Module Specifications..
Current Academic Year 2023  2024
Please note that this information is subject to change.
 
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Description This module provides a selfcontained treatment of the axiomatic theory of probability, including the theory of expectation (integration) and conditional expectation. While no prior knowledge of these topics is assumed, some familiarity with abstract mathematical reasoning is expected. The module contains also a brief introduction to discretetime martingales, concentrating on their role in the theory of discretetime finance, and especially in the theory of option pricing. No prior knowledge of finance is assumed. The end of semester examination is of two hours’ duration and a choice of questions is available for students.  
Learning Outcomes 1. State and interpret the main definitions relevant to advanced probability and asset pricing and demonstrate a masterty of the ideas underlying them through examples and counterexamples 2. Deduce important properties of probability measures from the axioms 3. Prove the main theorems of expectation theory 4. Derive the main properties of conditional expectations from their geometric characterisation 5. Apply martingales and arbitrage arguments to discretetime financial models 6. Calculate the price of American options in discretetime binomial models 7. Prove results in the theory of option pricing by applying the method of pricing by replication  
All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml 

Indicative Content and Learning Activities
Events Probability triple, the elementary approach, general sample space. The axioms of probability, sigma algebras, probability measures. Properties of probability measures. The necessity of the axiomatic construction. Borel sigma algebra, extension of probabilities. Random Variables Measurability, elementary (closure) properties, probability distribution functions. Expectation Simple random variables, approximation of positive random variables by simple ones. The main limit theorems: monotone convergence, dominated convergence, Fatou’s lemma. Properties of expectation. Variance, Chebyshev’s inequality. Expectation of functions of random variables, the moment generating function. Conditional Expectation Elementary definition, conditional expectation with respect to a decomposition of the sample space; optimal approximation property of conditional expectation. Conditional expectation with respect to a subsigma algebra as an orthogonal projection. Properties of the conditional expectation. Martingales. Simple Models of the Stock Market Arbitrage pricing of forward contracts. Simple binomial model. Options, pricing a call by replication. Pricing American put options by recursion. General Models of the Stock Market Trading strategies, arbitrage, replicating portfolio, complete and incomplete markets. Statement of the two fundamental theorems of asset pricing. Pricing American call options by arbitrage arguments.  
 
Indicative Reading List
 
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