[16.01.2019] A new version of Appendix A posted.
[21.12.2018] Solution 9 posted. Have a nice holiday!
[18.12.2018] Exercise sheet 9 and Soluton 8 posted. This is the last one. I shall discuss it on Thursday.
[14.12.2018] A new version of Exercise sheet 8 posted. In Exercise 8.2 (c), add the assumption \( U(\infty)<\infty\). Also, there is one last exercise sheet which will be posted next week. The exercise class will be arranged on Thursday instead of on Friday, during the usual lecture time.
[12.12.2018] Exercise sheet 8 and Solution 7 posted.
[28.11.2018] Solution E posted.
[21.11.2018] Exercise sheet 7 and Solution 6 posted. A new version of Exercise sheet E posted.
[14.11.2018] I posted an extra exercise sheet on \(\langle\cdot\rangle\) and \([\cdot]\). I shall discuss it next week.
[07.11.2018] Exercise sheet 6 and Solution 5 posted.
[02.11.2018] Solution 4 posted.
[01.11.2018] Exercise sheet 5 posted.
[22.10.2018] Solution 3 posted.
[18.10.2018] Exercise sheet 4 posted.
[12.10.2018] Next exercise sheet will be posted by Friday, 19.10.2018. It might be again a longer one.
[05.10.2018] Exercise 3 posted. It is slightly longer, so you will have more than a week before due.
[04.10.2018] Solution 2 posted.
[01.10.2018] Two more references about semimartingales and general stochastic integration added.
[27.09.2018] An appendix about Kreps-Yan theorem posted.
[26.09.2018] Exercise 2 posted. Solution 1 posted.
[20.09.2018] A typo in Exercise 1.1 is fixed. Use the updated version.
[19.09.2018] Exercise sheet 1 is posted. Also the reference list has been updated, which can be found at the bottom of this page.
This is an advanced course on mathematical finance for students with a good background in probability. We want to give an overview of main concepts, questions and approaches, and we do this mostly in continuous-time models.
Topics include
- semimartingales and general stochastic integration
- absence of arbitrage and martingale measures
- fundamental theorem of asset pricing
- option pricing and hedging
- hedging duality
- optimal investment problems
- and probably others
Prerequisites are the standard courses
- Probability Theory (for which lecture notes are available)
- Brownian Motion and Stochastic Calculus (for which lecture notes are available)
Those students who already attended "Introduction to Mathematical Finance" will have an advantage in terms of ideas and concepts.
This course is the second of a sequence of two courses on mathematical finance. The first course "Introduction to Mathematical Finance" (MF I), 401-3888-00, focuses on models in finite discrete time. It is advisable that the course MF I is taken prior to the present course, MF II.
For an overview of courses offered in the area of mathematical finance, see link.
| Date | Content | Reference | 18.09.2018 | goal, prerequisites, references; setup, assets, discounting; examples CRR, BS; strategy, value, costs; integrals; self-financing, L0.1 (equivalence and properties of self-financing) + proof | – |
|---|---|---|
| 20.09.2018 | remarks; assumptions; example stopped BM for arbitrage; topics / setup; strategy vs integrand; admissible; simple integrands; arbitrage opportunities, no-arbitrage conditions | – |
| 25.09.2018 | L1.1 + proof; ELMM, question; counterexample discrete time; counterexample continuous time | – |
| 27.09.2018 | finite discrete time, notations, T1.2, C1.3, overview proof ideas / local properties, semimartingale, \(b\mathcal{E}\), elementary stochastic integral, good integrator, goal, intuition, quasimartingale, remark | – |
| 02.10.2018 | P2.1, L2.2 + proof (only part a)); proof of P2.1; T2.3 (Rao) + proof; class (D); T2.4 (Doob-Meyer) + proof uniqueness | – |
| 04.10.2018 | T2.4, proof existence; T2.5 + proof; L2.6 (without proof); T2.7 + proof; remark | – |
| 05.10.2018 | proof of L2.6 / spaces \( \mathbb{L} \),\(\mathbb{D}\); remark; metrics \(d, d_E, d'_E\); L3.1 + proof, L3.2 + proof, C3.3 + proof, T3.1 + proof | – |
| 16.10.2018 | pointwise extension to \(\mathbb{L}\); T3.2 (extension to \(\mathbb{L}\) + proof; quadratic variation, L3.6 + proof; Def. \(\mathcal{H}^1_0\), remarks; T3.3 (Davis) + proof (sketch); C3.8; Def \(L^1(M)\), remarks | Beiglböck/Siorpaes (2015) |
| 18.10.2018 | L3.9 + proof; L3.10 + proof; C3.11 + proof; T3.12 (extension to \(b\mathcal{P}\) + proof; extension to semimartingales; Emery metrics and topology; T3.13 + proof; P3.14 + proof; definition \(L(S)\) | – |
