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Module Specifications..

Current Academic Year 2023 - 2024

Please note that this information is subject to change.

Module Title Probability and Finance II (Intermediate)
Module Code MS408
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 thorough introduction to Brownian motion, stochastic calculus and their application to finance. It builds on Probability and Finance I in that it deals with the problem of extending to continuous time the ideas first encountered in a discrete-time set-up. The Black-Scholes model is covered in detail. The end of semester examination is of two hours’ duration and a choice of questions is available for students.

Learning Outcomes

1. Demonstrate an understanding of the fundamental concepts of the theory of stochastic processes in continuous time through examples and counter-examples
2. Use the Optional Stopping Theorem to establish properties of various hitting times
3. Solve simple stochastic differential equations
4. Prove the basic results of utility theory and solve Merton's problem for CRRA utilities
5. Prove Girsanov's Theorem an apply it to selected problems in continuous-time finance
6. Derive the Black-Scholes formula and apply the method to a variety of extensions of the basic problem

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

The Limit Theorems of Probability Theory
The Borel-Cantelli lemmas; modes of convergence; Weak and Strong Laws of Large Numbers; the Central Limit Theorem.

Brownian Motion
Provisional definition, specification of a stochastic process through its finite order distributions, Daniell-Kolmogorov theorem; versions, difficulty with continuity, completion of the probability space, Kolmogorov's continuity criterion, modification of a process; properties of Brownian motion: scaling, nowhere differentiability of sample paths.

Martingales in Continuous Time
Filtrations, adaptedness, Brownian martingales; stopping times, optional stopping, hitting times.

Ito Calculus
Ito integral for simple adapted processes; Ito integral as an isometry; Ito processes, Ito's lemma, stochastic differential equations.

Optimal Portfolio Theory
The stochastic differential equation of stock prices; utility, Merton's problem.

Option Pricing
Girsanov's theorem and the equivalent martingale measure approach to option pricing; the arbitrage approach.

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

  • Lamberton, D., and Lapeyre, B: 1996, Introduction to Stochastic Calculus with Financial Applications, Chapman and Hall, London,
  • Bjork, T: 1998, Arbitrage Theory in Continuous Time, Oxford University Press,
  • Etheridge, A.: 2003, A course in Financial Calculus, Oxford 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)
Date of Last Revision18-JUN-08

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