97438 - Stochastic Models for Finance

Learning outcomes

The course aims at providing students with an introduction to affine modeling in finance. Thanks to their analytical tractability, affine models are largely used to parametrically describe the evolution of financial time series (returns, volatilities, jumps) and model the dynamics of interest rate term structures, survival probabilities, and, more recently, contagious markets. Affine modeling in continuous time represents a powerful tool to price and hedge options, interest rate derivatives, credit instruments and to capture the spreading of financial distress. In recent years, an affine approach in discrete time has been proven to be extremely successful to link time series econometrics (e.g. GARCH and Gamma (Realised) volatility modeling) to asset pricing. At the end of the course, the student is familiar with a flexible approach to modeling in finance and economics and with several applications of relevant interest for both regulators and the financial industry.

Course contents

1. Some preliminary notions in probability and finance
2. Discrete-time, discrete-space models

   2.1 The random walk process

   2.2 Time-limit distributions: Poisson and Gauss

   1. Monte Carlo simulations
   2. Stock markets and the random walk
   3. Markov chains and credit ratings
   4. Markov chains and pandemics
   5. Binomial trees and option pricing
3. Discrete-time, continuous-space models

   3.1 Asset returns and portfolio allocation

   • Markowitz (1958) model
   • Tobin (1958) demand for money model
   • Sharpe (1964) CAPM model
   • Ross (1976) APT model
   • Engle (1982) ARCH model and extensions
4. Continuous-time, continuous-space models

   4.1 The Wiener process

   4.2 The Wiener process in discrete time

   4.3 Monte Carlo simulations

   4.4 Option pricing

   • Black and Scholes model (1973)
     • Call, Put
     • Pricing
     • Hedging
   • Princing insurance and guarantees
   • Margrabe (1978) model
   • Heston (1993) stochastic volatility model
   • Extensions

   4.5 The term structure of default-free bonds

   • Vasicek (1977) model
   • Cox-Ingersoll-Ross (1985) model
   • Hull and White (1990) model
   • Affine models
   • Brace-Gatarek-Musiela (1997) Libor Market Model
   • Extensions

   4.6 The default structure of interest rates

   • Merton (1974) model
   • Extensions

   4.7 Copula models

   • Vasicek (1987)
   • Li (2000)
   • Extensions
5. Continuous-time, discrete-space models

1. The Poisson process

5.2 The Poisson process in discrete time

5.3 Monte Carlo simulations

5.4 Lundberg-Cramer (1930) insurer’s ruin model

5.5 Bonus/malus systems in non-life insurance

5.6 Jumps in the stock market

5.7 Modelling insurance of extreme events

Readings/Bibliography

X. Sheldon Lin, Introductory Stochastic Analysis for Finance and Insurance, Wiley, 2006

Teaching methods

classroom lessons

Assessment methods

written exam

Teaching tools

exercises

Office hours

See the website of Riccardo Cesari