period 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.
2 hours per week with a teaching assistant
Homework (30%), midterm exam (30%), final exam (40%)