In this course we will cover the following topics

1. Ito calculus (Ito formula and Ito integral, Ito processes, Feynman Kac formula and applications)
 2. Black Scholes model and option pricing via PDEs (Black Scholes model, pricing by replication Black-Scholes  PDE, risk-neutral pricing via Feynman Kac, properties of option prices, currency options)
 3. Change of measure and Girsanov theory, martingale approach to Black Scholes model
 4. Further mathematical finance (eg. short-rate models  for the term structure and affine models, change of numeraire, multidimensional Black Scholes model
 5. Optimal stopping and applications (eg. Heinkel Zechner or Leland model)
 6. Optimal control and applications (eg Merton model)

This course deepens the understanding of continuous-time finance and it covers a number of advanced topics of Continuous Time Finance.

The aim of this course is to:

• obtain a good  understanding of the main topics, such as stochastic calculus for Brownian motion, financial mathematics in continuous time and the financial applications
• understand and describe the properties of simple term-structure models, change of numéraire, optimal stopping and basic stochastic control.
• Study relevant financial applications

After completing this course the student will also:

• have deepened his/her ability for teamwork
• be able to formulate essential problems of CTF and propose possible solutions in a precise way (that is in a mathematical language). This skill is different from a purely intuitive understanding of the topics of this course.