- Lecturer
- Martin Larsson
- Coordinator
- Bálint Gersey

- First Lecture
- 18/02/2019
- First Exercise Class
- 27/02/2019
- Course Catalogue
- Introduction to Mathematical Finance

Time | Room | |
---|---|---|

Lectures |
Mondays 13-15 | HG G 3. |

Thursdays 08-10 | HG D 1.1 | |

Exercice classes |
Wednesdays 14-15 | HG F 26.3 |

Wednesdays 15-16 | HG F 26.3 |

[18.02.2019] Exercise sheets 1 and 2 posted

[28.02.2019] Exercise sheet 3 posted

[06.03.2019] Exercise sheet 4 posted

[13.03.2019] Exercise sheet 5 posted

[20.03.2019] Exercise sheet 6 posted

[27.03.2019] Exercise sheet 7 posted

[03.04.2019] Exercise sheet 8 posted

[10.04.2019] Exercise sheet 9 posted

[06.05.2019] Exercise sheet 10 posetd

[15.05.2019] Exercise sheet 11 posted

[22.05.2019] Exercise sheet 12 posted

This is an introductory course on the mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. We discuss arbitrage, completeness, risk-neutral pricing and utility maximisation. We prove the fundamental theorem of asset pricing and the hedging duality theorems, and also study convex duality in utility maximization.

A related course is 401-3913-01L Mathematical Foundations for Finance (3V+2U, 4 ECTS credits). Although both courses can be taken independently of each other, only one will be recognised for credits in the Bachelor and Master degree. In other words, it is not allowed to earn credit points with one for the Bachelor and with the other for the Master degree.

A knowledge of measure-theoretic probability theory (as taught e.g. in the course "Probability Theory").

This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II.
For an overview of courses offered in the area of mathematical finance, see link.

Date | Content |
---|---|

18.02.2019 | introductory words and admin / setup, assets, agents, consumption, preference order, example utility, budget set, interpretation, notation, allocation, equilibrium, feasible allocation, Pareto-efficiency, attainability |

21.02.2019 | completeness and equivalence; equilibrium is Parato when complete, comments / poof, arbitrage and two kinds, NA and 1st NA, NA in equilibrium, remark, NA implies independent pricing, application valuation, example call option |

25.02.2019 | numeraire, bond, interpretation, convention, EMM, remarks, LI.4.1 + proof, LI.4.2 + proof, TI.4.3 + proof, CI.4.4 + proof, remarks, TI.4.5 + proof of easy direction |

28.02.2019 | proof hard direction, price system, remark, consistent, interpretation, PI.5.1 + proof; application valuation, remark, example call option, TI.5.2 + proof, intuition and state prices, TI.5.3, beginning of proof |

04.03.2019 | finish the proof of TI.5.3, motivation, return, remark, goal, TI.6.1 + proof, remark, interpretation, remarks, CI.6.2 + proof, interpretation, remark / goal, contingent claim, question, attainable case, interpretation, not attainable case, possible approach, alternative, seller and buyer prices, interpretation |

08.03.2019 | LI.7.1, proof of part 1), question, example call, general problem, TI.7.2, comments, proof of TI.7.2, Linear Programming, weak duality, Farkas Lemma, strong duality |

11.03.2019 | Proposition I.7.5, comment, example // trading dates, information, interpretation, assets, numeraire, convention, remark, strategy, interpretation, value/wealth, comment, remark, self-financing, interpretation, notations, stochastic integral, gains, interpretation, LII.1.1 |

14.03.2019 | proof of LII.1.1, PII.1.2, interpretation, admissible, interpretation, notation, strategy, arbitrage, interpretation, remarks, PII.2.1, remark, proof of PII.2.1 (except 4 implies 5) |

18.03.2019 | end of proof, example, equivalent measures, example, LII.2.2 + proof, PII.2.3 |

21.03.2019 | setup, TII.3.1, CII.3.2 (Dalang-Morton-Willinger),proof of easy parts; proof that 4) implies 5); LII.3.3 + proof |

25.03.2019 | proof of TII.3.1 that 2) implies 3), PII.3.4 + proof, beginning of Chapter 4 (density process) |

