Basic probability theory
Stochastic models for financial markets. Binomial trees. Brownian motions. Martingales. Stochastic calculus and Itô formula. Lèvy and jump processes. Stochastic calculus for jump processes. Stochastic differential equations. Kolmogorov equations and Feynman-Kac formula.
Option pricing and hedging. Cox-Ross-Rubinstein and Black-Scholes models. Risk neutral valuation: European, American, and exotic option pricing. Dynamic hedging. Market premium and change of numeraire. Affine models in continuous time and valuation formulas. Merton and Bates models.
Volatility modeling. Volatility surfaces. Local volatility pricing models. Continuous-time stochastic volatility models. Non-parametric measures of volatility. Stable convergence and infill asymptotics: definitions and main results. Realized volatility measures: asymptotic properties.
Interest rate models. Forward rates, Zero Coupon Bonds. Term structures. Affine models. HJM models.
Discrete time processes. Price models with realized measures of volatility. Affine models under the real and risk-neutral measures: Esscher transform, no-arbitrage condition, Moment Generating Gunctions and recursive formulas for option pricing.
Numerical methods for parameter estimation.Maximum Likelihood: Estimation of stochastic differential equation coefficients. Generalized Method of Moments. Markov Chain Monte Carlo. Filtering and smoothing.
Educational goals: At the end of the course, the student is familiar with stochastic calculus and with models describing the random evolution of financial markets. The student is able to price and hedge financial derivatives and to critically assess the impact on price of the modelling assumptions. He/she knows how to measure and manage volatilities, implied volatilities, risk premia and interest rate term structures.
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Andrea Pascucci, PDE and Martingale Methods in Option Pricing, Bocconi and Springer Series, 2011.
Darrell Duffie, Jun Pan, and Kenneth Singleton, Transform Analysis and Asset Pricing for Affine Jump-Diffusions, Econometrica, Vol. 68, No. 6 (Nov., 2000), pp. 1343-1376
Mark Podolskij, and Mathias Vetter, Understanding limit theorems for semi martingales: a short survey, Statistica Neerlandica (2010) Vol. 64, nr. 3, pp. 329–351
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Yacine Aït-Sahalia, Maximum-likelihood estimation of discretely-sampled diffusions, Econometrica, Vol. 70, No. 1 (January, 2002), 223–262
Lars Peter Hansen, Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, Vol. 50, No. 4 (Jul., 1982), pp. 1029-1054