Summary module description:

Aims:
This module introduces to students the mathematical tools of probability, calculus and stochastic calculus needed for the valuation of financial derivatives. The course covers the basic concepts and methods of selected areas of modern probability, calculus and stochastic analysis placing emphasis on the possible applications in finance.

Assessable learning outcomes:
By the end of the module, it is expected that the students will be familiar with:
• the main concepts of probability (standard distributions, random variables, expectations, independence, conditional expectations etc.) and their use in financial applications;
• the concepts of stochastic process and different classes of stochastic processes and their distribution;
• the concept of ODE’s and PDE’s relevant for finance; students are expected to be able to solve basic standard ODE’s and PDE’s using transform methods (Laplace and Fourier);
• the concepts of stochastic integration and stochastic differential equations; students are expected to know how to perform stochastic integrations, solve stochastic differential equations, model the behaviour of financial derivatives and price derivatives in simple applications.

Students will get a synthesis of modern probability and stochastic analysis which will improve their ability to read and understand the relevant literature.

Outline content:
• Probability, Random Variables, their Distributions and Characteristics
• Transform methods (Laplace, Fourier)
• Joint Distribution, Conditional Probability and Expectation
• Independence
• Law of Large Numbers and Central Limit Theorem
• Stochastic Processes
• Different Classes of Stochastic Processes
• Brownian motion
• Martingales
• Integration theory
• Itô Stochastic Integration
• Itô Calculus
• ODE’s, PDE’s and their application in finance (including the heat equation)
• Stochastic Differential Equations used in finance
• Change of probability, change of numeraire, and their use in derivatives’ valuation

Brief description of teaching and learning methods:
Teaching is based on presentation supported by extended exercises. Compulsory homework assignments are set weekly for each part. In addition reference is made to the recommended textbooks.

Contact hours:
 Autumn Spring Summer Lectures 20 Seminars 10 Guided independent study 70 Total hours by term 100.00 Total hours for module 100.00

Summative Assessment Methods:
 Method Percentage Written exam 70 Written assignment including essay 10 Class test administered by School 20

Other information on summative assessment:

Formative assessment methods:

Penalties for late submission:
Penalties for late submission on this module are in accordance with the University policy. Please refer to page 5 of the Postgraduate Guide to Assessment for further information: http://www.reading.ac.uk/internal/exams/student/exa-guidePG.aspx

Length of examination:
2 hours closed book written examination

Requirements for a pass:
50% weighted average mark

Reassessment arrangements:
By written examination only, as part of the overall examination arrangements for the MSc programme.