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

  Upon course completion, the students can:

  Knowledge

  • Explain the foundations of financial modeling in single and a multi-period trading timeline, including the binomial model.
  • Understand the definition of arbitrage opportunity and the use of the law-of-one-price to define the concept of "fair" price.
  • Be able to identify and distinguish complete and incomplete markets models.
  • Characterise and compute the fair price in a complete market and bid-ask spread in the incomplete case.
  • Understand the concepts of attainability and perfect hedging of a financial claim or risk, and use them in the context of complete markets.
  • Construct hedging strategies of replicable claims.
  • Solve optimal portfolio problems single and multiperiod setting

  Skills

  • Critically analyse stochastic processes and the information flow.
  • Calculate probability measures, probability distributions,
  • Characterise the transformation rules to change of probabilities: find the state-price density and the risk-neutral pricing measures.
  • Compute expectations and conditional expectations, also under change of measure.
  • Recognise and use martingales processes and the martingale property.
  • Solve stochastic optimal control problems by the risk-neutral approach and dynamic programming

  General competence

  • Understand financial modelling from the non-arbitrage theory point of view.
  • Pricing theory of replicable and non-non replicable claims.
  • Understand the correct power of prediction as a result of the analysis.
  • Hedging concept and computational methods.
  • Write and solve problems of optimization in the single and multiperiod setting.

• Teaching

  The course will be delivered through a combination of lectures, in which both theory is presented, and a selection of exercises, which solved in plenum and/or in small groups. Other exercises with solutions are uploaded along with the progression of the course. The students are are expected to work through them. These proposed exercises are intended to reinforce the notions and improve on the mathematical formalisation.

  The course format is semi intensive, lectures are grouped according to schedule. The lectures are held in a mixture of physical, fully virtual and hybrid format. These are announced on Canvas.

• Restricted access

  The course is offered to students at NHH, and students taking MAT14 as an Online Exchange Course through the ENGAGE.EU alliance (University of Mannheim, the University of Toulouse Capitole, LUISS University, Tilburg University, University of National and World Economy in Sofia, Vienna University of Economics and Busines, Ramon Llull University and Hanken School of Economics).

• 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 be admitted to the final exam.

• Assessment

  4 hours written individual home exam.

  The examination can only be written in English.

  An assessment in MAT14 will not be organised in the the non-teaching semester. As of autumn 2023, only mandatory bachelor courses with an individual assessment will have an assessment in the non-teaching semester. This only applies to students with a valid course approval. The retake options that apply at all times are decided by the dean for the bachelor program and will be published in the course description.

• 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, selected material is uploaded on CANVAS, together with exercises and solutions.

• This is an ENGAGE-course

  The course is offered as an Online Exchange Course to students from the ENGAGE.EU alliance.