Stochastic Finance in Continuous Time
ISEG, 2025-2026
(This page will be updated regularly)

Contents

The course is part of the Mestrado em Mathematical Finance, Lisbon School of Economics & Management (Instituto Superior de Economia e Gestão, ISEG), Universidade de Lisboa.
The object of this course is to provide an introduction to arbitrage theory in continuous time and in particular to pricing and hedging theory for financial derivatives. Aims and objectives include the following.
  • Understanding the notions of: arbitrage portfolio, self-financing portfolio, risk-neutral measure, static or dynamics hedging, market completeness;
  • Deriving: the Black-Scholes model as well as it standard extensions to include dividends or currency derivatives;
  • Use the change of numeraire technique to simplify pricing expressions;
  • Distinguish between futures and forward contracts and prices;
  • Apply basic notions of Stochastic Optimal Control, in particular, concerning their use under a particular control function to maximize a value function.

Literature

  • The course is based on T. Björk, Arbitrage Theory in Continuous Time. The link leads to the 4th, expanded, edition of the book, but the slides (see below) are based on the 3rd edition as the reference text. Try to find a second hand copy. The 2nd edition can mostly be used as well.
  • The slides that have been produced by Tomas Björk in 2017 will be used as the main vehicle of the course, with chapters from the (3rd edition of the) book as background material (also the 2nd Edition will do for most purposes). Be aware of (rather minor) differences in notation between slides and book.
  • During the course, annotated slides will incrementally be uploaded. As time passes by you will find here the annotated slides of the lectures, and a (sketch) proof of the Verification theorem for the HJB equation.
  • Have a look at the file containing only(!) the exercises of the 3rd Edition.
  • We will sometimes use some additional exercises in the tutorials.
  • Other, very different, topics in Mathematical Finance for further studying after the course are treated in the video lectures by Lech Grzelak. Recommended!

Prerequisites

Basic concepts of probability and measure theory, for instance at the level of the Appendices A and B of the book by Tomas Björk.

Examination

Written exam, officially scheduled for 19 December 2025, with a retake at 9 January 2026; see also the official assessment schedule. You (students) are allowed to bring up to 2 pages (A4 format, written in normal print) with your own notes with you (think of important theorems, formulas).

The 19 December 2024 exam

Have a look at the (slightly corrected and shortened) exam questions and their very concise solutions.

People

Lectures by Peter Spreij.
Assistance and tutorials by Henrique Guerreiro.

General schedule

Lectures during four weeks in September 2025, usually in the mornings (11:00 - 13:00) in room F1 209 of September 9, September 10, September 11, September 12, September 16, September 17, September 18, September 19, September 22, September 25, September 26, September 29, September 30 (14:30 - 16:30, F1 202), and October 2. Tutorials are spread over Mondays in September and October of the Fall semester, and planned for September 8 (refresher on It&ohat; calculus), September 15, September 18, September 22, September 29, October 2, and October 6. See the timetable, or the alternative timetable (with FETC as the acronym for the course) for details. This timetable is part of the overview site, which contains further administrative information.

Planned programme of 2025-2026, still under construction
(may also see updates during the course; )

