Examination procedure
Oral exam and seminars
Prerequisites
Basic Notion of Probability
Syllabus
-- Introduction to portfolio optimization. Utility functions, Optimal portfolios, Consumption-Investment problems, Mean-variance portfolio problems
-- Stochastic Models for financial markets. Binomial models. Brownian Motion. Martingale. Stochastic Calculus, Itô's Formula. Levy processes and jump processes. Stochastic Calculus with jump processes. Stochastic Differential Equations (SDE). Kolmogorov's Equations. Feynman-Kac's theorem.
-- Evaluation of Options. Models of Cox-Ross-Rubinstein and of Black-Scholes. Risk Neutral evaluation (European Options, American Options, Exotic Options). Dynamic evaluations. Market premium and change of numeraire. Affine processes in continuous time and valuation formulae. Models of Merton and Bates.
-- Volatility. Volatility surfaces. Extensions of the Black and Scholes Formula and local volatility models. Stochastic Volatility models in continuous time. Rough Volatility models. Estimation of volatility. Stable convergence and infill asymptotic. Realized Measures of Volatility: Asymptotic properties.
-- Numerical methods for the estimation of models. Maximum Likelihood Methods: Estimation of coefficients of SDE.
Bibliographical references
Notes given by the Prof.s