Financial Mathematics

Description

The course consists of two components.

The first component concentrates on topics in time series econometrics, which may include
• univariate nonlinear stochastic models; stationary nonlinear models; nonlinear time series models; regression analysis of time series;
• discrete and continuous time series with long-range dependence. Applications include risk management and term structure dynamics, nonlinear estimation of interest rate models and nonparametric pricing
• derivatives, selection of time series models for detecting climate change, and trend detection in regional mean temperature series.

Regarding the second, topics will be selected from:
• Discrete time models I (Single period models, pricing a European option, characterizing no arbitrage, risk neutral probabilities);
• Discrete time models II (Multiperiod binary models, discrete parameter martingales, risk-neutral pricing, Cox-Ross-Rubinstein);
• Brownian motion (Definition and Lévy's construction); The reflection principle and hitting times (reflection principle, hitting times, scaling properties);
• Martingales in continuous time (filtrations, adapted processes, Optional Sampling Theorem);
• Stochastic integration and Ito's formula; The Martingale Representation Theorem, Lévy's characterisation of Brownian motion, Girsanov's Theorem;
• The Black-Scholes model; Pricing and hedging European options; Evaluation of price and hedging strategies for European calls and puts.