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 knowledge of proofs of certain deep theorems, the emphasis on interpretation of theoretical results in a financial context and on financial model building, as well as the appraisal of the strengths and weaknesses of this financial model, and the pricing of exotic options in this framework, are among the features which distinguish this module from a corresponding undergraduate module (MS437). The end of semester examination is of three hours’ duration and will require students to answer all questions on the paper.  
Learning Outcomes 1. State and interpret the main definitions relevant to advanced probability and asset pricing and demonstrate a mastery of the ideas underlying them through examples and counterexamples 2. Deduce important properties of probability measures from the axioms and defend the necessity of the axiomatic approach to probability theory for uncountable sample spaces 3. Prove measurability results for random variables and to interpret these results in a financial context 4. Prove the main theorems of expectation theory, and to construct examples and counterexamples to demonstrate the sharpness of these results 5. Derive the main properties of conditional expectations from their geometric characterisation and to compare and judge the effectiveness of this approach with more elementary definitions 6. Apply martingale methods and arbitrage arguments to discretetime financial models, and to construct novel portfolios to achieve arbitrage pricing 7. Calculate the price of American options in discrete time binomial models 8. Prove results in the theory of option pricing by applying the method of pricing by replication 9. Prove the fundamental theorems of asset pricing and interpret these results in the context of efficient market theory 10. Critique the assumptions underlying discrete time financial models  
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. Demonstrating the necessity of axiomatic construction. Borel sigma algebra, extension of probabilities. Random Variables Measurability, elementary (closure) properties, limit of sequences of random variables, probability distribution functions. Expectation Simple random variables, approximation of positive random variables by simple ones. Expectation as an integral over the sample space. 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, and other exotic options, by recursion. General models of the Stock Market Arbitrage pricing of forward contracts. Simple binomial model. Options, pricing a call by replication. Pricing American put options, and other exotic options, by recursion.  
 
Indicative Reading List
 
Other Resources None  
Module code is MS537  
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