[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.
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 | Reference |
|---|---|---|
| 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 | – |
| 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 |
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.
