Introductory description

This module runs 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.

Pre-requisites

• Statistics Students:

  • ST218 Mathematical Statistics A; or,
  • ST228 Mathematical Methods for Statistics and Probability and ST229 Probability for Mathematical Statistic.

• Non-Statistics Students:

  • ST220/ST226 Introduction to Mathematical Statistics; or,
  • ST232/ST233 Introduction to Mathematical Statistics.

It is strongly recommended to take either MA359 Measure theory or ST342 Mathematics of Random Events/ST3xx Measure Theory for Probability 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 web page

Module aims

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.

1. No-Arbitrage 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
2. Mean-Variance Portfolio Selection and the CAPM
   a) The return of an asset and of a portfolio
   b) Maximising the expected return
   c) The mean-variance 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) Mean-variance equilibria
   h) The Capital Asset Pricing Model (CAPM)
3. Utility Theory
   a) Preferences on lotteries
   b) Von Neumann-Morgenstern representation
   c) Concave functions and Jensen’s inequality
   d) Expected utility representation
   e) Measuring risk aversion
   d) A primer on utility maximisation
4. Introduction to Risk Measures
   a) Monetary measures of risk
   b) Value at Risk and Expected Shortfall
5. Pricing and Hedging in Finite Discrete Time
   a) Conditional expectations
   b) Filtrations and martingales
   c) Financial markets in finite discrete time
   d) Self-financing 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 mean-variance 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 self-financing 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

• Demonstrate facility with rigorous probabilistic methods.

• Evaluate, select and apply appropriate mathematical and/or probabilist techniques.

• Demonstrate knowledge of and facility with formal probability concepts, both explicitly and by applying them to the solution of finance problems.

• Create structured and coherent arguments communicating them in written form. 

• Construct logical mathematical arguments with clear identification of assumptions and conclusions.

• Reason critically, carefully, and logically and derive (prove) mathematical results.

Transferable skills

• Problem solving: Use rational and logical reasoning to deduce appropriate and well-reasoned conclusions. Retain an open mind, optimistic of finding solutions, thinking laterally and creatively to look beyond the obvious. Know how to learn from failure.

• Self awareness: Reflect on learning, seeking feedback on and evaluating personal practices, strengths and opportunities for personal growth.

• Communication: Present arguments, knowledge and ideas, in a range of formats.

• Professionalism: Prepared to operate autonomously. Aware of how to be efficient and resilient. Manage priorities and time. Self-motivated, setting and achieving goals, prioritising tasks.