# Introduction to Mathematical Finance Spring 2018

Lecturer
Martin Schweizer
Coordinator
Zhouyi Tan
Lectures
Mon 08-10 & 13-15 @ HG D 3.2
Thu 08-10 @ HG D 1.1, 13-15 @ HG D 3.2
Exercise Classes
Tu 12-14 @ HG E 33.1
We 14-16 @ HG F 26.3
First Lecture
Mon 09.04.2018
First Exercise Class
Tu 10.04.2018
Course Catalogue
401-3888-00L Introduction to Mathematical Finance

[31.05.2018] Solution 12 posted. Please find the main reference for Chapter IV in the appendix.
[28.05.2018] Solution 11 posted.
[23.05.2018] Es 12 posted. Solution 10 posted.
[17.05.2018] Es 11 posted. Solution 7,8,9 posted.
[11.05.2018] Es 8 due. Es 10 posted. Solution 7&8 should be posted shortly.
[27.04.2018] Es 5 due. Es 7, Solution 4 & 5 posted.
[25.04.2018] Es 4 due. Es 6 posted.
[20.04.2018] Es 3 due. Es 5 & Solution 3 posted.
[18.04.2018] Es 2 due. Es 4 & Solution 2 posted.
[13.04.2018] Es 1 due. Es 3 & Solution 1 posted.
[11.04.2018] Preliminary discussion for Exercise sheet 1. Exercise sheet 2 posted.
[09.04.2018] Lecture starts. Exercise sheet 1 typo corrected.

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.

## Prerequisites/Notice

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

DateContentReference
09.04.2018a introductory words and admin / setup, assets, agents, consumption, preference order, example utility, budget set, interpretation, notation, allocation, equilibrium w/. intuition, remark, feasible, intuition, Pareto-efficient w/. intuition, attainable and equivalence  –
09.04.2018b completeness and equivalence; equilibrium is Parato when complete, comments / basic idea of pf, arbitrage and two kinds, NA and 1st NA, NA in equilibrium, remark, NA implies independent pricing, application valuation, example call option & rmk  –
12.04.2018a interpretation strategy / goal, 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, comments  –
12.04.2018b goals, 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, remark, beginning of proof  –
16.04.2018a proof of TI.5.3, part 5 implies 1 / 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, LI.7.1  –
19.04.2018a LI.7.1, proof of part 1), question, example call, general problem, TI.7.2, comments, proof of TI.7.2, LI.7.3 + proof, proof of TI.7.2 (cont.), question, PI.7.4 + proof  –
19.04.2018b 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 + proof, interpretation, PII.1.2  –
23.04.2018a proof of PII.1.2, interpretation, admissible, interpretation / model, notation, strategy, arbitrage, interpretation, remarks, PII.2.1, remark, proof of PII.2.1, example  –
23.04.2018b example continued, equivalent measures, EMM, example, LII.2.2 + proof, PII.2.3 + proof, remark / sigma-field F-tau, remarks, value at tau and stopped process, LC.2 + proof, PC.3; local martingale, remarks, PC.4, TC.5  –
26.04.2018a proof of PC.4, CC.6, LC.7 + proof / setup, notations, TII.3.1, CII.3.2 (DMW), easy parts; TII.3.1: proof that 4) implies 5); LII.3.3 + proof  –
26.04.2018b proof of TII.3.1 that 2) implies 3), remark, PII.3.4 + proof, remarks / goal, locally equivalent, density process, PII.4.1 + proof, remark  –
30.04.2018a factorisation, conditions on D, remark, example i.i.d. returns, example discrete, example binomial, example multinomial / goals, setup, payoff, attainable, complete, interpretation, remark  –
30.04.2018b PII.5.1 + proof, interpretation, remark, TII.5.2 + proof, remark, TII.5.3 + proof, remark, summary; example binomial model, setup, PII.5.4  –
07.05.2018a setup, strategies with consumption, self-financing condition, wealth $$W$$, positive wealth and prices, criterion, utility functions, comment / goal, problem, idea of conditional problems  –
07.05.2018b LIII.2.1 + proof, PIII.2.2 + proof, remark supermartingale, TIII.2.3 + proof, remarks, conditional problems, CIII.2.4 + proof, notions for DP, rewriting MOP  –
14.05.2018a recursion for strategy/consumption candidate, algorithm, verification, remark / goal, positive wealth, construction of candidates, function at T-1, expression for $$\mathcal{O}_{T-1}$$, functions at $$T-1$$, candidates at $$T-1$$, general $$k$$ via induction, candidates for $$k$$, construction of wealth forward  –
14.05.2018b power utility, compute function $$F_T$$, recursion for $$h_k$$, candidates at $$k$$, remarks, verify optimality / goal, basic formulation, case $$T=1$$, EMM, remark  –
17.05.2018a Omega finite, allowed strategies and indirect utility, PIII.4.1 + proof, notation $$\mathcal G$$, TIII.4.2 + proof / goal, ideas of constraints, extremal points in P_a, Lagrange function, Lagrange multipliers  –
17.05.2018b rewriting $$L$$, minimax property, conditions on $$U$$, LIII.5.1 + proof, conjugate $$J$$ of $$U$$, LIII.5.2 + proof, primal and dual problems, TIII.5.3 + proof, TIII.5.4 + proof, remarks, conjugacy relation and comments  –
24.05.2018a setup, wealth, goal, $$\mathcal{V}(x)$$, utility function, remark, $$u(x)$$, assumption (1.1), conjugate $$J$$, LIV.1.1 + short proof, example, remark / abstract problem for $$u(x)$$, $$\mathcal{C}(x)$$, comments, abstract version, motivation for dual processes, $$\mathcal{Z}(z)$$, remarks, definition $$j(z)$$, abstract version, $$\mathcal{D}(z)$$, conjugacy with inequalities; important remark  –
24.05.2018b goal, idea, convexity for compactness, LIV.3.1, PIV.3.2 + proof, PIV.3.3 + proof, TIV.3.4 + proof, CIV.3.5 + proof  –
28.05.2018a goal, idea why equality gives optima, find maximisers, computation, reverse engineering recipe, optimality, conjugacy for $$u$$ and $$j$$  –
28.05.2018b goal, standing assumptions, LIV.5.1 (without proof), definition RAE, intuition, example  –
31.05.2018a LIV.5.2 + proof, remark, LIV.5.3 (without proof), remark, TIV.5.4 + proof  –
31.05.2018b LIV.5.5 (without proof), LIV.5.6 (without proof) / goal and conditions, step 1), step 2), step 3), LV.6.1 + proof, step 4), PIV.6.2 + proof, TIV.6.3 + proof, extra remarks  –

## 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

The exercise sheets will be posted before the lecture. In particular, exercise sheets $$2k+1, k=0,1,2,...$$ will be posted on Fridays and exercise sheets $$2k, k=1,2,...$$ will be posted on Wednesdays. If you would like to have your work graded, then please hand them in by the next Fridays and Wednesdays, respectively.

• 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