Probability and Finance II (SFI)

This class is a natural extension to Probability and Finance I taught by Prof. Fabio Trojani to continuous random variables along with an introduction to continuous-time stochastic processes. The focus will be on applications that are relevant in today´s market practice as well as academics. After having taken the course you will have good knowledge of the models that are around and also how to work with them.
For all topics covered many examples will deepen our intuition and un- derstanding. The main textbook is Shreve (2004). We will occasionally use Bjork (1998), Brigo and Mercurio (2006), Øksendahl (2000) as well as original research papers.

Exam
There will be a final closed-book written exam

Topics in Detail
Some of the below topics you will already have covered in Probability and Finance I. In that case we will concentrate on the extensions necessary to accommodate our more elaborate sample spaces. I will also take the liberty to include and exclude topics as we go along.

• Random variables and distributions
  • Expectation
  • Conditional expectation
  • Characteristic function
  • Change of measure and likelihood ratio
• Continuous-time processes
  • Wiener processes
  • Diffusion Processes
  • Functions of Wiener processes and Itˆo calculus – Girsanov´s theorem and change of measure
  • Affine models
• No-arbitrage pricing
  • The fundamental theorem of asset pricing
  • Short rate models
• Trading strategies
  • Variance and skew trading
  • Trading conditional moments

References
Bjork, T. (1998). Arbitrage Theory in Continuous Time. Oxford University Press.
Brigo, D. and Mercurio, F. (2006). Interest Rate Models – Theory and Prac- tice. With Smile, Inflation and Credit. Springer Finance. Springer-Verlag, 2nd edition.
Øksendahl, B. (2000). Stochastic Differential Equations. Springer.
Shreve, S. E. (2004). Stochastic Calculus for Finance II, Continuous-Time Models. Springer, Berlin.