We use the book Arbitrage Theory in Continuous Time (third edition) by T. Bjork, and the book Stochastic Calculus for Finance, volume I (The binomial asset pricing model) by S.E. Shreve.
week 36: Shreve, Section 1.1.
week 37: Shreve, Section 1.2.
week 38: Shreve, Sections 2.1, 2.2, 2.3, and 2.4 until (and including) Thm. 2.4.4.
week 39: Shreve, Section 2.4 until (and including) Thm. 2.4.7, started with Bjork, Ch. 4. HOMEWORK1.
week 40: Bjork Section 4.1, 4.2, 4.3; Example 4.14 done using approximation by simple processes.
week 41: Bjork Section 4.4, Part of Section 4.5; Solutions to homework problems explained (by D. Kiss).
week 42: Bjork Section 4.5, and some examples. Started to derive Black-Scholes equation (7.31) for special cases.
week 43: no class.
week 44: Material corresponds roughly to Bjork Sections 7.1-7.3, but with a somewhat different explanation (and with r and `sigma' constant). Briefly explained Theorem 7.8. HOMEWORK2.
week 45: Material corresponds to Bjork, Sections 5.1, 5.2 and 5.5; also showed connections with the Black-Scholes equations in Sections 7.3-7.5.
week 46: Corresponds to Bjork Theorem 7.7 and Theorem 7.8 (now also for non-constant alpha and sigma), and Sections 7.7.1 and 7.7.2.
week 47: Bjork Section 7.4 reviewed, in particular Theorem 7.8, and compared with discrete case; Solutions to homework problems explained.
week 48: Corresponds to Bjork Sections 9.1 and 18.5. Further, some discussion concerning Feynman-Kac. HOMEWORK3.
week 49: Bjork, Proposition 18.4 (proof presented for a special case); Section 8.2, Proposition 8.6 (Remark: The rest of Section 8.2, and Section 8.1, have essentially already been treated during the discussions on Chapter 7 in the past few weeks); Section 9.2.
week 50: Solutions of homework 3 explained; some extra exercises discussed. Also made:
SOME REMARKS CONCERNING THE EXAM: The exam will consist of about six exercises (of which one concerns the discrete model), of various level of difficulty, comparable with the homework exercises. Use of textbooks, personal notes and calculator is not allowed. The students evidently have to know by heart the risk neutral pricing formula (for the discrete as well as the continuous model), the basic properties of stochastic integrals, the rules of stochastic calculus (Ito formulas), the notion of Martingales, and the definition of the Black-Scholes dynamics. But ALSO the explicit formula (as a geometric Brownian motion) for the stock price in the Black-Scholes model, and the Feynman-Kac theorem.