2019/20 Taught Postgraduate Module Catalogue

MATH5734M Advanced Stochastic Calculus and Applications to Finance

20 creditsClass Size: 25

Module manager: Elena Issoglio
Email: E.Issoglio@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2019/20

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

Delivery type	Number	Length hours	Student hours
Lecture	44	1.00	44.00
Practical	1	2.00	2.00
Private study hours			154.00
Total Contact hours			46.00
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

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)	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).

Reading list

The reading list is available from the Library website

Last updated: 20/05/2019

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