| 19.10.2018 | remark; P3.15 + proof; remark; metric \(d_S\), L3.16 + proof; T3.17 (Memin) + proof; remark | Cherny/Shiryaev (2002) |
| 23.10.2018 | goal, setup, L1.1 as recall; P4.1 (Ansel-Stricker); L4.2, proof: proof of Ansel-Stricker and of L1.1; definition NFLVR; P4.3, L4.4 (without proof) | – |
| 25.10.2018 | proof of P4.3; \(\sigma\)-martingale, E\(\sigma\)MM, ESM, remarks, example; FTAP; proof outline for FTAP; T4.5, main steps for proof | Delbaen/Schachermayer (2006) Section 8.3, 14.3, 14.4 |
| 30.10.2018 | T4.6, notation, proof outline; step 1, Fatou-closed, step 2, maximal, step 3, step 4; T4.7, comments / NUPBR | Cuchiero/Teichmann (2015) T5.1 |
| 01.11.2018 | E\(\sigma\)MD, ELMD, remarks; numeraire portfolio, intuition, remark; T5.1, comment; continuous case: P5.2 + proof; parametrisation in C5.3 + proof; minimal ELMD, MVT | Karatzas/Kardaras (2007) T4.12 Takaoka/Schweizer (2014) T2.6 |
| 06.11.2018 | Ito processes, results; continuous S: L5.4 + proof; example Black-Scholes, C5.5 + proof, P5.6 + proof | – |
| 08.11.2018 | C5.7; L5.8 + proof; Levy processes; P5.9 + proof; P5.10 + proof / basic question; valuation by replication; valuation by risk-neutral expectation | – |
| 13.11.2018 | comments; attainable, complete; for finite discrete time L6.1, T6.2 (characterisation attainable), T6.3 (characterisation complete); illustration Black-Scholes formula | – |
| 15.11.2018 | question, setup, idea; definition; L7.1 + proof; P7.2 + proof; example, generalised strategies; T7.3 (without proof); T7.4 + proof | – |
| 16.11.2018 | proof of T7.3 (for continuous filtration); recall T7.4; comments, remark / goal, recall; feasible weight function, remarks, notation | Kramkov (1996) Föllmer/Kabanov (1997) |
| 20.11.2018 | w-admissible, L8.1 + proof; comments, C_w; T8.2; C8.3 + proof; notation; T8.4 (without proof); C8.5 + proof; remark | Delbaen/Schachermayer C15.4.11 |
| 22.11.2018 | proof of T8.4 / goals; superreplicable, prices, intuition, remark; T9.1 + proof, remark; buyer vs seller price, T9.2 + proof, price interval, L9.3 | – |
| 04.12.2018 | proof of L9.3, remarks; payoffs with one price, comments; maximal, hedgeable, remark; P9.4 + proof; comments; T9.5 + proof up to 2) implies 3) | – |
| 06.12.2018 | rest of proof of T9.5; remark; example / goal, setup, wealth, goal, remark, \( \mathcal{V}(x)\), utility function \(U\), primal problem | Kramkov/Schachermayer (1999) |
| 07.12.2018 | interpretation, natural assumption, remarks, questions; \(\mathcal{C}(x)\), \(u(x)\) from \(\mathcal{C}(x)\), remark, L10.1 + proof; dual problem, \(\mathcal{Z}(z), \mathcal{D}(z), J(y), j(z)\), dual problem, \(j(z)\) from \(\mathcal{D}(z)\), pre-conjugacy, L10.2 + proof | Kramkov/Schachermayer (1999) |
| 11.12.2018 | goal, Legendre transform \(J\), L11.1, example; goal; idea; P11.2 + proof; P11.3, without proof; T11.4 + proof; C11.5, without proof; idea with inequalities; equalities | Kramkov/Schachermayer (1999) |
| 13.12.2018 | reverse engineering recipe / L12.1, without proof; definition RAE, intuition; example, L12.2, without proof; L12.3, without proof; T12.4 + idea of proof; L12.5, without proof; L12.6, without proof; L12.7, without proof | Kramkov/Schachermayer (1999) |
| 18.12.2018 | goal, recipe in steps worked out; T13.1 + proof; remark conjugacy; remark necessity RAE; existence approach, condition on U; L13.2 + proof; remark; P13.3 + proof; P13.4 (without proof) | Kramkov/Schachermayer (1999) |
| Content | Reference |
|---|---|
| The Kreps-Yan theorem | Appendix A |
| The Komlos lemma | Appendix B |
| Essential supremum | Appendix C |
| The bipolar theorem | Appendix D |
The exercise sheets will be posted on Wednesdays. If you would like to have your work graded, please hand them in by the next Wednesday 12pm.
| Exercise sheets | Due dates | Solutions |
|---|---|---|
| Exercise sheet 1 | 26.09.2018 | Solution 1 |
| Exercise sheet 2 | 03.10.2018 | Solution 2 |
| Exercise sheet 3 | 17.10.2018 (Note the unusual due date!) | Solution 3 |
| Exercise sheet 4 | 31.10.2018 (Note the unusual due date!) | Solution 4 |
| Exercise sheet 5 | 07.11.2018 | Solution 5 |
| Exercise sheet 6 | 14.11.2018 | Solution 6 |
| Exercise sheet E | – | Solution E |
| Exercise sheet 7 | 05.12.2018 | Solution 7 |
| Exercise sheet 8 | 19.12.2018 | Solution 8 |
| Exercise sheet 9 | – | Solution 9 |
