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 non-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, i.e., we study how to obtain a strategy maximizing the expected utility of the final wealth.

  The mathematical tools presented in the course belong to the theory of probability and stochastic processes. They include the conepts 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. For example, the techniques of optimal control of this course can be used for studying optimal use of resources to the benefit of a sustainable world. 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 probability 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, bid-ask spread,Snell envelope
  • Hedging: the concepts of perfect replication versus imperfect replication
  • Optimal portfolio problems via martingale methods

• Learning outcome

  Upon course completion, the students:

  Knowledge

  • Can explain the foundations of financial modeling in single and a multi-period trading time, 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.
  • Are able to identify and distinguish complete and incomplete markets models.
  • Can characterise and compute the fair price in a complete market and bid-ask spread in the incomplete case.
  • Understand the concepts of replicability and perfect hedging of a financial claim or risk, and use them in the context of complete markets.
  • Can construct hedging strategies of replicable claims.
  • Can solve optimal portfolio problems single and multiperiod setting
  • Balance between satisfaction in investments and risk, in Markovitz portfolio theory

  Skills

  • Understand how the information flow enters decision making and strategies.
  • Can use of probability measures and probability distributions to compute expectations, conditional expectations for prediction.
  • Can characterise the transformation rules to change of probabilities to find the state-price density and the risk-neutral pricing measures.
  • Are able to recognise and use martingales processes and the martingale property.
  • Can compute hedging strategies.
  • Can 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.
  • Understand the benchmarks given by the non-arbitrage pricing theory of replicable and non-replicable claims.
  • Understand the correct power of prediction as a result of the analysis.
  • Understand the concept of hedging, the role of information, and perfume the related computations to define the hedging strategy.
  • Can write and solve problems of optimal control in the single and multi-period setting aimed to financial applications, and see how these can be used also for optimal extraction of resources according to different preferences.

• Teaching

  The course will be delivered in a combination of lectures, in which the theory is presented and a selection of exercises and examples are given in full details. The exercise parts will be both arranged in a combination of moments of individual (or small groups) work with solution in plenum. Furthermore, additional exercises with full solutions are uploaded in Canvas along with the progression of the course. The students are are expected to work through these exercises individually. These proposed exercises are intended to reinforce the notions of the economics principles entering the modelling and improve on the mathematical formulation.

  The format of the course is semi intensive and lectures are grouped according to schedule. The lectures are held in a mixture of hybrid and fully digital classes, this to allow for differentiated participation of Engage.eu students as well as students on site at NHH. The schedule is 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

  Useful background comes from linear algebra, probability and statistics. Particularly, linear systems of equations, knowledge of the concepts of random variable, stochastic processes, probability distribution, expectation and variance.

  Example of courses providing useful background: 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.

• 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)
  • An Introduction to Mathematical Finance with Applications, by A.O. Petters and X. Dong, Springer 2016. ISBN: 978-1-4939-8137-3

  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. Both books present how the economic principles are entering the mathematical formulation aim at quantitative benchmarking of prices and decisions. The last reference is a newer book that helps building the financial intuition while still dealing with the same topics. Please note that only the first reference covers portfolio otpimisation in full.

  Suggestion is to consider the first and the third book as priority, keeping the second as alternative for a softer (and slower) introduction.

  Additional material in the form of notes is uploaded on CANVAS, together with exercises completed by their solutions.

• Retake

  Retake in MAT14 will not be offered during the non-teaching semester (autumn). Only mandatory bachelor courses with an individual written school exam or a home exam lasting up to one day will have a retake assessment in the non-teaching semester.

  For detailed information regarding the retake policy, please visit our website: https://www.nhh.no/en/for-students/examinations/retake-of-exams (copy URL).