# 2020/21 Taught Postgraduate Module Catalogue

## MATH5734M Advanced Stochastic Calculus and Applications to Finance

### 20 creditsClass Size: 25

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

Taught: 1 Jan to 31 May View Timetable

Year running 2020/21

### Pre-requisite qualifications

(MATH1710 or MATH2700) and MATH2750

Basic knowledge of Excel spreadsheets

### This module is mutually exclusive with

 MATH3734 Stochastic Calculus for Finance MATH5320M Discrete Time Finance MATH5330M Continuous Time Finance

Module replaces

MATH3733 Stochastic Financial Modelling

This module is not approved as an Elective

### Module summary

This module provides a rigorous exposition of fundamental mathematical aspects of stochastic calculus in continuous time and its applications to finance. Students will learn materials from mathematical analysis and probability theory which will be combined to derive key concepts in stochastic analysis as, e.g., stochastic differential equations. Further, the module will review applications of stochastic calculus in actuarial and financial models and will address some examples of stochastic control problems.

### 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 exposition of the fundamental results from this theory. Students will acquire a solid understanding of advanced concepts as, e.g., martingales, stochastic integration and stochastic differential equations. Further, this module will review some applications of the theory in the context of stochastic control problems and mathematical finance.

Learning outcomes
1. Obtain an overview of modern probability theory via basic measure theory and basic functional analysis (including L2-spaces)
2. Understand the following mathematical concepts: martingales, stopping times, Brownian motion, Itô's formula and Ito-Tanaka formula, local times of Brownian motion, diffusion theory
3. Understand key results concerning stochastic differential equations (SDEs): existence, uniqueness, concepts of strong and weak solution
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. Understanding of the mainstream stochastic control models in actuarial and financial mathematics, e.g., the dividend problem and American option pricing
8. Use of Excel spreadsheet for simulation of SDEs and applications to option pricing

### Syllabus

1. Preliminaries: Probability spaces with sigma-algebras and elements of measure theory.

2. Brownian motion: construction and properties of its trajectories.

3. Martingales and stopping times: optional sampling theorem, Doob's inequality.

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

5. Stochastic differential equations (SDEs): existence and uniqueness of solutions; difference between strong and weak solutions; Itô's formula.

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

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).

9. Elements of stochastic control.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Lecture 22 1.00 22.00 Practical 1 2.00 2.00 Private study hours 176.00 Total Contact hours 24.00 Total hours (100hr per 10 credits) 200.00

### Private study

Study course material and complete assignments. Attempt exercise sheets in advance of tutorial classes. Reading as directed. Review of Excel basic commands as advised in preparation for practical sessions and coursework.

### Opportunities for Formative Feedback

Regular problem sheets

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

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 Open Book exam 3 hr 00 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).