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.

Common requirements for the summer program

A course in Probability and Statistics


On completion of the course, students will

  • have a broad view of the type of derivatives and structured contracts delivered in the financial markets.
  • be able to define a methodology to value this class of derivatives for complete financial markets in discrete time, implementing concepts like hedging price and absence of arbitrage opportunity. 
  • be able to propose different models to describe the evolution of stock prices, in discrete time, based on Markov chains.
  • be able to describe the hedging strategy for derivatives in complete markets, in particular in the binomial model. 
  • be able to describe the evolution of the fundamental assets in the risk-neutral framework. 
  • be able to develop numerical techniques to calculate the arbitrage-free price of derivatives using Monte Carlo methods. 

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.


  1. Baxter, Martin; Rennie, Andrew (1996). Financial Calculus, An Introduction to derivative pricing, Cambridge University Press.
  2. Duffie, Darrell (2001). Dynamic Asset Pricing Theory, 3rd Edition, Princeton University Press.
  3. Lamberton, Damien; Lapeyre Bernard (2007).  Introduction to Stochastic Calculus Applied to Finance, 2nd Edition, Chapman & Hall/CRC.
  4. Pliska, Stanley R. (1997). Introduction to Mathematical Finance, Wiley.
  5. Lapeyre, Bernard; Sulem, Agnes; Taley, Denis. Simulation of Financial Models: Mathematical Foundations and Applications, Cambridge University Press.
  6. Shreve, Steve (2004). Stochastic Calculus for Finance. Volume I – The Binomial Asset Pricing Model, Springer.
  7. Shreve, Steve (2010). Stochastic Calculus for Finance. Volume II – Continuous Time Models, New York: Springer- Verlag.
  8. Wilmott, Peter (1998). Derivatives. The Theory and Practice of Financial Engineering, Wiley.


Homework (30%), midterm exam (30%), final exam (40%)