# MSc Quantitative Finance / Course details

Year of entry: 2024

## Course unit details:Stochastic Calculus for Finance

Unit code BMAN71541 15 FHEQ level 7 – master's degree or fourth year of an integrated master's degree Semester 1 No

### Overview

The course takes a general probability theory-view to introduce students to fundamental concepts in finance, such as (Black-Scholes) option pricing, risk-neutral pricing, etc. The course starts off with an overview of general probability theory. For example, the first sessions deal with infinite probability spaces, filtrations, expectations, conditioning (i.e., sigma-algebras), etc. Having discussed this material, students are ready to learn about Levy processes, in particular, Brownian motion. Topics in this part of the course include: random walks, quadratic variation, the martingale property, first passage time, etc. The course then turns to stochastic calculus, for example, the Ito integral. These sessions will also offer a rigorous proof of the Black-Scholes model. The final important topic is risk-neutral pricing.

The course will be delivered via ten lectures. There are three (optional) tutorials. In the first tutorial, we practice the Ito-Doeblin formula. In the second, we use risk-neutral valuation techniques to derive the Black-Scholes-Merton formula. In the final tutorials, we derive important properties of a multi-asset market. The final lecture will be spent revisiting the material, discussing what topics are particularly relevant for the exam. Half of the sixth lecture will be used to discuss the assignment.

### Pre/co-requisites

BMAN71541 Programme Req: BMAN71541 is only available as a core unit to students on MSc Quantitative Finance

### Aims

In this course unit, students learn about the basic mathematical tools necessary to understand fundamental concepts in quantitative finance. These mathematical tools include: the Markov property, Brownian motion, first passage time, the Ito-Doeblin formula, etc. Using these tools, students learn how to price plain-vanilla European call and put options. The course also demonstrates that every asset can be valued using risk-neutral pricing techniques.

The course aims to introduce the above concepts from a rigorous perspective: Mathematical proofs will almost always be offered. Despite this aim, the course instructor will also spend a great amount of time on discussing the underlying intuition behind the mathematical tools, in many cases using examples from the realm of finance. The main objective of the course is to introduce a small number of challenging mathematical propositions in such a way that (a) students can understand them and (b) know why they are relevant for Finance.

### Learning outcomes

On completion of this unit successful students will have achieved the following learning outcomes:

·  Understand the basic mathematical tools necessary to derive fundamental results in Finance, such as, for example, the Black-Scholes (1973) model:

• Ito’s Lemma;
• Brownian motion;
• Change of Measure;
• Etc.

·  Independently use the above mathematical tools to value non-standard financial assets, such as exotic (say look-back) options.

·  Simulate values from stochastic processes introduced during the course, and use these to price options via Monte-Carlo techniques.

### Assessment methods

Coursework Assignment (50%)

Written Examination (50%)

### Feedback methods

Informal advice and discussion during a lecture or workshop.

Responses to student emails and questions from the course director.

Detailed written comments (generic and specific) on assessed assignment.

Generic feedback posted on Blackboard regarding overall performance.

The course will be entirely based on the following textbook:

Shreve, Steven (2004): Stochastic Calculus for Finance II: Continuous-time Models. Springer Finance Series, Springer, New York (ISBN: 144192311X).
Note that there is a second revised edition from 2010. The second edition covers the same material as the 2004 edition, but some mistakes in the earlier version have been corrected.

### Study hours

Scheduled activity hours
Assessment written exam 2
Lectures 30
Practical classes & workshops 3
Independent study hours
Independent study 115

### Teaching staff

Staff member Role
Kevin Aretz Unit coordinator