Scientific program and abstracts
Monday 16 December 2024, room 2BC30
14:30-15:10 Fabio Antonelli: Evaluation in finance and insurance for random time contracts under stochastic volatility
15:10-15:50 Marco Mastrogiovanni: A model for forward prices in the electricity markets with fractional dynamics
15:10-15:50 Coffee Break
15:50-16:30 Giorgia Callegaro: From elephant to goldfish (and back): memory in stochastic Volterra processes
16:30-17:10 Beatrice Ongarato: Optimal Cyber-Security Investment in a Dynamic Version of the Gordon-Loeb Model
Tuesday December 2024, room 7B1
9:30-10:10 Yuliya Mishura: Gaussian processes with Volterra kernels
10:10-10:50: Giacomo Ascione: TBA
10:50-11:30: Tiziano Vargiolu: A model for insolation and temperature with (fractional) CARMA models
ABSTRACTS:
Fabio Antonelli: Evaluation in finance and insurance for random time contracts under stochastic volatility.
Due to the recurrent credit crises, the interest in properly evaluating financial derivatives by taking into account default risks (and also risks of other kinds) has grown immensely, generating a thorough theory about Value Adjustments, initially studied in the seminal form of the Credit Value Adjustment (CVA) used to “adjust” portfolio values when investors may default. The default event is usually represented by the occurrence of a default time, not necessarily measurable with respect to the market filtration, as it often encases some independent factors. A very efficient methodology to deal with the problem was the intensity approach (used for instance by Brigo et al., Antonelli et al.), when the default time exhibits a stochastic intensity λ. When the market is represented by a stochastic (either rough or not) volatility model, the increase in the dimensionality of the problem and correlations among the several sources of randomness make this evaluation extremely difficult. Mutatis mutandis, the same mathematical setting applies when considering life insurance products such as the Guaranteed Minimum Maturity Benefits (GMAB), linked to some investment fund and affected by the insured’s random time of death. By exploiting Malliavin Calculus techniques (introduced by Alòs and Ewald), we develop representation formulas that allow manageable and efficient analytical approximations, when specializing to a Markovian setting and affinity for the intensities. When restricting to rough volatilities, we compare the numerical efficiency of our method with Monte Carlo simulations for different values of the model parameters and we analyze the formula sensitivity with respect to the Hurst parameter. Based on joint works with Elisa Alòs (University Pompeu Fabra), Alessandro Ramponi and Sergio Scarlatti (University of Rome Tor Vergata).
Marco Mastrogiovanni: A model for forward prices in the electricity markets with fractional dynamics
In the last decade, a new generation of stochastic models, known as rough models, has emerged in finance to describe stochastic volatility. We aim to apply this class of models to capture same characteristics of electricity markets. Empirical observations indicate that energy spot prices exhibit autocorrelation and non-Markovian behavior, which may be modeled using fractional dynamics. It is important to note that spot energy is not a tradeable asset due to limited storage possibilities and significant seasonality. Instead, the primary tradeable instruments in electricity markets are forward contracts, which are defined by averaging over a delivery period rather than being associated with a single maturity. To value these derivatives, we adopt the no-arbitrage Heath-Jarrow-Morton (HJM) framework, which ensures the generation of suitable martingales. Furthermore, it is essential to incorporate the mean-reverting behavior of forward prices for more accurate modeling. Considering these requirements, we propose a modification of the well-known Lucia-Schwartz model, where the spot price dynamics are driven by factors, X1 and X2, hidden in the forward price process. The first factor, X1, is stationary and mean-reverting, capturing short-term fluctuations in the price curve. In contrast, X2 represents long-term price behavior and is modeled using a combination of fractional martingales. Unlike the classical model, our approach does not fully account for the non-Markovian nature of spot prices. However, it preserves the autoregressive properties of forward prices and reveals the influence of the Hurst parameter on these prices. Within this framework, we derive a closed-form expression for forward prices and provide pricing formulas for call and put options on forward contracts.
Giorgia Callegaro: From elephant to goldfish (and back): memory in stochastic Volterra processes
We propose a new theoretical framework that exploits convolution kernels to transform a Volterra path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. This transformation is achieved by embedding a Markovian "memory process" within the dynamics of the non-Markovian process. We discuss existence and path-wise regularity of solutions for the stochastic Volterra equations introduced and we provide a financial application to volatility modelling. We also propose a numerical scheme for simulating the processes. The numerical scheme exhibits a strong convergence rate of 1/2, which is independent of the roughness parameter of the volatility process. This is a significant improvement compared to Euler schemes used in similar models.
Beatrice Ongarato: Optimal Cyber-Security Investment in a Dynamic Version of the Gordon-Loeb Model
In this work, we aim at determining the optimal cyber-security investment strategy for an entity (e.g. a corporation or government) subject to cyberattacks. Inspired by the Gordon-Loeb model, we assume that the success rate of cyber-attacks depends on the vulnerability of the entity’s security system which can be reduced investing in security measures. We develop a dynamic version of the Gordon-Loeb model, incorporating the attack dynamics using Hawkes processes. The (Markovian) problem is framed as a 2-dimensional stochastic control problem with jumps and is addressed using dynamic programming techniques. The optimal value is characterized by a partial-integro differential equation (PIDE) that we solve numerically. As a possible extension, we aim at investigating a non-Markovian Hawkes setting processes (i.e., with non-exponential kernels). This is a joint work with Giorgia Callegaro, Claudio
Fontana, Caroline Hillairet.
Yuliya Mishura: Gaussian processes with Volterra kernels
We consider the class of Gaussian processes admitting the integral representation via generating kernel and a Wiener process. Such processes arise e.g. in finance or in the modelling of functioning of technical devices if we want to insert memory in the model. They are the natural extension of fractional Brownian motion (fBm) which admits the integral representation via the Wiener process. We study some properties of such Gaussian processes that turned out to be directly related to the analytical properties of the generating kernel.
Giacomo Ascione: TBA
Tiziano Vargiolu: TBA