MATH52230: Financial Mathematics

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

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

• None

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

• 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.

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.

• 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.

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