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Discrete Time Modelling & Derivative Securities (Online Version) - MAT00024M

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  • Department: Mathematics
  • Module co-ordinator: Prof. Tomasz Zastawniak
  • Credit value: 20 credits
  • Credit level: M
  • Academic year of delivery: 2022-23

Module will run

Occurrence Teaching period
A1 Autumn Term 2022-23 to Spring Term 2022-23
A2 Autumn Term 2022-23 to Summer Term 2022-23
B1 Spring Term 2022-23 to Summer Term 2022-23
B2 Spring Term 2022-23 to Spring Term 2023-24

Module aims

The aim of the module is to explain in simple - namely discrete time - settings the fundamental ideas of modelling of financial markets and pricing of derivative securities using the principle of no arbitrage. Even the simplest of all models with only one time step allows several important notions to be illustrated. The module progresses with more complex models - involving many time steps and several stocks - which are developed along with the corresponding theory of pricing and hedging derivative securities such as options or forwards. Relatively simple mathematical considerations lead to powerful notions and techniques underlying the theory - such as the no-arbitrage principle, completeness, self-financing and replicating strategies, and equivalent martingale measures. These are directly applicable in practice, particularly in the continuous time limiting theory developed in a subsequent module. The general methods are applied in detail in particular to pricing and hedging European and American options within the Cox-Ross-Rubinstein (CRR) binomial tree model. The Black-Scholes model as the limit of CRR models is discussed to pave the way for continuous time theory.

Module learning outcomes

By the end of this module students should

  • understand the basic ideas, principles and assumptions for building simple discrete time financial models, and associated notions (portfolios, trading strategies, discounted prices etc);
  • be able to test a discrete-time model for the absence of arbitrage;
  • understand the notion of a derivative security and the common examples;
  • understand the principle of no arbitrage (NAP) for pricing derivatives;
  • understand the notions of self-financing and replicating strategies, and their role in pricing using the NAP;
  • understand the notion of an equivalent martingale measure and its use in pricing;
  • understand the notion of completeness of a market;
  • be able to apply the above theory to particular examples, especially European options, and explicitly calculate replicating (hedging) trading strategies, equivalent martingale measures and derivative prices, using Excel spreadsheets as well as hand calculation;
  • understand how to price and hedge American options in a discrete model and do explicit calculations.


Task Length % of module mark
Coursework - extensions not feasible/practicable
N/A 100
Oral presentation/seminar/exam
Online Viva
N/A 0

Special assessment rules



Task Length % of module mark
Coursework - extensions not feasible/practicable
N/A 100
Oral presentation/seminar/exam
Online Viva
N/A 0

Module feedback

Information currently unavailable

Indicative reading

1. M. Capinski and E. Kopp, Discrete Models of Financial Markets, Cambridge University Press 2012.
2. M. Capinski and T. Zastawniak, Mathematics for Finance: An Introduction to Financial Engineering, 2nd edition, Springer 2011.
3. R. Elliott, E. Kopp, Mathematics of Financial Models, Springer 2005.
4. J. van der Hoek, R. Elliott, Binomial Models in Finance, Springer 2005
5. D. Lamberton, B. Lapayere, Introduction to Stochastic Calculus Applied to Finance, Chapman & Hall 2008.
6. A. Etheridge, A Course in Financial Calculus, Cambridge University Press, 2002.

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.