TAs: -

The aim of this course is to present and deepen the various mathematical concepts, techniques and intuition necessary for modern financial modelling, derivative pricing, portfolio optimization and risk management. This course provides the foundations for a sufficiently rigorous mathematical treatment of these topics.

Students should have basic knowledge of probability theory, and of derivatives valuation and use.

40% project + 60% final exam (open book)

There is no required textbook for the course. However, there are some reference books that are recommended:

- Ciucu, G si C. Tudor, 1979, Probabilitati si procese stocastice, Editura Academiei
- Hull, J., 2006, Options, Futures, and other Derivatives, Prentice Hall
- Karatzas, I and S. Shreve, (1991), Brownian Motion and Stochastic Calculus, Springer-Verlag
- Necula, C., 2009, Evaluarea optiunilor financiare. Volumul I - Modelul Black-Scholes-Merton, Editura ASE
- Necula, C., 2009, Evaluarea optiunilor financiare. Volumul II - Modele multifactoriale, Editura ASE
- Oksendal, B, 2000, Stochastic Differential Equations. An Introduction with Applications, Springer-Verlag
- Shreve, S., 1997, Stochastic Calculus and Finance, lecture notes, Carnegie Mellon University
- Stoica, G, 1999, Introducere in studiul miscarii browniene, Tipografia Universitatii Bucuresti
- Zbaganu, Gh, 1998, Curs de Teoria masurii si a probabilitatilor, Tipografia Universitatii Bucuresti
- Wilmott, P, Howison, S, and J. Dewynne, 1995, The Mathematics of Financial Derivatives. A student introduction , Cambridge U. Press

1. A Review of Probability Theory

- measure and probability, measurable spaces
- discrete, continuous and absolutely continuous random variables
- pdf and cdf
- Lebesgue integral, the expected value of a random variable
- random vectors, independence
- conditional mean

2. Stochastic Processes

- martingales
- the Brownian motion
- the stochastic integral
- Ito processes, diffusion processes
- the change of variables formula
- the change of probability theorem
- martingales representation theorem

3. The Framework for Asset Pricing in Continuous Time

- self-financing portfolios, arbitrage portfolio
- viable and complete market models
- the risk neutral measure
- martingale measures
- the fundamental theorem of asset pricing
- risk neutral valuation, change of numeraire

4. Black-Scholes-Merton Market Model

- the viability of the BSM market model
- the completeness of the BSM market model
- BSM fundamental valuation equation
- European options, valuation using the risk neutral measure and the change of numeraire
- path dependent options

5. Black-Scholes-Merton Market Model Extensions

- stochastic interest rate
- stochastic volatility, Heston model, GARCH models
- jump diffusion models

6. Stochastic Optimization and Applications in Finance

- Hamilton-Jacobi-Bellman equation
- optimal portfolio allocation, Merton problem
- optimal hedging
- optimal pension fund contribution rate

Course Materials