## MAS452 Stochastic Processes and Finance

Note: This is an old module occurrence.

You may wish to visit the module list for information on current teaching.

 Both semesters, 2017/18 20 Credits Lecturer: Dr Nic Freeman Home page Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

Stochastic processes are models that reflect the wide variety of unpredictable ways in which reality behaves. In this course we study several examples of stochastic processes, and analyze the behavior they exhibit. We apply this knowledge to mathematical finance, in particular to arbitrage free pricing and the Black-Scholes model.

Prerequisites: MAS113 (Introduction to Probability and Statistics); MAS221 (Analysis); MAS223 (Statistical Inference and Modelling) [recommended]; MAS275 (Probability Modelling)
Not with: MAS352 (Stochastic Processes and Finance)
No other modules have this module as a prerequisite.

## Outline syllabus

• Stochastic Processes: We introduce conditional expectation and martingales, which are used to study the behavior of stochastic processes such as random walks, urn models, branching processes, Brownian motion and diffusions. Stochastic integration with respect to Brownian motion is introduced.
• Stochastic Finance: We study the key concept of arbitrage and arbitrage free pricing, both in finite markets and in the continuous time Black-Scholes model. We also introduce the idea of modelling debt contagion using branching processes.

## Aims

• Introduce probability spaces, σ-fields and conditional expectation.
• Introduce martingales and study their basic properties.
• Analyse the behaviour of different types of stochastic process, such as random walks, urn models and branching processes.
• Explain the role of arbitrage and arbitrage free pricing.
• Use finite market models to price and hedge a range of financial derivatives.
• Introduce Brownian motion and study its basic properties.
• Introduce stochastic calculus, Ito's formula and stochastic differential equations.
• Derive the Black-Schole's formula in continuous time and use it to price a range of financial derivatives.
• Study extensions of the Black-Scholes model.
• Use branching processes to model debt contagion.

## Learning outcomes

• Study various examples of stochastic processes.
• Use martingales and related tools to study the behavior of stochastic processes.
• Price and hedge financial derivatives, using both finite market models and the continuous time Black-Scholes formula.

## Teaching methods

Lectures, with a complete set of printed notes, plus exercises. Some parts of the lecture notes are for independent study.

44 lectures, no tutorials

## Assessment

One three hour closed book exam.

## Full syllabus

Stochastic Processes

• Probability and measure
• Conditional expectation
• Martingales and related theory
• Examples of stochastic processes
• Brownian motion and stochastic calculus
Stochastic Finance
• Arbitrage pricing
• Finite markets models, options and hedging
• Black-Scholes Theory in continuous time
• Extensions of the Black-Scholes model
• Contagion through branching processes

Type Author(s) Title Library Blackwells Amazon
C Bjork, T Arbitrage Theory in Continuous Time Blackwells Amazon
C Etheridge, A. A Course in Financial Calculus 332.0151922 (E) Blackwells Amazon
C Williams, D. Probability with Martingales 519.236 (W) Blackwells Amazon
C Wilmott, P., Howison, S., Dewynne, J. The Mathematics of Financial Derivatives

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

Most books on reading lists should also be available from the Blackwells shop at Jessop West.

## Timetable (semester 1)

 Wed 09:00 - 09:50 lecture Hicks Lecture Theatre 7 Thu 10:00 - 10:50 lecture Hicks Lecture Theatre 2