Backward Stochastic Differential
Equations and Applications to Derivatives Pricing
Modern financial markets rely heavily on mathematical models to price derivatives and manage risk. Many of these models are built on stochastic processes that evolve randomly over time. While classical approaches such as the Black–Scholes model provide elegant solutions in idealised settings, more sophisticated frameworks are required to capture nonlinear effects and more realistic market behaviour.
Backward stochastic differential equations (BSDEs) provide a powerful and flexible tool for modelling such problems. Originally introduced by Jean-Michel Bismut in the linear case and later generalised by Étienne Pardoux and Shige Peng, BSDEs play a central role in modern theories of nonlinear expectations and financial pricing. This project explores both the mathematical foundations of BSDEs and their applications to derivative pricing.
The group project will develop the necessary background in stochastic calculus before introducing backward stochastic differential equations and their role in financial modelling. The project will begin with fundamental concepts such as Brownian motion, Itô’s formula, and stochastic differential equations, and will progress towards the classical Black–Scholes model for option pricing. Building on this, students will be introduced to BSDEs and their applications in modelling pricing mechanisms for financial derivatives.
By the end of the group project, students will have developed a solid understanding of core concepts in stochastic calculus, including Brownian motion, stochastic integration, Itô’s formula, and stochastic differential equations. They will also understand the derivation and interpretation of the Black–Scholes pricing framework.
In addition, students will have been introduced to backward stochastic differential equations, including questions of existence and uniqueness, and their role in modelling pricing mechanisms. In particular, they will study nonlinear pricing via g-expectations, where the price of a contingent claim is defined through the solution of a BSDE. They will understand how classical models such as Black–Scholes arise as special linear cases within this broader framework.
Throughout the project, students will practise applying theoretical ideas to financial modelling problems and develop the ability to communicate complex mathematical concepts clearly within a group setting.
The project will be based primarily on guided reading, with a strong emphasis on conceptual understanding and mathematical rigour. Students will work collaboratively to study the material, discuss key ideas, and solve relevant problems.
Understanding will be demonstrated through problem solving, exploration of examples in derivative pricing, and clear communication of mathematical arguments. Evidence of learning will include regular participation in discussions, contributions to a group diary, a group presentation in the form of a mini-lecture, and a final oral examination assessing both understanding and collaborative work.
In the individual project, students will build on the material developed in the group project and undertake a self-directed investigation into a more advanced aspect of BSDEs or their applications in financial mathematics.
Possible directions include:
Students may also explore further developments in mathematical finance or propose related topics in consultation with the supervisor.
The individual project will centre on independent study supported by regular meetings with the supervisor. Students will engage with advanced literature, deepen their understanding of their chosen topic, and explore theoretical or applied aspects in detail.
Evidence of learning will be provided through a written project report, a project diary documenting progress and challenges, and an oral examination in which students will discuss their work and demonstrate their understanding.
Indicative resources include: