Module: Probability and Finance II (Intermediate)

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-ordinator	Semester 1: John Appleby Semester 2: John Appleby Autumn: John Appleby
Module Teachers	John Appleby
NFQ level	8	Credit Rating	7.5
Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

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Array

Description

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

*Type*	*Hours*	*Description*
Workload	Full-time hours per semester
Lecture	36	No Description
Tutorial	12	No Description
Independent Study	150	No 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 Assessment	20%	Examination Weight	80%

Type	Description	% of total	Assessment Date
Course Work Breakdown

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

None

Programme or List of Programmes

ACM	BSc in Actuarial Mathematics
AFU	Age Friendly University Programme
BSSA	Study Abroad (DCU Business School)
BSSAO	Study Abroad (DCU Business School)
FIM	B.Sc. Financial Mathematics
HMSA	Study Abroad (Humanities & Soc Science)
HMSAO	Study Abroad (Humanities & Soc Science)
IESA	Study Abroad (Institute of Education)
IESAO	Study Abroad (Institute of Education)
SHSA	Study Abroad (Science & Health)
SHSAO	Study Abroad (Science & Health)

Date of Last Revision

18-JUN-08

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