Module: Stochastic Finance (Advanced)

Latest Module Specifications

Current Academic Year 2025 - 2026

Module Title	Stochastic Finance (Advanced)
Module Code	MTH1072 (ITS: MS526)
Faculty	Mathematical Sciences	School	Science & Health
NFQ level	9	Credit Rating	7.5

Description

This postgraduate course covers asset pricing, with emphasis on a rigorous analysis of continuous-time models. Arbitrage, trading strategies, market completeness. Portfolio choice in complete and incomplete markets. Myopic and hedging demand. Derivatives pricing: risk-neutral pricing and risk premia.

Learning Outcomes

1. Prove the main results in Mathematical Finance
2. Solve portfolio choice problems rigorously
3. Price assets and derivatives with advanced models
4. Design hedging strategies for new financial products
5. Develop customized pricing methods
6. Critically evaluate the assumptions underlying different asset pricing models

*Type*	*Hours*	*Description*
Workload	Full time hours per semester
Lecture	36	Classes
Directed learning	3	Final Exam
Seminars	5	Attendance to Research Seminars
Independent Study	150	Independent work on textbooks and related papers
Total Workload: 194

Section Breakdown
CRN	20799	Part of Term	Semester 2
Coursework	0%	Examination Weight	0%
Grade Scale	40PASS	Pass Both Elements	Y
Resit Category	RC1	Best Mark	N
Module Co-ordinator	Paolo Guasoni	Module Teacher

Type	Description	% of total	Assessment Date
Assessment Breakdown
Formal Examination	End-of-Semester Final Examination	100%	End-of-Semester

Reassessment Requirement Type
Resit arrangements are explained by the following categories; RC1: A resit is available for both^* components of the module. RC2: No resit is available for a 100% coursework module. RC3: No resit is available for the coursework component where there is a coursework and summative examination element. ^* ‘Both’ is used in the context of the module having a coursework/summative examination split; where the module is 100% coursework, there will also be a resit of the assessment

Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

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

Arbitrage
Assets, Payoffs, Simple and Gross Returns, Strategies, Self-financing portfolios, Arbitrage, Stochastic Discount Factors, First and Second Fundamental Theorems of Asset Pricing, Law of one price.

Hedging
Hedging Arbitrage bounds, Replication Bounds, Superhedging, Superreplication Theorem, Market Completeness, Perfect Replication, Redundancy of Assets.

Optimality
Utility Functions, Absolute and Relative Risk Aversion, Allais and Ellsberg paradoxes, Savage Representation, Arbitrage and Utility, First-order condition, Investment and Consumption. Logarithmic, Power, and Exponential Utilities.

Mean-Variance Analysis
Expected Return and Risk, State Price-Beta Representation, Hansen-Jagannathan Bound, Mean-Variance Frontier, Two-fund Separation

Duality
Legendre Transforms and their properties, Duality Method for Verification, Pricing by Marginal Utility. Duality Bounds.

Continuous Time
Bachelier and Samuelson models. Continuous trading. Doubling strategies. Admissible Strategies. Local Martingales and Supermartingales. No arbitrage and Admissibility. Stochastic Discount Factors and Martingale Measures. Bayes' formula.

Diffusion Models
Instantaneous returns and covariances. Stochastic exponential. Discount factors in diffusion models and risk premia. Representation of payoffs.

Portfolio Choice
Utility maximization from terminal wealth and consumption. Solution by Duality in Bachelier and Samuelson models with exponential and power utility. Merton and Markowitz formulas. General solution in complete markets. Duality with consumption.

Stochastic Control
Dynamic programming. Value Function. Hamilton-Jacobi-Bellman equations. Verification. Homogeneity.

Stochastic Investment Opportunities.
Asset Prices and State Variables. Incompleteness. Intertemporal Hedging. Logarithmic and Power Transformations. Long-run limits. Martingale Measures and Risk-Neutral Dynamics. Long-run optimality. Finite-horizon Bounds.

Indicative Reading List

Books:

Hans Follmer, Alexander Schied: 0, Stochastic finance, 3110183463
Darrell Duffie: 2001, Dynamic asset pricing theory, Princeton University Press, Princeton, N.J., 978-0691090221
Freddy Delbaen, Walter Schachermeyer: 0, The mathematics of arbitrage, 3540219927

Articles:
None

Other Resources

None