ST33915 Introduction to Mathematical Finance
Introductory description
This module runs in Term 1 and is available for students on a course where it is a listed option and as an Unusual Option to students who have completed the prerequisite modules.
Prerequisites:
Statistics Students: ST218 Mathematical Statistics A
NonStatistics Students: ST220 Introduction to Mathematical Statistics
It is strongly recommended to take either MA359 Measure theory or ST342 Mathematics of Random Events alongside this module.
Results from this module will be partly used to determine exemption eligibility in the Institute and Faculty of Actuaries (IFoA) module CM2.
This module serves as a prerequisite for ST401 Stochastic Methods in Finance, IB357 Investment Management
IB359 Derivatives and Risk Management, IB394 International Financial Management and EC334 Topics in Financial Economics: Corporate Finance and Markets.
Module aims
To provide an introduction to Mathematical Finance in discrete time and cover the discrete part of the actuarial syllabus.
To be able to evaluate and interpret the theory of mathematical finance in discrete time and to apply theoretical concepts to construct stochastic models of financial markets.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
 NoArbitrage and the Fundamental Theorem of Asset Pricing
a) Mathematical model for one period financial markets
b) Trading strategies and arbitrage opportunities
c) Discounting and Equivalent Martingale Measures
d) The Fundamental Theorem of Asset Pricing  MeanVariance Portfolio Selection and the CAPM
a) The return of an asset and of a portfolio
b) Maximising the expected return
c) The meanvariance problems
d) The case without a riskless asset
e) The case with a riskless asset
f) The Markowitz tangency portfolio and the capital market line
g) Meanvariance equilibria
h) The Capital Asset Pricing Model (CAPM)  Utility Theory
a) Preferences on lotteries
b) Von NeumannMorgenstern representation
c) Concave functions and Jensen’s inequality
d) Expected utility representation
e) Measuring risk aversion
d) A primer on utility maximisation  Introduction to Risk Measures
a) Monetary measures of risk
b) Value at Risk and Expected Shortfall  Pricing and Hedging in Finite Discrete Time
a) Conditional expectations
b) Filtrations and martingales
c) Financial markets in finite discrete time
d) Selffinancing strategies
e) The Fundamental Theorem of Asset Pricing revisited
f) Valuation of contingent claims
g) Complete markets
h) Pricing and hedging in the binomial model
Learning outcomes
By the end of the module, students should be able to:
 Understand key notions of arbitrage and equivalent martingale measures in a one period financial market; calculate the set of equivalent martingale measures in a financial market
 Solve the meanvariance problems, understand the concept of the capital market line, describe the Capital Asset Pricing model including the principal results and assumptions
 Describe preference orders of financial investors, explain the concept of risk aversion, solve simple utility maximisation problems
 Explain the modern concept of monetary measures of risk, Calculate the Value at Risk and Expected Shortfall for given distributions
 Model financial markets in finite discrete time, describe selffinancing strategies and absence of arbitrage, hedge derivate products in complete and incomplete markets in discrete time
Indicative reading list
H. Föllmer and A. Schied: Stochastic Finance. An Introduction in Discrete Time, 4th ed., de Gruyter, 2016.
S.F. LeRoy and J. Werner: Principles of Financial Economics, 2nd ed., Cambridge University Press, 2014.
S.E. Shreve: Stochastic Calculus for Finance 1: The Binomial Asset Pricing Model, Springer, 2003.
J. Jacod and P. Protter: Probability Essentials, Springer, 2003.
View reading list on Talis Aspire
Subject specific skills
TBC
Transferable skills
TBC
Study time
Type  Required  Optional 

Lectures  30 sessions of 1 hour (20%)  2 sessions of 1 hour 
Tutorials  5 sessions of 1 hour (3%)  
Private study  115 hours (77%)  
Total  150 hours 
Private study description
Weekly revision of lecture notes and materials, wider reading, practice exercises and preparing for examination.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group B1
Weighting  Study time  

Online Examination  100%  
The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade. ~Platforms  Moodle

Assessment group R
Weighting  Study time  

Online Examination  Resit  100%  
The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade. ~Platforms  Moodle

Feedback on assessment
Solutions and cohort level feedback will be provided for the examination.
Postrequisite modules
If you pass this module, you can take:
 EC33415 Topics in Financial Economics: Corporate Finance and Markets
Antirequisite modules
If you take this module, you cannot also take:
 EC33315 Topics in Financial Economics: Theories and International Finance
 IB25312 Principles of Finance 1
 IB25315 Principles of Finance 1
Courses
This module is Core for:
 Year 3 of USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

USTAG301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
 Year 3 of G30E Master of Maths, Op.Res, Stats & Economics (Actuarial and Financial Mathematics Stream) Int
 Year 4 of G30E Master of Maths, Op.Res, Stats & Economics (Actuarial and Financial Mathematics Stream) Int
This module is Optional for:
 Year 3 of UCSAG4G1 Undergraduate Discrete Mathematics
 Year 3 of UCSAG4G3 Undergraduate Discrete Mathematics
 Year 4 of UCSAG4G2 Undergraduate Discrete Mathematics with Intercalated Year

USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
 Year 3 of G300 Mathematics, Operational Research, Statistics and Economics
 Year 4 of G300 Mathematics, Operational Research, Statistics and Economics
This module is Option list A for:

UMAAG100 Undergraduate Mathematics (BSc)
 Year 3 of G100 Mathematics
 Year 3 of G100 Mathematics

USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 3 of G1G3 Mathematics and Statistics (BSc MMathStat)
 Year 4 of G1G3 Mathematics and Statistics (BSc MMathStat)
 Year 4 of USTAG1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)
 Year 3 of USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 4 of USTAGG17 Undergraduate Mathematics and Statistics (with Intercalated Year)
 Year 4 of UMAAG101 Undergraduate Mathematics with Intercalated Year
 Year 3 of USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics
 Year 4 of USTAY603 Undergraduate Mathematics,Operational Research,Statistics,Economics (with Intercalated Year)
This module is Option list B for:
 Year 3 of USTAG302 Undergraduate Data Science
 Year 3 of USTAG304 Undergraduate Data Science (MSci)
 Year 4 of USTAG303 Undergraduate Data Science (with Intercalated Year)

UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)
 Year 3 of G105 Mathematics (MMath) with Intercalated Year
 Year 5 of G105 Mathematics (MMath) with Intercalated Year

UMAAG103 Undergraduate Mathematics (MMath)
 Year 3 of G103 Mathematics (MMath)
 Year 3 of G103 Mathematics (MMath)
 Year 4 of G103 Mathematics (MMath)
 Year 4 of G103 Mathematics (MMath)

UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 3 of G106 Mathematics (MMath) with Study in Europe
 Year 4 of G106 Mathematics (MMath) with Study in Europe