0
No lecture, only a refresher tutorial (8 Sep 2025) on Itô calculus and related topics. Among them Wiener process, filtrations, martingales, Itô isometry, Itô formula, simple SDEs, Feynman-Kac formula. Make Exercises (from the 3rd Edition) 4.1, 4.3, 4.8, 5.1, 5.5, 5.9, and 5.8 if there is sufficient time left.
1
 9 Sep 2025, F1 209
 Lecture: Main topics are Portfolio dynamics, self financing strategies and arbitrage pricing; Slides 42-81; related material from the book in Sections 6.1, 6.2, 7.1-7.5.
Tutorial 1: Exercises 7.1, 7.2, 7.3, 7.4, 7.5, 7.6; and give an example of a (very simple) derivative where the approach of slides 63-66 fails. How to remedy this? Is the Black-Scholes PDE also valid in this example?
2
 10 Sep 2025, F1 209
 Lecture: Main topics are Completeness, hedging; Slides 82-109; background in the book in Chapter 8.
Tutorial 1: Exercises 8.2, 8.3.
3
 11 Sep 2025, F1 209
 Lecture: Main topics are parity relations and the Greeks; Slides 110-127; background in the book in Chapter 9.
Tutorial 2: 9.1, 9.5, 9.10. In order to compute the Delta and Gamma for a European call (or put), show first that $s\phi(d_1)-Ke^{-r(T-t)}\phi(d_2) = 0$, where $\phi$ denotes the density of the standard normal distribution. Then it becomes easier to compute the Delta and Gamma.
4
 12 Sep 2025, F1 209
 Lecture: Recap of probability theory based on Sections B.1, B.5, B.6, and Theorem A.52.; Slides 128-153.
Tutorial: No particular exercises for this theoretical part.
5
 16 Sep 2025, F1 209
 Lecture: Main topic is the mathematical background of the martingale approach to arbitrage theory; Slides 154-182. More background probability theory based on Sections C.1, C.3. The `dynamic' version of all this (use of processes, filtrations, martingales) in Sections 11.1-11.5.
Tutorial 2: Additional exercises 2, 3.
6
 17 Sep 2025, F1 209
 Lecture: Main topics are martingale approach to pricing an hedging, completeness of a market; Slides 183-205, based on parts of Chapters 10, 12. Take good notice of the summary Section 10.7.
Tutorial 2: Additional exercises 4 (and, only if there is enough time left, 5).
7
 18 Sep 2025, F1 209
 Lecture: Dividends, forwards; Slides 206-236, based on Sections 7.6, 16.2, 16.3.1, 29.1.
Tutorial 3: Exercises 7.8 (very easy), 7.9, 16.2, 16.4 (slide 221), 16.5 (slide 222), 16.6, 16.8 (only if there is enough time), 16.9.
8
 19 Sep 2025, F1 209
 Lecture: Futures, currencies; Slides 237-261, based on Sections 29.2, 17.1;
Tutorial 4: 29.1, and (if there is enough time left) 29.2, 29.3.
9
 22 Sep 2025, F1 209
 Lecture: Currency derivatives, change of numéraire; Slides 262-288, based on Sections 17.3, 17.4, 26.1-26.5.
Tutorial 4: 17.1, 17.2, 17.3, 17.5 (if there is sufficient time).
10
 25 Sep 2025, F1 209
 Lecture: Incomplete markets I; Slides 289-313, based on Sections 15.1, 15.2.
Tutorial 5: Exercises 15.1, 15.3 (for a univariate model), 15.4, Additional exercises 8, 9, 10, 11.
11
 26 Sep 2025, F1 209
 Lecture: Incomplete marktes II, Stochastic control theory; Slides 314-329, based on Sections 15.3, 19.1, 19.2, beginning of 19.3.
Tutorial: No exercises for this part.
12
 29 Sep 2025, F1 209
 Lecture: Stochastic control theory, HJB equation; Slides 330-347, based on Sections 19.3, 19.4, 19.5.
Tutorial 6: Exercises 19.5, 19.8.
13
 30 Sep 2025 (14:30 - 16:30, F1 202)
 Lecture: Investment theory I; Slides 348-367, based on Sections 19.6, 19.7 (first part).
Tutorial 6: Exercises 19.1, 19.6 (if there is enough time).
14
 2 Oct 2025, F1 209
 Lecture: Investment theory II, martingale approach; Slides 368-386, based on Sections 19.7 (second part), 20.1-20.4 and 20.7.
Tutorial 6: Exercises 20.1, 20.2 and (if there is enough time) 20.3.

Planned schedule of the tutorials
(may be updated during the course; )

0
 8 Sep 2025, F1 209
 Refresher tutorial on Itô calculus and related topics. Among them Wiener process, filtrations, martingales, Itô isometry, Itô formula, simple SDEs, Feynman-Kac formula. Background on Slides 2-40 and Chapters 4,5. Make Exercises (from the 3rd Edition) 4.1, 4.3, 4.8, 5.1, 5.5, 5.9, and 5.8 if there is sufficient time left.
1
 15 Sep 2025, F1 209 Exercises 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.2, 8.3. Also give an example of a (very simple) derivative where the approach of slides 63-66 fails. How to remedy this? Is the Black-Scholes PDE also valid in this example?
2
 18 Sep 2025, F1 202
 Exercises 9.1, 9.5, 9.10. In order to compute the Delta and Gamma for a European call (or put), show first that $s\phi(d_1)-Ke^{-r(T-t)}\phi(d_2) = 0$, where $\phi$ denotes the density of the standard normal distribution. Then it becomes easier to compute the Delta and Gamma. Extra exercises 2, 3, 4 (and, only if there is enough time left, 5).
3
 22 Sep 2025, F1 209
 Exerices 7.8 (very easy), 7.9, 16.2, 16.4 (slide 221), 16.5 (slide 222), 16.6, 16.8 (only if there is enough time), 16.9.
4
 29 Sep 2025, F1 209
 Exercises 17.1, 17.2, 17.3, 17.5 (if there is sufficient time), 29.1, and (if there is enough time left) 29.2, 29.3;
5
 2 Oct 2025, F1 202
 Exercises 15.1, 15.3 (for a univariate model), 15.4 and Additional exercises 8, 9 , 10, 11.
6
 6 Oct 2025, F1 209
 Exercises 19.1, 19.5, 19.6 (if there is enough time), 19.8, 20.1, 20.2 and (if there is enough time) 20.3.



Links

Departamento de Matemática do ISEG, Universidade de Lisboa
Mestrado em Mathematical Finance, Lisbon School of Economics & Management (ISEG), Universidade de Lisboa