Period of duration of course
Basic Notion of Probability
-- 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.
The student will have the familiarity with the elements of the stochastic calculus and with the main models describing the random evolution of the financialprices. He/She will be able to compute the price of derivative options and to discuss the assumptions of the different modelling choices. He/She will be able to estimate volatility, implied volatility and risk premia.
Notes given by the Prof.s