Module Code - Title:
MS4528
-
MATHEMATICAL AND STATISTICAL MODELS OF INVESTMENTS
Year Last Offered:
2025/6
Hours Per Week:
Grading Type:
N
Prerequisite Modules:
Rationale and Purpose of the Module:
The aim of this module is to equip the student with the necessary analytical
and quantitative skills required for the pricing and hedging of contingent claims,
as well as of interest rate products, credit default swaps, and analyze the risk
and return of individual assets and portfolios.
Syllabus:
• The Black-Scholes Model as a limit of the Binomial Model. Definition
and Properties of Brownian motion. Stochastic Integration, Ito Calculus and
Stochastic Differential Equations for continuous-time models in finance.
Option pricing and hedging in the Black-Scholes model.
• Fixed Income securities and interest rate derivatives, including Swaps,
Caps, Floors, and Black's Formula.
• Credit risk and Credit derivatives such as Credit default swaps,
Collateralised debt obligations. Credit spreads, implied default probabilities
and the pricing of simple derivatives.
• What is volatility? Black-Scholes implied volatilities, realized volatilities,
Volatility Swaps. Time Series models for volatility estimation and forecasting (e.g. using GARCH).
• Portfolio optimization with the Markowitz approach. The Capital Asset
Pricing Model.
Learning Outcomes:
Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)
On successful completion of this module, students should be able to
• hedge contingent claims within the Black-Scholes model
• understand the term-structure concept of interest rates and give the definition of
basic interest rate derivatives
• analyze credit spreads and price credit derivatives such as CDS and CDO
tranches.
• build stochastic time-series models and forecast volatility of asset returns.
• understand basic portfolio optimization problems as trade-offs between
risk and return.
• price assets using the beta of the asset with the market portfolio.
Affective (Attitudes and Values)
N/A
Psychomotor (Physical Skills)
N/A
How the Module will be Taught and what will be the Learning Experiences of the Students:
Tutorials and Labs alternate. Trading exercises with spreadsheet software are used
to facilitate a deeper understanding of the content.
Some of the labs are motivated by current research in portfolio optimization at the department.
Research Findings Incorporated in to the Syllabus (If Relevant):
Prime Texts:
Hull, J.C. (2007)
Options, Futures and Other Derivatives (6ed)
, Pearson
Schonbucher P.J. (2003)
Credit Derivatives Pricing Models
, Wiley
Tsay R.S. (2005)
Analysis of Financial Time Series
, Wiley Blackwell
Luenberger D.G. (1997)
Investment Science,
, Oxford UP.
Other Relevant Texts:
Cvitanic J. and Zapatero F. (2004)
Economics and Mathematics of Financial Markets
, MIT Press
Kennedy D. (2010)
Stochastic Financial models
, CRC Press
Clewlow L. and Strickland C. (1998)
Implementing Derivatives Models
, John Wiley & Sons
Jarrow R. (2002)
Modelling Fixed Income Securities and Interest Rate Options (2nd Edition)
, Stanford UP
Programme(s) in which this Module is Offered:
BSFIMAUFA - FINANCIAL MATHEMATICS
BSMSCIUFA - MATHEMATICAL SCIENCES
MSMAMOTFA - MATHEMATICAL MODELLING
Semester(s) Module is Offered:
Spring
Module Leader:
eugene.gath@ul.ie