School of Mathematics and Statistics (SoMaS)

## MAS6052 Stochastic Processes and Finance

 Both semesters, 2011/12 20 Credits Lecturer: Prof John Biggins Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

A stochastic process is a mathematical model for phenomena unfolding dynamically and unpredictably over time. This module studies two classes of stochastic process particularly relevant to financial phenomena: martingales and diffusions. The module develops the properties of these processes and then explores their use in Finance. A key problem considered is that of the pricing of a financial derivative such as an option giving the right to buy or sell a stock at a particular price at a future time. What is such an option worth now? Martingales and stochastic integration are shown to give powerful solutions to such questions.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.

## Outline syllabus

• Semester 1: Stochastic Processes
• Probability and measure
• Conditional expectation
• Martingales and stopping times
• Brownian motion
• Stochastic integration
• Itô calculus
• Stochastic differential equations
• Semester 2: Stochastic Finance
• The binomial model revisited
• The fundamental theorem of asset pricing
• Asset pricing in continuous time

## Aims

• extend the student's knowledge of stochastic processes by studying martingales, stochastic calculus and diffusions;
• extend the student's ability to perform stochastic modelling in continuous time and continuous space;
• introduce Brownian motion, an important example of a stochastic process;
• introduce the basic ideas and concepts of financial markets;
• show how stochastic processes may be applied to the study of financial markets, in particular to the pricing and hedging of financial derivatives;
• develop the theory of the Black-Scholes formula in discrete and continuous time, risk-neutral valuation and the Fundamental Theorem of Asset Pricing.

## Learning outcomes

• the martingale property in discrete and continuous time;
• basic properties of martingales;
• the use of martingales for stochastic modelling in a range of settings;
• the definition and basic properties of diffusions;
• the use of diffusions for stochastic modelling in a variety of applications;
• the basic types of financial derivatives, particularly options;
• how to price and hedge options;
• how to derive and use the Black-Scholes formula in discrete and continuous time;
• volatility and other parameters measuring sensitivity in financial models.

## Teaching methods

Lectures, with a complete set of printed notes, plus task and exercise sheets. Some outside reading is also expected.

40 lectures, no tutorials

## Assessment

One three hour closed book exam.

## Full syllabus

Semester 1: Stochastic Processes

• Probability and measure (2 weeks)
• Conditional expectation (1 week)
• Martingales and stopping times (2 weeks)
• Brownian motion (1 week)
• Stochastic integration (1 week)
• Itô calculus (2 weeks)
• Stochastic differential equations (2 weeks)
• Revision (1 week)
Semester 2: Stochastic Finance
• The binomial model revisited (4 weeks)
• Fundamental theorem of asset pricing (3 weeks)
• Asset pricing in continuous time (5 weeks)

Type Author(s) Title Library Blackwells Amazon
B Billingsley, P. Probability and Measure 519.2 (B) Blackwells Amazon
B Bingham, N. H. and Kiesel, R. Risk-neutral valuation: Pricing and hedging of financial derivatives 332.6457 (B) Blackwells Amazon
B Etheridge, A. A Course in Financial Calculus 332.0151922 (E) Blackwells Amazon
B Williams, D. Probability with Martingales 519.236 (W) Blackwells Amazon
B Williams, D. Weighing the Odds 519.2 (W) Blackwells Amazon
B Williams, R. J. Introduction to the Mathematics of Finance 332.0151 (W) Blackwells Amazon
C Klebaner, F. C. Introduction to stochastic calculus with applications 519.2 (K) Blackwells Amazon
C Mikosch, T. Elementary stochastic calculus, with finance in view 519.2 (M) Blackwells Amazon
C Nefti, S. N. An Introduction to the Mathematics of Finance Blackwells Amazon
C Rosenthal, J. S. A first look at rigorous probability theory 519.2 (R) Blackwells Amazon
C Steele, J. M. Stochastic calculus and financial applications 519.2 (S) Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop on Mappin Street.