DCU Home | Our Courses | Loop | Registry | Library | Search DCU

Module Specifications..

Current Academic Year 2023 - 2024

Please note that this information is subject to change.

Module Title Probability and Finance I (Intermediate)
Module Code MS437
School School of Mathematical Sciences
Module Co-ordinatorSemester 1: John Appleby
Semester 2: John Appleby
Autumn: John Appleby
Module TeachersJohn Appleby
NFQ level 8 Credit Rating 7.5
Pre-requisite None
Co-requisite None
Compatibles None
Incompatibles None

This module provides a self-contained 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 discrete-time martingales, concentrating on their role in the theory of discrete-time 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 counter-examples
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 discrete-time financial models
6. Calculate the price of American options in discrete-time binomial models
7. Prove results in the theory of option pricing by applying the method of pricing by replication

Workload Full-time hours per semester
Type Hours Description
Lecture36No Description
Tutorial12No Description
Independent Study150No Description
Total Workload: 198

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

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.

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 sub-sigma 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.

Assessment Breakdown
Continuous Assessment20% Examination Weight80%
Course Work Breakdown
TypeDescription% of totalAssessment Date
Reassessment Requirement Type
Resit arrangements are explained by the following categories;
1 = A resit is available for all components of the module
2 = No resit is available for 100% continuous assessment module
3 = No resit is available for the continuous assessment component
This module is category 3
Indicative Reading List

  • Williams, D: 1991, Probability with Martingales, Cambridge University Press,
Other Resources

Programme or List of Programmes
ACMBSc in Actuarial Mathematics
AFUAge Friendly University Programme
BSSAStudy Abroad (DCU Business School)
BSSAOStudy Abroad (DCU Business School)
FIMB.Sc. Financial Mathematics
HMSAStudy Abroad (Humanities & Soc Science)
HMSAOStudy Abroad (Humanities & Soc Science)
IESAStudy Abroad (Institute of Education)
IESAOStudy Abroad (Institute of Education)
SHSAStudy Abroad (Science & Health)
SHSAOStudy Abroad (Science & Health)

My DCU | Loop | Disclaimer | Privacy Statement