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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 (Advanced)
Module Code MS508
School School of Mathematical Sciences
Module Co-ordinatorSemester 1: John Appleby
Semester 2: John Appleby
Autumn: John Appleby
Module TeachersJohn Appleby
NFQ level 9 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 and particular emphasis is placed on learning to adapt it to new situations. 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 are among the features which distinguish this module from a corresponding undergraduate module (MS408). 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 fundamental concepts of the theory of stochastic processes in continuous-time and demonstrate a mastery of these concepts through examples and counter-examples
2. Employ the Optional Stopping Theorem to establish properties of various hitting times
3. Apply stopping time methods to solve problems in continuous time financial models
4. Solve simple stochastic differential equations using Ito calculus
5. Model financial phenomena using stochastic differential equations and interpret the solutions of these equations
6. Prove and interpret the basic results of utility theory
7. Solve Merton's problem for constant relative risk aversion (CRRA) utilities and interpret and appraise the results
8. Prove Girsanov's Theorem an apply it to selected problems in continuous-time finance
9. Derive the Black-Scholes formula and apply the method to a variety of extensions of the basic problem
10. Critique the assumptions underlying simple continuous time stock price models

Workload Full-time hours per semester
Type Hours Description
Lecture36No Description
Tutorial12No Description
Independent Study175No Description
Total Workload: 223

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

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. Necessity of non-differentiable paths of stock prices in arbitrage free markets.

Martingales in Continuous Time
Filtrations, adaptedness, Brownian martingales. Stopping times, optional stopping, hitting times. Applications of hitting times to exchange rate bands, timing of buying or selling risky assets.

Ito Calculus
Ito integral for simple adapted processes; Ito integral as an isometry; Ito processes, Ito's lemma, stochastic differential equations. Applications to the evolution of stock prices and interest rates.

Optimal Portfolio Theory
The Black-Scholes stochastic differential equation of stock prices. Significance of the equation. Utility theory. Merton's optimal investment problem.

Option Pricing
Girsanov's theorem and the equivalent martingale measure approach to option pricing; the arbitrage approach. Derivation of the Black-Scholes Formula. Extensions of the formula. Interpretation and significance of the Black-Scholes formula. The strengths and weaknesses of the Black-Scholes model.

Assessment Breakdown
Continuous Assessment20% Examination Weight80%
Course Work Breakdown
TypeDescription% of totalAssessment Date
In Class Testn/a20%Week 11
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 UP,
  • Etheridge, A.: 2003, A course in Financial Calculus, Oxford University Press,
Other Resources

Programme or List of Programmes
CHPMMaster of Science
GTSHGraduate Training Visitor Program (S&H)
MFMMSc in Financial Mathematics
PYPMMaster of Science

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