The webinars will take place on Zoom and a link to the virtual room will be sent out to all those who registered at the registration page.
Speaker at 13:00: Emanuela Rosazza Gianin (University of Milano Bicocca)
Title: Are Shortfall Systemic Risk Measures One Dimensional?
Abstract: In this talk we show that shortfall systemic (multivariate) risk measures \(\rho\)
defined through an N-dimensional multivariate utility function 
\(U\) and random allocations can be represented as classical (one dimensional) shortfall risk measures associated to an explicitly determined 1-dimensional function constructed from \(U\). The study of several properties of shortfall systemic (multivariate) risk measures 
\(\rho\), such as law invariance and a Law of Large Numbers-type result, is then simplified by applying the findings above.
Joint work with Alessandro Doldi and Marco Frittelli
Speaker at 14:00: Massimiliano Moda (University of Antwerp)
Title: A finite-difference approach for European option pricing under the tempered stable process
Abstract: In this seminar, a numerical approach for approximating the value of European options will be presented, assuming that the dynamics of the underlying asset price is described by a tempered stable process. Following Küchler & Tappe (2013), such processes are Lévy processes with infinite activity and, therefore, give rise to an integral (singular) term in the differential equation that holds for the option value function. Note that the most famous processes Variance Gamma and CGMY are special cases of this. Using the method of lines for solving the PIDE (partial integro-differential equation), the numerical scheme is divided into two general steps: spatial discretization, in which the (spatial) integral-differential operator is replaced with a finite-difference version, converting the PIDE to a system of linear ODEs, and next temporal discretization, in which this system is numerically solved by a suitable time-stepping method.
A special feature of the numerical approach under consideration, that is strongly based on the work of Wang et al. (2007) and Cont & Voltchkova (2005), is the fact that jumps smaller than a given threshold are replaced by a diffusive term. This makes it possible to remove the singularity from the integral term, and hence, to resort in principle to well-known numerical schemes for finite activity Lévy processes. A subsequent challenge, however, is that the PIDE is convection-dominated and therefore a correction based on the method of characteristics, known as the semi-Lagrangian technique, is proposed. Finally, the integral term in the PIDE corresponds to a dense matrix in the system of linear ODEs. To avoid the inversion of this matrix, an IMEX (implicit-explicit) time-stepping method is proposed, which allows for an efficient evaluation.
Speaker at 14:30: Mustapha Regragui (University of Gent)
Title: Numerical valuation of swing options: discrete and continuous exercise rights
Abstract: Swing options are widely used derivative contracts in the energy markets, especially the electricity market. They give the holder the right to, dynamically, buy electricity at a predetermined, fixed price and hence reducing exposure to strong price fluctuations.
In this talk, we study the valuation of swing options under volume constraint when the electricity price is driven by an Ornstein-Uhlenbeck process with a finite activity jump. We will focus on two types of swing options: those with discrete, fixed-time exercise rights and those with continuous-time exercise rights. The former will lead us to the study of multiple parabolic partial integro-differential equations[1] and the latter to a Hamilton-Jacobi-Bellman equation [2,3]. We will employ finite difference methods to solve them numerically with adequate boundary conditions and special attention for the treatment of the integral term.
- Kjaer, M., Pricing of swing options in a mean reverting model with jumps, Applied Mathematical Finance 15(5-6), p. 479-502 (2008).
- Benth, F. E., Lempa, J. and Nilssen, T. K., On the optimal exercise of swing options in electricity markets, The Journal of Energy Markets 4(4), p. 3-28 (2012).
- Eriksson, M., Lempa, J. and Nilssen, T. K., Swing options in commodity markets: a multidimensional Lévy diffusion model, Mathematical Methods of Operations Research 79, p. 31-67 (2013).
This series of webinars addresses all interested people in probability, stochastic analysis, control, risk evaluation, statistics, with a view towards applications, in particular to renewable energy markets and production. This series brings together the major research themes of the projects STORM, SCROLLER, and SPATUS.