Course Unit Page


This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Quality education Decent work and economic growth

Academic Year 2019/2020

Learning outcomes

The course aims at providing students with an introduction to affine modelling 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 modelling 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 modelling) to asset pricing. At the end of the course, the student is familiar with a flexible approach to modelling in finance and economics and with several applications of relevant interest for both regulators and the financial industry.

Course contents

Introduction to securities markets: Model specifications, arbitrage and other economic considerations, risk neutral probability measures, valuation of contingent claims, complete and incomplete markets, risk and return, the binomial model, options, futures, and other derivatives.

Continuous-time models for option pricing: the Black-Scholes formula, stochastic volatility, the Heston model, jumps, Merton and Bates models. Jump-diffusion and affine pricing.

Affine models for interest rate derivatives: Zero coupon bonds, short and forward rates, term structures. Short rate models. The Heath-Jarrow-Morton forward rate framework and the Ritchen-Sankarasubramanian model.

Distressed markets: Pricing and hedging of derivatives in contagious markets. Modelling financial contagion using mutually exciting jump processes.

Econometric Asset Pricing: Financial volatility and jumps: non parametric realised measures. Observation driven and parameter driven models. Stochastic discount factors: absolute and relative asset pricing. Esscher transform. GARCH and Gamma models. Moment generating functions and recursive option pricing formulas.


Pliska, Stanley R. Introduction to mathematical finance. Oxford: Blackwell publishers, 1997.

Gatheral, Jim. The volatility surface: a practitioner's guide. Vol. 357. John Wiley & Sons, 2011.

Brigo, D., and F Mercurio. Interest rate models-theory and practice: with smile, inflation and credit. Springer Science & Business Media, 2007.

Duffie, D., Pan, J., & Singleton, K. (2000). Transform analysis and asset pricing for affine jump‐diffusions. Econometrica, 68(6), 1343-1376.

Ritchken, P. and Sankarasubramanian, L., Volatility structures of forward rates and the dynamics of the term structure. Math. Finance, 1995, 7, 157–176.

Kokholm, T. (2016). Pricing and hedging of derivatives in contagious markets. Journal of Banking & Finance, 66, 19-34.

Aït-Sahalia, Y., Cacho-Diaz, J., & Laeven, R. J. (2015). Modeling financial contagion using mutually exciting jump processes. Journal of Financial Economics, 117(3), 585-606.

Christoffersen, P., Jacobs, K., Ornthanalai, C., & Wang, Y. (2008). Option valuation with long-run and short-run volatility components. Journal of Financial Economics, 90(3), 272-297.

Corsi, F., Fusari, N., & La Vecchia, D. (2013). Realizing smiles: Options pricing with realized volatility. Journal of Financial Economics, 107(2), 284-304.

Teaching methods

Lectures at the blackboard.

Assessment methods

Final oral exam

Teaching tools

Classroom lectures.

Office hours

See the website of Giacomo Bormetti