Quantitative Finance
Prerequisiti
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.
Programma
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.
Obiettivi formativi
The student will have the familiarity with the elements of the stochastic calculus and with the main models describing the random evolution of the financial prices. He/She will be able to compute the price of derivative options and to discuss the assumptions of the different modelling choices. The student will be able to use stochastic calculus tools to model financial assets, derivatives, and portfolios.
Riferimenti bibliografici
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.
Moduli
| Modulo | Ore | CFU | Docenti |
|---|---|---|---|
| Quantitative Finance | 50 | 7 | Giacomo Bormetti, Fabrizio Lillo, Giorgio Rizzini |
| Supplementary Teaching | 15 | 0 | Giorgio Rizzini |