1 |
Class: Random walk, Central limit theorem, Brownian motion,
martingale, quadratic variation; from Shreve, most (but not all!) of sections 3.1-3.4.
TA session: Exercises 3.4, 3.6 Homework: make Exercises 3.1, 3.2. Read Section 3.2.7, a motivation for the use of Brownian motion in finance (but you have to know the Binomial model for stock prices...). |
2 |
Class: Integral of simple functions, construction of stochastic
integral w.r.t. Brownian motion, properties (isometry), general
integrands; from Shreve, most of sections 4.1 - 4.3
TA session: Exercises 4.2, 4.4 Homework: Exercises 4.1, 4.3 |
3 |
Class: Itô-formula, product rule, Lévy's
characterization; from Shreve, theory of Section 4.4, parts of Section 4.6
TA session: Exercises 4.5, 4.8 and perhaps 4.6 Homework: Understand the examples in Section 4.4, pay attention to the parts of Section 4.6 that I skipped and make exercises 4.7, 4.14 |
4 |
Class: Change of measure, Girsanov's theorem, Martingale
representation theorem, relevance
for risk neutral pricing; from Shreve, Sections 5.2.1-5.2.4, 5.3
TA session: 5.1, 5.8 Homework: Read Section 5.4.1 and (if you like) more on derivative pricing in Section 5.4 to get an idea why the change of measure is useful and necessary in finance, have a look at the Summary (Section 5.7) and the Notes (Section 5.8) as well. Make Exercises 5.2, 5.5 and Exercise 8.3(a,b,c) from the lecture notes of another course in which you have to find the process $\Gamma$ of Theorem 5.3.1 for the martingales in the exercise (do this for $t < T$ only). Don't get confused by the notation $X\cdot W$. It means stochastic integral, not a product (this notation is not uncommon, by the way, see also Williams for a discrete time version). |
5 |
Class: Stochastic differential equations, connections to Partial
differential equations, Feynman-Kac formula, stopping times; from Shreve, Sections 6.2 - 6.4, 8.2 and Definition 8.3.1
TA session: 6.8 and 6.2 or 6.7 Homework: Read Sections 6.5, 6.7, 6.8 (as far as applicable); make Exercises 6.1, 6.3, 6.9; extra question: suppose that $X$ is an adapted continuous process. Define $\tau$ as the first moment $t$ that $X(t)$ is at least equal to some level $m$, $\tau = \min\{t\geq 0: X(t)\geq m\}$. Give a simple argument (no complicated computations and use what you already know) that shows that $\tau$ is a stopping time. (Distinguish between $X(0)\geq m$ and $X(0) < m$.) |
6 |
Class: (Compound) Poisson process, integrals
w.r.t. jump processes; from Shreve, Chapter
11 up to Theorem 11.4.5
TA session: (1) Example 11.4.6 in detail. (2) Given a compound Poisson process (notation as in Shreve), find a left continuous process $Y$ such that $Q(t)=\int_0^t Y(s)\,\mathrm{d}N(s)$. Show (argue) that $\mathbb{E}\int_0^t Y(s)\,\mathrm{d}s=\beta t$ and relate this to Theorem 11.3.1. Homework: Exercises 11.1 (first (ii), than (i)), 11.2, 11.3 |
7 |
Class: quadratic variation, Ito-formula for processes with jumps, change of measure
for (compound) Poisson process; from Shreve, Chapter 11, Theorem 11.4.5 - Corollary 11.5.3, Lemma 11.6.1 - Theorem 11.6.5 with a very informal discussion of Theorems 11.6.9, 11.6.10.
TA session: Prove Theorem 11.6.10 and make exercise 11.6. Homework: Exercises 11.4, 11.5, 11.7, Watanabe's characterization of a Poisson process. |