# 2022/23 Undergraduate Module Catalogue

## MATH3734 Stochastic Calculus for Finance

### 15 creditsClass Size: 140

Module manager: Dr Konstantinos Dareiotis
Email: K.Dareiotis@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2022/23

### Pre-requisite qualifications

(MATH1710 or MATH2700) and MATH2750

Basic knowledge of Excel spreadsheets

### This module is mutually exclusive with

 MATH5320M Discrete Time Finance MATH5330M Continuous Time Finance MATH5734M Advanced Stochastic Calculus and Applications to Finance

Module replaces

MATH3733 Stochastic Financial Modelling

This module is not approved as a discovery module

### Module summary

This module provides a mathematical introduction to stochastic calculus in continuous time with applications to finance. Students will learn material in areas of mathematical analysis and probability theory. This knowledge will be used to derive expressions for prices of derivatives in financial markets under uncertainty.

### Objectives

Stochastic calculus is one of the main mathematical tools to model physical, biological and financial phenomena (among other things). This module provides a rigorous introduction to this topic. Students will develop a solid mathematical background in stochastic calculus that will allow them to understand key results from modern mathematical finance.

Learning outcomes
1. Obtain an overview of modern probability theory via basic measure theory and basic functional analysis
2. Understand the following mathematical concepts: martingales, stopping times, Brownian motion, Itô's formula and diffusion theory
3. Understand key results concerning stochastic differential equations (SDEs)
4. Draw links between SDEs and partial differential equations
5. Use SDEs to model financial assets and price simple derivatives, e.g., European vanilla options
6. Use SDEs to model markets with stochastic interest rates and, in this context, price Zero Coupon Bonds
7. Use of Excel spreadsheet for simulation of SDEs and applications to option pricing

### Syllabus

1. Preliminaries: Elements of measure theoretic probability.

2. Brownian motion and its properties.

3. Martingales and stopping times.

4. Itô calculus: Construction of Itô's integral and its properties. Itô's formula.

5. Stochastic differential equations (SDEs): existence and uniqueness of solutions.

6. Links between Ito calculus and PDE theory: Feynman-Kac formula.

7. Applications of SDEs to mathematical finance (part 1): Black and Scholes model and European vanilla options.

8. Applications of SDEs to mathematical finance (part 2): stochastic models of interest rates (CIR and Vasicek models for spot rates).

### Teaching methods

 Delivery type Number Length hours Student hours Lectures 33 1.00 33.00 Practical 1 2.00 2.00 Private study hours 115.00 Total Contact hours 35.00 Total hours (100hr per 10 credits) 150.00

### Opportunities for Formative Feedback

Regular problem sheets

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Computer Exercise To be based on the use of spreadsheet software 15.00 Assignment To be based on a set of questions based on the course material 5.00 Total percentage (Assessment Coursework) 20.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 80.00 Total percentage (Assessment Exams) 80.00

Examination material for level 3 (MATH3734) and level 5 (MATH5734M) module is partly shared. Exams should be timetabled at the same time (but level 5 exam is longer).