This module aims to introduce students to theadvanced mathematical techniques underlying financial markets and theassociated stochastic analysis.

(LO1) To provide an introduction to key financial derivatives and their properties

(LO2) To provide an understanding of the modelling processes used in financial derivatives e.g. binomial tree, Brownian motion

(LO3) To provide a general foundation to the stochastic theory behind financial derivatives.

(LO4) To provide a general methodology for the pricing and hedging of financial derivatives

(LO5) A critical awareness of issues in the field of financial derivatives

(S1) Problem solving skills

(S2) Numeracy

Measure Space, Probability Space, Measurable Function and Random Variables:   Introductory set theory, algebra and σ-algebra, σ-algebra generated by a collection of subsets, monotone class theorem, measurable space, measure and probability (definitions, properties), measure spaces, probability space.Measurable mappings; measurable functions, random variables, independence.   Integrals and Expectations:   Definitions;properties; Fatou lemma; monotone convergence theorem; dominant convergence theorems; Fubini theorem.   Conditional Expectations and Conditional Probabilities:   conditional expectations given random variables; conditional probability; properties of conditionalexpectations . Introduction to Derivatives:   Futures, forwards and options; no arbitrage; commodity forwards andfutures; swaps. Put-call parity. ExoticOptions: asian options; barrieroptions; compound options and exchange options. Binomial Option Pricing:   Aone-period binomial tree; two or more periods binomial model; valuation ofEuropean and American Options using the binomial model; early exercise; stockspaying discrete dividends. Brownian Motion and Ito’s Lemma:   Brownian motion, geometric Brownian motion, changing the numeraire,risk-neutral process, Sharpe ratio . Ito’s formula: statementand application in simple problems. The Black-Scholes Equation:   Black-Scholesequation, option pricing when the stock price can jump. Partial derivatives(Greeks) of an option price: first and second derivative, delta-hedging. Martingalesand derivative pricing model Arbitrage,complete market, introduction to martingales, risk-neutral pricing andequivalent martingale measure, price and hedge simple derivative contractsusing the martingale approach.