MMath Mathematics and Statistics / Course details

Derivative securities (such as options) depend on the values of primary securities (such as stock or bond prices). During the last thirty years trading in derivative securities have undergone a tremendous development, and nowadays derivative securities are traded on markets all over the world in large numbers. The purpose of the course is to exhibit basic features of advanced financial derivatives, starting with basic model specifications, introducing the concept of arbitrage, and ending with a risk-neutral valuation formula and its analysis.

Students are not permitted to take, for credit, MATH49102 in an undergraduate programme and then MATH69012 in a postgraduate programme at the University of Manchester, as the courses are identical.

The unit aims to provide a concise mathematical formulation of the main characteristics of financial instruments, with an emphasis on quantitative aspects of stock price, options, and other financial derivatives.

On successful completion the students will be able to: 

• Apply modern Probability Theory, including martingale theory, stochastic calculus and stochastic differential equations, to stochastic models in finance.
• Define and apply discrete time financial market models, in particular to find the hedging and fair prices of various options in the CRR model.
• Define and apply continuous time financial market models, in particular to find the hedging and fair prices of various options in the Black-Scholes model.
• Define and apply interest rate models, in particular to analysis various affine term structure models in short rate models and to construct risk-neutral forward rate models.

1. Discrete-time market models: The Cox-Ross-Rubinstein model; self-financing portfolios; contingent claims; arbitrage opportunity; risk-neutral measure; fundamental theorem of asset pricing; risk-neutral pricing. [6]

2. Continuous-time market models: A review of continuous martingales, Brownian motion and stochastic calculus; the Black-Scholes models; self-financing portfolios; contingent claims; arbitrage opportunity; risk-neutral measure; fundamental theorem of asset pricing; risk-neutral pricing. [12] 

3. American (put and call) options: Optimal stopping and free boundary problems; exotic options: Knock-out barrier option; lookback option; Asian option; chooser option; digital option; forward-Start option; basket option. [8]

4. Interest rate models: Short rate models; the Vasicek model; the Ho-Lee model; Cox-Ingersoll-Ross model; Heath-Jarrow-Morton framework; Hull-White models. [7]

End of semester examination: weighting 100%.

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• Lamberton, D. and Lapeyre, B., Introduction to Stocastic Calculus Applied to Finance, Chapman and Hall 1996.
• BjÃrk, T., Arbitrage Theory in Continuous Time, Oxford University Press 1998.
• Etheridge, A., A Course in Financial Calculus, Cambridge University Press, 2002.
• Musiela, M. and Rutkowski, M., Martingale Methods in Financial Modelling, Springer 2005.
• Shiryaev, A. N., Essentials of Stochastic Finance, World Scientific 1999.

This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  

Please see Blackboard / course unit related emails for any further updates.