# Introduction to Mathematical Finance Spring 2019

Lecturer
Coordinator
Bálint Gersey
First Lecture
18/02/2019
First Exercise Class
27/02/2019
Course Catalogue
Introduction to Mathematical Finance
TimeRoom
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.

## Prerequisites

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.

## Lecture Summary

>
DateContent
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

## Appendices

ContentReference
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