Deep Learning in Finance, 7.5 credits

Contents

Module 1 (3.5 hp): Theory

The module starts with an introduction to feed-forward neural networks (FFN) and the backpropagation algorithm. Thereafter, the (stochastic) gradient descent algorithm (SGD) is investigated and applied to train FFN in the context of financial mathematics. In particular, this will be applied to solving the Black-Scholes PDE with an FFN and more general high-dimensional parabolic PDEs. This is followed by an investigation of the universal approximation theorem. After that, and in the context of financial mathematics, autoencoders (AE), generative adversarial networks (GAN), and binary classifiers alongside various layer types such as dropout layers and batch normalization layers are introduced.

In the next part of the course, all the concepts are extended by recurrent neural networks (RNN). In particular, Long-Short-Term-Memory (LSTM) and Gated-Recurrent-Units (GRU) are used.

At the end of the course, we further investigate advanced topics and choose among topics like delta hedging with possible transaction costs, stochastic optimal control, or non-Markovian modelling.

Module 2 (3 hp): Computer labs

This module covers the implementation of neural networks using Python with the PyTorch library. In particular, the applications that are studied are a selection of option pricing, optimal stopping, anomaly detection, data generation, and hedging.

Expected learning outcomes

After successful completion of the course, the student should be able to:

Knowledge and Understanding

• describe the mathematical foundations of feed-forward and recurrent neural networks, gradient descent, and backpropagation.
• analyze how different network structures (e.g., AE, GAN, RNN, LSTM, GRU) function and their role in modeling complex financial systems.
• critically assess the suitability of different deep learning models for specific financial applications, including their assumptions and limitations.

Skills and Abilities

• implement and train neural networks using PyTorch to solve problems such as option pricing, optimal stopping, and anomaly detection,
• design and develop tailored neural network architectures for financial applications.

Judgement and Approach

• critically evaluate model interpretability, transparency, and risk implications in financial decision-making,
• select appropriate modeling approaches based on problem structure, data availability, and computational considerations,
• Reflect on the strengths and limitations of machine learning methods in financial contexts.

Required Knowledge

The course requires 90 ECTS, including at least 7.5 ECTS in linear algebra, and a course at advanced level in financial mathematics (at least 7.5 ECTS) in which Itô calculus is covered. Proficiency in English equivalent to the level required for basic eligibility for higher studies. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

Form of instruction

The teaching takes the form of lectures, lessons and a numerical group project.

Examination modes

Modules 1 and 2 are assessed through written or oral examinations. In all modules, one of the following judgements is awarded: Fail (U), Pass (G) or Pass with distinction (VG). For the whole course, one of the following grades is awarded: Fail (U), Pass (G) or Pass with distinction (VG). To pass the whole course, all modules must be passed.

A student who has been awarded a passing grade for the course cannot be reassessed for a higher grade. Students who do not pass a test or examination on the original date are given another date to retake the examination. A student who has undergone two examinations for a course or a part of a course, without passing either examination, has the right to have another examiner appointed, provided there are no specific reasons for not doing so (Chapter 6, Section 22, HEO). The request for a new examiner is made to the Head of the Department of Mathematics and Mathematical Statistics. Examinations based on this course syllabus are guaranteed to be offered for two years after the date of the student's first registration for the course.

Examiners may decide to deviate from the modes of assessment in the course syllabus. Individual adaption of modes of assessment must give due consideration to the student's needs. The adaption of modes of assessment must remain within the framework of the intended learning outcomes in the course syllabus. Students who require an adapted examination must submit a request to the department holding the course no later than 10 days before the examination. The examiner decides on the adaption of the examination, after which the student will be notified.

Credit transfers

Students are entitled to an assessment of whether previous education or equivalent knowledge and skills acquired in professional experience can be accredited for equivalent studies at Umeå University. Applications for credit transfers must be sent to Student Services/Degree Evaluation Office. More information on credit transfers can be found on Umeå University's student website, www.student.umu.se, and in the Higher Education Ordinance (Chapter 6). Rejected applications for credit transfers can be appealed (Higher Education Ordinance, Chapter 12) to the Higher Education Appeals Board. This applies regardless of whether the rejection relates to all or parts of the credit transfer application.

Other regulations

In a degree, this course may not be included together with another course with a similar content. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics.

Transitional provisions

In the event that the syllabus ceases to apply or undergoes major changes, students are guaranteed at least three examinations (including the regular examination opportunity) according to the regulations in the syllabus that the student was originally registered on for a period of a maximum of two years from the time that the previous syllabus ceased to apply or that the course ended.