Mathematical methods in modern finance
(African Institute for Mathematical Sciences, 29 January-16 February 2007)
All the computer practicals in this course will be done by using the spreadsheet package Gnumeric, which is available on all the computers at AIMS (in the Office menu). A quick introduction to Gnumeric can be found here.
Part I: Asset pricing
(offered by Alet Roux)
Lecture 1 provided a short introduction to financial markets, derivative securities and how they are used.
Single-period models
In Lecture 2 we studied a one-step binary model with a risky asset (stock) and a risk-free asset (a bank account) and investigated conditions under which it admits arbitrage. Lecture 3 introduced replication and risk-neutral probabilities and how these quantities may be used to price derivative securities. In Lecture 4 we priced and replicated derivative securities in one-step models with many scenarios. We also studied conditions under which such models are complete.
Assignment 1 was due on Thursday 8 February. The questions and model solutions of the written part can be found here, and a model solution for the computer assignment here.
Multi-period models
In Lecture 5 we discussed the general modelling assumptions of a finite multi-period model with a bank account and a stock. We defined a self-financing trading strategy, an arbitrage opportunity and stated a theorem that gives a characterisation of a viable model. We introduced replicating strategies for derivative securities in Lecture 6, and applied the Law of One Price to find the fair price of a call option in a two-period binary model. In Lecture 7 we discussed the conditions under which a model is complete. We also studied the flow of information in a discrete-time model, in preparation for the risk-neutral pricing method.
In Lecture 8 we defined and gave examples of adapted and predictable processes, and discussed the one-to-one relationship that there exists between probability measures and collections of one-step conditional probabilities. Lecture 9 was devoted to the risk-neutral pricing method. We studied the properties of equivalent martingale measures and used them to calculate the prices of attainable derivative securities. In Lecture 10 we discussed the application of the risk-neutral method to the pricing of chooser and American options.
Assignment 2 is due on Thursday 15 February.
Part II: Portfolio theory
(offered by Ekkehard Kopp)
In Lecture 1 we covered the concept of return on investment (see Ex 1) and variance/covariance. See Ex 2-4 for an example using options. In the Friday lectures we developed the theory of the Markowitz bullet for markets with just 2 underlying assets, and we are now ready to generalise this to markets with n assets. The Lecture Notes below now cover pages 1-24 and include Exercises 1-7, which should be handed in by Tuesday 20 February. Your final assignment will be a gnumeric spreadsheet to illustarte the genral theory we can now discuss. This should be handed in by Monday 26 February. The full text of Lectuire Notes is now posted, together with Assignment 2. We have so far (Tuesday 13th) covered up to page 35, and you now have two parametrisations of the MVL in the general case.
A separate file with the 7 Exercises and one with Summary Notes is also posted below, as is a file marked ''Extras' which outlines some calculus and linear algebra background