Module manager: TBC
Email: TBC
Taught: Semester 1 (Sep to Jan) View Timetable
Year running 2026/27
None
| MATH2501 | Financial Mathematics |
| MATH2702 | Stochastic Processes |
MATH3734 Stochastic Calculus for Finance
This module is not approved as an Elective
This module introduces students to the mathematical foundations and applications of stochastic calculus, focusing on modelling uncertainty in continuous time. It covers key concepts such as Brownian motion, Itô calculus, and financial models like Black–Scholes. Students will develop analytical and computational skills relevant to pricing derivatives and interest-rate products. Ideal for those interested in quantitative finance, applied mathematics, or any field where randomness and dynamic modelling play a central role.
This module provides a rigorous introduction to stochastic calculus, a fundamental mathematical framework for modelling uncertainty in continuous time. Students will gain an understanding of measure-theoretic probability and then move on to develop an understanding of stochastic processes, leading to martingales and Brownian motion—the central object in stochastic calculus. With Brownian motion established, students will be introduced to stochastic integrals and Itô’s formula, the stochastic counterpart to the classical chain rule. Students progress to stochastic differential equations (SDEs), studying existence, uniqueness, and numerical approximation of solutions and how they arise in applications to finance.
On successful completion of the module students will have demonstrated the following learning outcomes relevant to the subject: 1. Obtain an overview of modern probability theory including basic measure theory. 2. Understand the fundamentals of continuous time stochastic processes. 3. Demonstrate the understanding of the necessity of Itô calculus. 4. Be able to apply martingale techniques to stochastic integrals, use Ito formula, solve explicitly linear SDEs 5. Demonstrate an understanding of the Black–Scholes derivative-pricing model: completeness, risk-neutral pricing, replication, explicit valuation of vanilla options 6. Establish links between solutions to SDEs and PDEs, in particular, in the context of Black-Scholes model 7. Apply the risk-neutral approach to the pricing of zero-coupon bonds and interest-rate derivatives in a general one-factor diffusion model for the risk-free rate of interest 8. Demonstrate awareness of the Vasicek, Cox–Ingersoll–Ross and Hull–White models for the term structure of interest rates. 9. Use appropriate computer software for numerical computations in the context of stochastic analysis and pricing of stock and interest-rate derivatives
Elements of measure theoretic probability: probability spaces, random variables, expectation; convergence of random variables; conditional expectation 2. Stochastic processes: filtration; martingales; Brownian motion and its properties 3. Itô calculus: construction of Itô's integral and its properties; Itô's formula; stochastic differential equations (SDEs): existence and uniqueness of solutions; numerical approximation of solutions to SDEs; Girsanov theorem; links between Ito calculus and PDE theory, Feynman-Kac formula 4. Black and Scholes model and pricing of European options; risk-neural measure and market completeness; pricing of derivatives including Black-Scholes formula for vanilla call/put options; Black–Scholes partial differential equation; the Greeks 5. Short-rate models; fundamentals of term structure models of interest rates; valuation of zero-coupon bonds and derivatives; discussion of Vasicek, CIR, Hull-White models
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Practicals | 1 | 2 | 2 |
| Lecture | 44 | 1 | 44 |
| Private study hours | 154 | ||
| Total Contact hours | 46 | ||
| Total hours (100hr per 10 credits) | 200 | ||
Formative feedback will be provided on regular example sets or other similar learning activity.
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Last updated: 30/04/2026
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