Numerical Methods for Finance and Risk Management

Sigma algebra and filtration
Convergences
Introduction to continuous time stochastic processes
Simple and quadratic variations
Wiener process
Stochastic differential equations
Ito stochastic integral and Ito formula
Martingales
Principles of options pricing and the Black&Scholes market
Contingent T-Claims and derivation of the general Black&Scholes pricing formula
Derivation of explicit formula for European call/put options under the Black&Scholes market
Martingale measures and their relations to pricing
Fundamental theorem of option pricing
Simulation of stochastic differential equations
Monte Carlo approach to option pricing
Variance-reduction techniques
Parameter estimation from discretely observed stochastic differential equations
Historical and implied volatility
Monitoring volatility though change point analysis
Explorative data analysis: clustering and lead-lag estimation
Quasi MLE, AIC and Lasso model selection
Introduction to Lévy processes: properties, simulation and parametric estimation
Estimated (MC) expected payoffs under different physical measures
Pricing for the multidimensional Black&Scholes model

Prerequisites.

Overview of Basel 2, Basel 3 and Solvency 2. Basic Concept in Risk Management: Risk Measures (VaR and ES).

Light tailed versus Heavy tailed distributions. Regularly varying distributions, EVT: the POT method.

Modeling dependence with copulas.

Multivariate Modelling: ''if Only the World Were Elliptical'' - Coherent Measures of Risk .

Standard methods for Market Risk .

Risk Aggregation and Model Uncertainty.

Operational Risk: some case studies.