## 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: 2018/9

Module Convenor: Dr Horatio Boedihardjo

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 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 0 Tutorials 10 Guided independent study 170 Total hours by term 200.00 Total hours for module 200.00

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 in January

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:
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

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.