Integrative teaching
Giorgio Rizzini
Examination procedure
Oral exam and seminar on a scientific paper.
Prerequisites
Basic notion of probability theory topics. The course is intended for student of the fourth and fifth year in mathematics and physics and for PhD students in mathematics, physics, and computer science.
Syllabus
Stochastic models for financial markets. Binomial models. Brownian motion. Martingales. Stochastic calculus, Itô's formula. Stochastic differential equations. Kolmogorov equations. Feynman-Kac theorem. Lévy processes. Jump models. Option pricing and hedging. Cox-Ross-Rubinstein and Black-Scholes models. Risk-neutral pricing (European, American, and exotic options). Dynamic hedging. Volatility. Volatility surfaces. Extension of the Black-Scholes formula to local volatility models. Continuous-time stochastic volatility models. Rough volatility models. Stochastic optimal control. Stochastic optimization problems. Solution methods: the PDE approach and the dynamic programming approach. Optimal switching and free-boundary problems. Applications in finance. Introduction to portfolio optimization. Utility functions. Optimal portfolios. Portfolio problems with mean-variance tradeoffs.
Bibliographical references
Notes and slides provided by the teachers
Pham, Huyên. Continuous-time stochastic control and optimization with financial applications. Vol. 61. Springer Science & Business Media, 2009.
Peskir, Goran, and Albert Shiryaev. Optimal stopping and free-boundary problems. Birkhäuser Basel, 2006.
Gatheral, Jim. The volatility surface: a practitioner's guide. John Wiley & Sons, 2011.
Bayer, Christian, et al., eds. Rough volatility. Society for Industrial and Applied Mathematics, 2023.