Description

Financial mathematics revolves around the unpredictability of financial markets. The cumulative effect of the large number of participants in a market is prices which fluctuate extremely quickly, and this apparently random behaviour can be captured by suitable stochastic models. The dynamics of, e.g., stock prices, portfolios and volatility, are then described by stochastic differential equations, and can be analysed using the associated stochastic calculus.

In the course Mathematical Finance III, you can explore these mathematical foundations, and their application to the pricing of financial derivatives. The purpose of this project would be to look at further problems in the realm of financial mathematics, particularly those relating to the optimisation of a stochastic system subject to suitable constraints. This could include, e.g., portfolio allocation, optimal liquidation of assets, statistical arbitrage, or problems of optimal stopping (such as in the pricing of American options).

You would begin by getting to grips with some of the fundamental principles required to understand the stochastic models used in financial mathematics, such as martingales, stochastic integration, and stochastic differential equations. You would then explore some of the theory of stochastic optimisation, as well as one or more of its applications in finance. There is scope for a more stochastic analytical project, looking at, e.g., backward stochastic differential equations, viscosity solutions, or stochastic volatility models, or a more applied project focusing on specific applications in finance or economics.

Prerequisites

For this project, it is essential that you have taken Probability II, and that you also intend to take Mathematical Finance III. Taking Analysis III or Stochastic Processes III alongside may also be helpful.

Resources

There are a number of books on the subject, such as:

• Huyên Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer, 2009.
• E. Robert Fernholz, Stochastic Portfolio Theory, Springer, 2002.
• Ernst Eberlein and Jan Kallsen, Mathematical Finance, Springer, 2019.
• Jiongmin Yong and Xun Yu Zhou, Stochastic Controls, Hamiltonian Systems and HJB Equations, Springer, 1999.

You can also find various sets of lecture notes online, such as:

• Gonçalo dos Reis and David Siska, Stochastic Control and Dynamic Asset Allocation, Lecture notes, 2018.
• H. Mete Soner, Stochastic Optimal Control in Finance, Lecture notes, 2004.