MAT14 Mathematical Finance

• Topics

  Introductory Mathematical Finance is a semi-intensive course that equips the students with the foundation of financial modeling in a single and multi-period trading framework. The focus is on the rigorous understanding of how the principles of finance are merged into the models. The focus is on the pricing of financial derivatives.

  The objective of the course is to introduce the theory of mathematical finance and the mathematical tools on which this is based. The focus chosen is on the pricing of financial assets via arbitrage theory. We will concentrate on discrete time models, e.g. Cox-Ross-Rubinstein and multinomial models, and classical assets as European call and put options.

  In the above market models we will also study optimal portfolio problems, that is we study how to obtain a strategy maximizing the expected utility of the final wealth. For this we will concentrate on complete markets.

  The mathematical tools presented in the course belong to the theory of probability and stochastic processes. They include the concepts of probability measures, conditional expectations, and martingales. The concepts, methods, and models discussed have a value by themselves and can be applied beyond the focus of this course. They also constitute the base for follow-up courses at master level.

  Topics

  • Elements of probability theory and stochastic processes: probability measures, conditional expectations, convergences, filtrations, martingales, change of measure
  • Financial assets: derivatives of European type
  • Discrete time models: Cox-Ross-Rubinstein and multinomial models as preparation to the classical Black-Scholes model
  • Pricing methods: concept on arbitrage, risk-neutral evaluation, Snell envelope
  • Optimal portfolio problems via martingale methods

• Learning outcome

  At the end of the course the students will acquire:

  KNOWELEDGE

  1. The foundations of financial modeling in single and a multi-period trading time, including the binomial model.
  2. The definition of arbitrage opportunity and the use of the law-of-one-price to achieve the concept of "fair" pricing.
  3. The concepts of attainability and perfect hedging of a financial claim or risk. The associated models for complete and incomplete markets.
  4. The construction of hedging strategies.
  5. The study of optimal portfolios and optimal consumption and investment schemes in the different market models of single and multiperiod type.

  SKILLS

  The course will provide specific skills in quantitative methods, these include:

  1. Stochastic processes and the information flow.
  2. Probability measures, probability distributions, change of probabilities: to find the state-price density and the risk-neutral pricing measures.
  3. Expectations and conditional expectations also under change of measure.
  4. Martingales processes.
  5. Stochastic optimal control problems: risk-neutral approach and dynamic programming

  GENERAL COMPETENCE

  The course provides a basic competence in quantitative methods for finance. This enables the students to be aware of the correct models to be applied in the different financial modeling contexts, to be aware of the techniques involved, and to be able to understand the correct power of prediction as a result of the analysis. The students will be able to write problems of optimization in the single and multiperiod setting and to solve them in the case of complete markets.

• Teaching

  The course will be delivered through a combination of lectures in which both theory is presented and a selection of exercises is solved in plenum with comments. The exercises are uploaded along with the progression of the course. The exercises proposed are intended to reinforce the notions and improve on the mathematical formalisation. These exercises together with the theory constitute a strong basis in preparation for the exam. The lectures are grouped in a semi-intensive fashion according to schedule.

  The course is going to be held with a mixture of physical, fully virtual and hybrid format.

• Recommended prerequisites

  There are no formal requirements for this course, however any background knowledge or insight in the mathematical directions can facilitate the participation.

  Very useful background is the ability to solve linear systems of equations. Good knowledge of the concept of random variable, probability distribution, stochastic process, expectation and variance is also helpful.

  Example of courses providing useful background can be: MAT10 Analyse og lineær algebra; MAT12 Matematisk statistikk; MAT13 Optimering.

• Credit reduction due to overlap

  VOA038, FOR10 (expired course codes)

• Compulsory Activity

  2 assignments given during the course.An approval in both assignments is a prerequisite to sit at the final exam.

• Assessment

  4 hours written individual home exam.

  If nothing else is determined by the course start, the examination can only be written in English.

• Grading Scale

  A-F

• Literature

  The course will present selected topics that can be retrived in the following recommended reference books:

  • Introduction to Mathematical Finance, Discrete Time Models, by S.R. Pliska, Blackwell Publishers 1997. ISBN: 978-1-55786-945-6.
  • Derivative Pricing in Discrete Time, by N. Cutland and A. Roux, Springer 2012. ISBN: 978-1-4471-4407-6 (This also available as e-book: ISBN: 978-1-4471-4408-3)

  The first reference is complete in all topics and includes the parts on stochastic optimisation. The second reference is a simpler introduction and reading, it does not cover the whole program.

  Moreover, lecture notes on selected topics will be made available.