Module MATH52230: Financial Mathematics

Department: Mathematical Sciences

MATH52230: Financial Mathematics

Prerequisites

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

Corequisites

• PHYS51915 Core Ia: Introduction to Machine Learning and Statistics; PHYS52015 Core Ib: Introduction to Scientific and High-Performance Computing

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

• 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.

• 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.

• 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

Summative Assessment

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