28.03.2019 | factorisation, conditions on D, remark, examples, beginning of Chapter 5, goals, setup, payoff, attainable, complete, interpretation, remark, PII.5.1 + proof, interpretation, TII.5.2 (proof except 3 implies 1) |

01.04.2019 | TII.5.2 (proof of 3 implies 1), remark, TII.5.3 + proof, remark, summary; example binomial model, PII.5.4, beginning of section II.6, goal, setup, PII.6.1 + proof |

04.04.2019 | Lemma II.6.2 + proof, Theorem II.6.3, remark + beginning of proof |

08.04.2019 | end of proof of Theorem II.6.3 |

11.04.2019 | section II.7: hedging duality |

15.04.2019 | Beginning of Chapter III: Utility Maximisation, goal, setup + problem formulation, beginning of section III.2 on martingale optimality principle |

18.04.2019 | Martingale Optimality Principle: conditional problems, Lemma III.2.1 + proof, Proposition III.2.2 + proof, Theorem III.2.3 + proof, remarks, Corollary II.2.4 + proof |

29.04.2019 | recall of the setup, use of previous results to derive dynamic programming principle (PIII.2.5) + proof, example: iid returns and power utility |

02.05.2019 | end of the example of iid returns, remark on trading involved even if deterministic proportions, beginning of Chapter IV: basics, def of utility function + examples, no optimal solution if NA fails, Lemma IV.1.2 (caution only true in discrete time) |

06.05.2019 | existence of solution for complete markets |

09.05.2019 | existence of solution in incomplete markets for bounded utility functions + beginning of duality |

13.05.2019 | duality, solving the dual problem |

16.05.2019 | digression on essential supremum: existence and main properties |

20.05.2019 | solving the primal problem |

23.05.2019 | Summary of the course |

The exercise sheets will be posted before the lecture.

Exercise Sheet | Due Date | Solutions | Solutions with questions |
---|---|---|---|

Exercise sheet 1 | 27/02/2019 | Solution 1 | Questions + Solutions 1 |

Exercise sheet 2 | 06/03/2019 | Solution 2 | Questions + Solutions 2 |

Exercise sheet 3; Facebook; S&P500; | 13/03/2019 | Solution 3 | Questions + Solutions 3 CAPM.py CAPM.r |

Exercise sheet 4 | 20/03/2019 | Solution 4 | Questions + Solutions 4 |

Exercise sheet 5 | 27/03/2019 | Solution 5 | Questions + Solutions 5 |

Exercise sheet 6 | 03/04/2019 | Solution 6 | Questions + Solutions 6 |

Exercise sheet 7 | 10/04/2019 | Solution 7 | Questions + Solutions 7 |

Exercise sheet 8 | 17/04/2019 | Solution 8 | Questions + Solutions 8 |

Exercise sheet 9 | 24/04/2019 | Solution 9 | Questions + Solutions 9 |

Exercise sheet 10 | 15/05/2019 | Solution 10 | Questions + Solutions 10 |

Exercise sheet 11 | 22/05/2019 | Solution 11 | Questions + Solutions 11 |

Exercise sheet 12 | 29/05/2019 | Solution 12 | Questions + Solutions 12 |

Content | Reference |
---|---|

Separation theorems for convex sets in \(\mathbb{R}^d\) | Separation theorems |

Optimisation under constraints, the Kuhn–Tucker theorem | Kuhn-Tucker |

Martingales and stochastic integrals in discrete time | Martingale-results |

The Kreps–Yan theorem | Kreps–Yan |

The essential supremum | Ess-sup |

The Komlós-type lemma | Komlos |

Main reference for Chapter IV | D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Annals of Applied Probability 9 (1999), 904-950 |

- Michael U. Dothan, "Prices in Financial Markets", Oxford University Press
- Hans Föllmer and Alexander Schied, "Stochastic Finance: An Introduction in Discrete Time", de Gruyter
- Marek Capinski and Ekkehard Kopp, "Discrete Models of Financial Markets", Cambridge University Press
- Robert J. Elliott and P. Ekkehard Kopp, "Mathematics of Financial Markets", Springer