Module manager: Dr Peter Gracar
Email: P.Gracar@leeds.ac.uk
Taught: Semester 1 (Sep to Jan) View Timetable
Year running 2026/27
| MATH1700 | Probability and Statistics |
| MATH3500 | Stochastic Calculus and Derivative Pricing |
| MATH3734 | Stochastic Calculus for Finance |
MATH5320M
This module is not approved as an Elective
This module introduces students to the mathematics of discrete-time finance and its core models. It starts with single-period markets, including the coin-toss model and a general one-period framework. It then builds multi-period market models with the binomial model as the main case. Using these settings, students study option pricing along with arbitrage and replication. The module develops risk-neutral valuation and the fundamental theorem of asset pricing. Students analyse European and American options and compute prices and hedging strategies. It also covers optimal portfolios via the Markowitz problem and the capital asset pricing model. Students develop analytical and computational skills for pricing and portfolio choice. Ideal for quantitative finance and applied mathematics, or any field that needs a rigorous view of risk and return in discrete time.
The aim of this module is to develop a general methodology for the pricing of financial assets in risky financial markets based on discrete-time models.
The aim of this module is to study the mathematical foundations of discrete-time financial models, beginning with simple single-period markets and extending to multi-period frameworks. Students will learn how to model asset prices, evaluate derivatives, and understand the role of randomness and risk in financial decision-making.
The objectives are to:
Introduce single-period models, including the coin-toss and general one-period market, as a basis for discrete-time finance.
Develop multi-period models, with emphasis on the binomial model as a discrete analogue of continuous-time dynamics.
Study arbitrage, replication, and the principle of risk-neutral pricing, leading to the fundamental theorem of asset pricing.
Analyse European and American options, deriving pricing formulas and hedging strategies in discrete time.
Investigate portfolio optimisation, including the Markowitz mean-variance framework and the capital asset pricing model (CAPM).
Equip students with a rigorous understanding of discrete-time methods in finance, preparing them for advanced courses in stochastic calculus, quantitative finance, or related applications in economics, actuarial science, and applied probability.
Subject specific learning outcomes:
On successful completion of the module students will be able to:
Formulate and analyse single-period financial models, including the coin-toss and general one-period framework.
Construct and interpret multi-period discrete-time models, with emphasis on the binomial market model.
Identify and explain conditions for arbitrage, and apply replication arguments in discrete-time settings.
Apply the concept of risk-neutral measures and state and use the Fundamental Theorem of Asset Pricing.
Price and hedge European and American options within discrete-time models.
Solve problems involving optimal portfolio selection, including the Markowitz mean-variance framework.
Demonstrate an understanding of the Capital Asset Pricing Model (CAPM) and its implications for equilibrium asset returns.
Skills learning outcomes:
On successful completion of the module students will be able to:
Apply problem-solving and analytical skills to discrete-time financial models.
Evaluate and interpret financial data and model outputs critically, recognising assumptions, limitations, and sources of risk.
Make informed decisions under uncertainty and ambiguity, applying risk-neutral and optimisation principles.
Communicate financial arguments and results clearly and concisely in a structured written form supported by evidence.
Manage time effectively to balance analytical, computational, and written tasks and meet deadlines.
Demonstrate initiative and perseverance in approaching complex financial problems.
Maintain academic integrity and use appropriate referencing when engaging with financial mathematics literature.
Simple two state market model, trading strategy, arbitrage, replicating principle, European options
General single period market model, value process, risk neutral probability measure, fundamental theorem of asset pricing, stochastic volatility model, attainable claims, no-arbitrage principle, market completeness
Multi-period market model, sigma-algebra, partition, random variables, measurability, filtrations, self-financing condition, conditional probability, conditional expectation, fundamental theorem of asset pricing for multi-period models
Exotic options, American options, binomial asset pricing model, Cox-Ross-Rubenstein model
Risk averse utility functions, Arrow-Pratt coefficients of risk aversion, two-step approach to finding optimal investment strategy
Returns, state price density, mean-variance analysis, Markowitz model, mutual fund principle, capital asset pricing model
Optimal portfolios in multi-period model, dynamic programming approach
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 33 | 1 | 33 |
| Private study hours | 117 | ||
| Total Contact hours | 33 | ||
| Total hours (100hr per 10 credits) | 150 | ||
Feedback on problem sheets will be provided orally to students during lectures/seminars. Individual feedback will be provided during office hours as required.
Check the module area in Minerva for your reading list
Last updated: 30/04/2026
Errors, omissions, failed links etc should be notified to the Catalogue Team