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Discrete Time Modelling & Derivative Securities - MAT00023M

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• Department: Mathematics
• Module co-ordinator: Dr. Alex Daletskii
• Credit value: 20 credits
• Credit level: M
• Academic year of delivery: 2022-23

Module will run

Occurrence Teaching cycle
A Autumn Term 2022-23

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 with 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 viability, completeness, self-financing and replicating strategies, arbitrage 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

At the end of the module you should be able to:

• 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 viability;
• 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
• understand how to price American options in a discrete model and do explicit calculations

Module content

Indicative Content:

1. Basic assumptions and definitions for discrete-time market models.
2. The 1-step binary model with 1 stock and bond:viability, completeness, replicating portfolios, risk-neutral probabilities, pricing derivatives.
3. Viability and incompleteness for more than binary branching.
4. General discrete-time models - many time steps, several stocks, one bond.General notions of trading strategies, self-financing strategies, value and gains processes, arbitrage.
5. Fundamental Theorem of Asset Pricing in discrete time.
6. Derivative securities - definitions and examples especially European options.
7. Pricing of attainable derivatives.
8. Specialisation to the 1 stock binomial (CRR) model
9. American options, pricing, hedging and optimal exercise time.

Assessment

Task Length % of module mark
Online Exam - 24 hrs (Centrally scheduled)
Discrete Time Modelling & Derivative Securities
3 hours 100

None

Reassessment

Task Length % of module mark
Online Exam - 24 hrs (Centrally scheduled)
Discrete Time Modelling & Derivative Securities
3 hours 100

Module feedback

Individual feedback and advice on assessed coursework will be offered to students during scheduled office hours.