## ICM127-Stochastic Calculus and Probability

Module Provider: ICMA Centre
Number of credits: 20 [10 ECTS credits]
Level:7
Terms in which taught: Autumn term module
Pre-requisites:
Non-modular pre-requisites:
Co-requisites:
Modules excluded:
Current from: 2021/2

Module Convenor: Dr Patrick Ilg

Type of module:

Summary module description:

This module consists of two main parts. The first part discusses the tools in Probability needed for derivatives pricing. The second part introduces the basic concepts in Stochastic Calculus (Brownian motions, martingales and Ito Calculus) used to price derivative products.

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 able to discuss and apply:

• 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 PD E’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&n bsp;

• Brownian motion

• Martingales

• Integration theory

• Itô Stochastic Integration

• Itô Calculus

• 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 5 Tutorials 10 Guided independent study: Wider reading (independent) 50 Exam revision/preparation 60 Essay preparation 55 Total hours by term 200 0 0 Total hours for module 200

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

Summative assessment- Examinations:

2 hours closed book written examination

Summative assessment- Coursework and in-class tests:

Coursework: Assignments to be submitted in every week of the term except the first week. The assignments will involve solving problems within the scope of the course. The problem sheets will be handed out during the lectures in the Autumn Term.

In-class test: A single 1 hour closed book written in-class test will take place in the 8th week of the Autumn Term.

Formative assessment methods:

Penalties for late submission:

The below information applies to students on taught programmes except those on Postgraduate Flexible programmes. Penalties for late submission, and the associated procedures, which apply to Postgraduate Flexible programmes are specified in the policy “Penalties for late submission for Postgraduate Flexible programmes”, which can be found here: http://www.reading.ac.uk/web/files/qualitysupport/penaltiesforlatesubmissionPGflexible.pdf
The Support Centres will apply the following penalties for work submitted late:

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.
The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Assessment requirements for a pass:

50% weighted average mark

Reassessment arrangements:

By written examination only, to be taken in August/September as part of the overall examination arrangements for the MSc programme.