Module manager: Dr Nadhir Ben Rached
Email: N.BenRached@leeds.ac.uk
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2026/27
Please note that only one pre requisite module is required. MATH3500 or MATH5320M is acceptable.
| MATH3500 | Stochastic Calculus and Derivative Pricing |
| MATH5320M | Discrete Time Finance |
This module is not approved as an Elective
As financial products grow increasingly complex, advanced numerical methods are essential to price options and assess risk where closed-form solutions are unavailable. This module introduces key computational techniques—binomial models, Monte Carlo simulation, and finite difference methods—for the accurate and efficient valuation of financial derivatives. With a strong emphasis on practical implementation and numerical simulation, it bridges theory and application, preparing students to meet real-world challenges in quantitative finance and derivative pricing.
The primary objective of this module is to introduce students to the standard computational methods used in financial mathematics for pricing both plain vanilla and exotic non-standard options. As financial instruments become increasingly complex, closed-form analytical solutions are often unavailable, making advanced numerical techniques essential for valuation. Students will focus on three core approaches—binomial models, Monte Carlo simulation, and finite difference methods—which are widely used across the financial industry.
The module provides students with Python programming skills required to implement numerical methods for financial derivative pricing and hedging. Lectures provide the mathematical foundations and modelling techniques, while practical sessions offer opportunities to apply these methods, investigate algorithmic performance, and develop analytical thinking.
Subject specific learning outcomes:
On successful completion of the module students will have demonstrated the following learning outcomes:
1. Ability to develop and apply Python programming skills to implement, structure, and test numerical algorithms for mathematical and finance-related problems.
2. Explain the basic modelling tools for financial options.
3. Implement and analyse the binomial model for option pricing, including techniques to improve convergence.
4. Apply standard sampling methods to generate random numbers with specified distributions.
5. Solve stochastic differential equations using numerical schemes and study their convergence properties.
6. Describe and implement Monte Carlo methods for derivative pricing and apply variance reduction techniques.
7. Compute price sensitivities (the Greeks) and explain their application in hedging strategies.
8. Derive the Black–Scholes partial differential equation (PDE) and interpret its financial significance.
9. Implement numerical methods to solve the Black–Scholes PDE for pricing options.
10. Analyse the accuracy, efficiency, and computational complexity of different numerical algorithms for derivatives pricing.
Skills learning outcomes:
On successful completion of the module students will be able to:
a. Communicate and present technical work effectively in both written reports and programming code.
b. Use digital tools and Python programming techniques to design and effectively implement financial algorithms.
c. Manage time effectively and work independently to meet deadlines.
d. Apply analytical thinking and technical knowledge to solve problems and interpret results.
e. Critically assess computational methods and reflect on their practical applications and limitations.
f. Identify and evaluate academic resources to support problem-solving.
1. Introduction to Python programming: fundamentals of syntax and program structure, control flow statements, data types, simple user interaction, loops, and functions.
2. Recap of key concepts behind option pricing.
3. Binomial models: algorithms, calibration, computational complexity, convergence, and techniques to improve convergence.
4. Monte Carlo methods: fundamentals, Monte Carlo integration, option pricing applications, and variance reduction techniques including antithetic variates and control variates.
5. Random number generation: sampling from specific distributions, inverse transform method, rejection sampling, and sampling from the normal distribution.
6. Numerical methods for SDEs: basic SDE concepts, Itô’s formula and applications, Feynman–Kac formula, Brownian motion simulation, numerical schemes and convergence properties, multidimensional SDEs, and standard Monte Carlo method for option pricing.
7. Greeks (price sensitivities) and hedging strategies: hedging concepts and numerical methods for computing Greeks.
8. Black–Scholes PDE: derivation, boundary conditions, and its relationship to the heat equation.
9. Numerical methods for PDEs: finite difference schemes, stability, and convergence analysis.
Additional topics that build on these may be covered as time allows. Such topics may be drawn from the following, or similar:
10. Advanced variance reduction techniques like importance sampling, conditional Monte Carlo, and multilevel Monte Carlo for option pricing
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 33 | 1 | 33 |
| Practical | 6 | 1 | 6 |
| Private study hours | 111 | ||
| Total Contact hours | 39 | ||
| Total hours (100hr per 10 credits) | 150 | ||
111
Students receive formative feedback through seminars, practical sessions, and written feedback on coursework.
Check the module area in Minerva for your reading list
Last updated: 06/07/2026
Errors, omissions, failed links etc should be notified to the Catalogue Team