MACF491(MAST679): Stochastic Calculus and Finance
                                                      Winter 2016
                      Lecture times:     TBC
                      Lecture location:  TBC
                      Textbook:          Stochastic Calculus for Finance II: Continuous Time Models by Steven
                                         Shreve. Springer Finance Textbook.
                      Topics covered:    This course will cover an introduction to stochastic calculus and applications
                                         to mathematical finance. See overleaf for a more precise schedule.
                      Assessment:        Grades in this course will be determined by a mid-term, a final exam and
                                         regular assignments. Your final mark will be composed as follows:
                                            • assigments TBC%
                                            • midterm TBC%
                                            • final exam TBC%
                                                            1
                    Schedule
                         week    mathematics                             finance                                   book
                           1     probability spaces, random variables,   –                                        Ch.1
                                 expectation.
                           2     Convergence theorems.     Change of     –                                       Ch. 1, 2
                                 measure, Radon-Nikodym´    derivative.
                                 Independence.
                           3     Conditional Expectation. Filtrations,   –                                        Ch. 2
                                 martingales.
                           4     Brownian motion.      Discretisations,  –                                        Ch. 3
                                 Brownianmotionasamartingale. Ex-
                                 ponential martingale.
                           5     BrownianmotionasaMarkovprocess.         –                                        Ch. 3
                                 First passage time. Reflection princi-
                                 ple, joint distribution of Brownian mo-
                                 tion and its maximum.
                           6     Quadratic variation of Brownian mo-     Volatility of geometric BM.             Ch. 3, 4
                                 tion. The Itˆo stochastic integral. Itˆo
                                 isometry.
                           7     Itˆo processes. Itˆo’s formula.         –                                        Ch. 4
                           8     The Black-Scholes equation. Stochas-    Pricing European (‘vanilla’) options.    Ch. 4
                                 tic calculus in higher dimensions.      Put-call parity.
                                 L´evy’s theorem.
                           9     Girsanov’s theorem. Martingale rep-     The risk-neutral measure.       Pric-    Ch. 5
                                 resentation theorem.                    ing derivative securities for European
                                                                         calls.
                          10     Girsanov and Martingale representa-     Multimarket models.     Fundamental      Ch. 5
                                 tion theorem in higher dimensions.      theorems of asset pricing
                          11     SPDEs. Markov property. Feynman-        Interest rate model. Asian options       Ch. 6
                                 Kac formula.
                          12     –                                       Exotic options: Knock-out and look-      Ch. 7
                                                                         back options.
                          13     Stochastic calculus for jump processes  European options in a jump model        Ch. 11
                                                                     2