Mathematical Finance Fall 2018

Martin Schweizer
Zhouyi Tan
Tue 08-10 @ HG D 1.1
Thu 08-10 @ HG D 1.1
Exercise Classes
Fri 10-12 @ HG D 1.1
First Lecture
Tue 18.09.2018
First Exercise Class
Fri 21.09.2018
Course Catalogue
401-4889-00L Mathematical Finance

[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.

Lecture Summary

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)


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


  • M. Beiglböck; M. Siorpaes. Pathwise versions of the Burkholder–Davis–Gundy inequality. Bernoulli 21 (2015), no. 1, 360--373.
  • C Cuchiero; J. Teichmann. A convergence result for the Emery topology and a variant of the proof of the fundamental theorem of asset pricing. Finance Stoch (2015) 19: 743.
  • H. Föllmer; Y. M. Kabanov. Optional decomposition and Lagrange multipliers. Finance Stoch (1997) 2: 69.
  • I. Karatzas; C. Kardaras. The numéraire portfolio in semimartingale financial models. Finance Stoch (2007) 11: 447.
  • D.O. Kramkov. Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Th. Rel. Fields (1996) 105: 459.
  • D. Kramkov; W. Schachermayer. The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Annals of Applied Probability 9 (1999), 904-950
  • A. Shiryaev; A. Cherny. Vector Stochastic Integrals and the Fundamental Theorems of Asset Pricing. Proceedings of the Steklov Institute of Mathematics, 2002, 237, 6–49.
  • K. Takaoka; M. Schweizer. A note on the condition of no unbounded profit with bounded risk. Finance Stoch (2014) 18: 393.
  • Books

  • T. Björk. "Arbitrage Theory in Continuous Time", Oxford University Press, 2009.
  • J. Cvitanić; F. Zapatero. "Introduction to the Economics and Mathematics of Financial Markets", MIT Press, 2004.
  • F. Delbaen; W. Schachermayer. "The Mathematics of Arbitrage", Springer Science & Business Media, 2006.
  • R. Elliott; R. Kopp. "Mathematics of Financial Markets", Springer Science & Business Media, 2005.
  • P. Hunt; J. Kennedy. "Financial Derivatives in Theory and Practice", John Wiley and Sons, 2004.
  • M. Jeanblanc; M. Yor, M. Chesney. "Mathematical Methods for Financial Markets", Springer Science & Business Media, 2009.
  • G. Kallianpur; R. Karandikar. "Introduction to Option Pricing Theory", Birkhäuser Boston, 1999.
  • I. Karatzas; S. Shreve. "Methods of Mathematical Finance", Springer, 2008.
  • D. Lamberton; B. Lapeyre. "Introduction to Stochastic Calculus Applied to Finance", CRC Press, 2007.
  • A. Shiryaev. "Essentials of Stochastic Finance: Facts, Models, Theory", World Scientific, 1999.