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Durham University

Postgraduate Module Handbook 2021/2022

Archive Module Description

This page is for the academic year 2021-22. The current handbook year is 2022-23

Department: Mathematical Sciences

MATH52230: Financial Mathematics

Type Tied Level 5 Credits 30 Availability Available in 2021/22
Tied to G5K609 Scientific Computing and Data Analysis [Final intake in October 2022]

Prerequisites

  • • Some undergraduate-level mathematics, covering calculus, integration, ordinary and partial differential equations, and some basic probability theory.

Corequisites

  • None

Excluded Combination of Modules

  • None

Aims

  • To provide an introduction to the mathematical theory of financial products.
  • Provide advanced knowledge and critical understanding of pricing of financial products and derivatives.

Content

  • Introduction to options and markets: the probabilistic basis for valuation of financial products. Arbitrage.
  • Background in basic probability theory. Random variables, conditional expectation, moment generating functions, modes of convergence, the normal distribution and the central limit theorem.
  • Modelling financial markets in discrete time. Binomial tree models. Arbitrage-free pricing. Portfolios. Risk-neutral probabilities. Discrete-time martingales.
  • Modelling financial markets in continuous time. Brownian motion, quadratic variation, continuous-time martingales.
  • Refresher on key calculus concepts: the Riemann integral, the heat equation.
  • Introduction to stochastic calculus: the Ito integral, Ito processes, Ito's formula. Stochastic differential equations.
  • The Black-Scholes market: pricing contingent claims via replicating portfolios. The Black-Scholes partial differential equation. Change of measure, Girsanov's theorem, and applications to pricing. The risk-neutral valuation formula.
  • Numerical and computational methods for pricing: Monte Carlo methods, finite-difference methods.
  • Further topics to be chosen from: Delta and Gamma hedging, exotic options, Feynman-Kac formula, limitations of the Black-Scholes model.

Learning Outcomes

Subject-specific Knowledge:
  • Advanced understanding of the principles and practice of probabilistic pricing methods for financial products.
  • Advanced understanding of the concepts of arbitrage, risk-neutral measures, and market equilibirum used in the pricing of financial derivatives.
Subject-specific Skills:
  • By the end of the module, students should have developed highly specialised and advanced technical, professional and academic skills that enable them to:
  • formulate and solve problems in asset allocation and portfolio management;
  • develop trading strategies and use appropriate models to evaluate performance.
  • Ability to apply arbitrage-free pricing theory to models of financial markets formulated either in discrete or continuous time.
  • Ability to derive mathematical properties of stochastic models for financial systems formulated via stochastic differential equations, using the methods of stochastic calculus.
  • Ability to select and apply appropriate probabilistic reasoning to developing pricing theories for models in appropriate financial markets, using the concepts of arbitrage, risk-neutral measures, and portfolio theory.
  • Ability to apply appropriate analytical, Monte Carlo, or numerical methods to price financial products.
Key Skills:
  • Students will have basic mathematical skills in the following areas: problem solving, modelling, computation.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

  • Lectures demonstrate what is required to be learned and give a thorough justification of the theoretical developments, with appropriate examples.
  • Problems classes will demonstrate the application of the theory to extended examples.
  • Computer practicals will, through guided worksheets, enable the students to apply some of the computational methods developed in the course to concrete problems.
  • Take-home examinations will assess students' ability to develop mathematically sound arguments in the context of financial models, to apply probabilistic reasoning and methods to analyse financial products, and to employ a variety of tools to correctly price financial products.

Teaching Methods and Learning Hours

Activity Number Frequency Duration Total/Hours
Lectures 46 3 per week, weeks 1-4, 6-9 (term 1) and 11-14, 16-17 (term 2); 2 per week, weeks 18-19 (term 2) 1 hour 46
Problem Classes 14 1 per week, weeks 1-4, 6-9 (term 1) and 11-14, 16-17 (term 2) 1 hour 14
Computer Practicals 4 2 per week, weeks 18-19 (term 2) 1 hour 4
Preparation and Reading 236
Total 300

Summative Assessment

Component: Take-home Examination Component Weighting: 80%
Element Length / duration Element Weighting Resit Opportunity
Take-home examination 48 hours 50%
Take-home examination 48 hours 50%
Component: Continuous Assessment Component Weighting: 20%
Element Length / duration Element Weighting Resit Opportunity
Written assignments to be assessed and returned 100%

Formative Assessment:


Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University