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
15:50-16:10 Coffee Break
16:10-16:50 Beatrice Ongarato: Semi-static variance-optimal hedging with self-exciting jumps.
16:50-17:30 Giorgia Callegaro: From elephant to goldfish (and back): memory in stochastic Volterra processes
Tuesday December 2024, room 7B1
9:30-10:10 Yuliya Mishura: Gaussian processes with Volterra kernels
10:10-10:50 Tiziano Vargiolu: A model for insolation and temperature with (fractional) CARMA models
10:50-11:30 Giacomo Ascione: Adding noise to time-nonlocal Gompertz 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.
Beatrice Ongarato: Semi-static variance-optimal hedging with self-exciting jumps
The aim of this work is to study a hedging problem in an incomplete market model where the underlying log-asset price is driven by an affine diffusion process with self-exciting jumps of Hawkes type. More precisely, we aim at hedging a claim at time T>0, using a basket of available contingent claims, to minimize the variance of the residual hedging error at time T. In order to improve the replication of the claim, we look for a hybrid hedging strategy of semi-static type: one part has to be dynamic (i.e., continuously rebalanced) and another one will be static (i.e., buy-and-hold). We discuss in detail a specific example in which the approach proposed is applied, i.e. a variance swap hedged by means of European options. As a possible extension, we aim at investigating a semi-static variance-optimal hedging problem in non-Markovian framework, e.g. with a model driven by Volterra processes. This is a joint work with Giorgia Callegaro, Paolo Di Tella and Carlo Sgarra
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.
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.
Tiziano Vargiolu: A model for insolation and temperature with (fractional) CARMA models
CARMA (Continuous Auto-Regressive Moving Average) models are a particolar example of multidimensional Ornstein-Uhlenbeck processes, and they have been proposed as a version of ARMA models (based on discrete times) in continuous time. Among other applications, they have been used for the modelling of temperature. In these models, the leading process represent the temperature itself; however, this representation does not give deeper meaning to the other components, other than the derivatives of the leading process.
In this talk, we try to fill this gap by representing the two-dimensional process given by temperature and insolation for a given place: intuitively, temperature is influenced by insolation, and this can be incorporated in a 2-dimensional Ornstein-Uhlenbeck process which in principle generalizes the CARMA model mentioned above, also giving an explicit meaning to the component(s) other than temperature in the process. We will also present possible extension of this model in the direction of fractional/Volterra processes.
Giacomo Ascione: Adding noise to time-nonlocal Gompertz models
Population dynamics models have been a fertile field for mathematics for more than two centuries. Lately, these models, together with a growing interest in fractional calculus, lead to the development of fractional (or more general time-nonlocal) growth laws. This is the case, for instance, of [1], in which the results of some dark fermentation and photofermentation experiments have been successfully modelled by means of a fractional Gompertz law. It is clear, however, that one needs to consider random fluctuations in the model to better describe the behaviour of such phenomena. However, the simple addition of a white noise is not sufficient, since this would lead to possibly negative values of the population density. Hence one needs to find a coherent way to introduce noise in such models. To do this, arguing exactly as in [1], one could decompose the fractional Gompertz model into a system of two possibly time-nonlocal (non-autonomous) linear equations describing separately the population density and the growth rate and then only perturb the rate with a suitable process. In this talk, we will discuss this construction in the framework of the generalized fractional calculus, as done in [2], and we will study the relation with the improved fractional Gompertz law introduced in [1]. An important class of interesting noise processes include a quite large family of Volterra processes, such as fractional Brownian motions or their generalization as in [3], together with the Liouville-fractional Brownian motion and its generalizations, leading to fractional SDEs as in [4], for which further regularity properties of the generalized fractional stochastic Gompertz law can be explored.
References:
[1] Frunzo, L., Garra, R., Giusti, A., & Luongo, V. (2019). Modeling biological systems with an improved fractional Gompertz law. Communications in Nonlinear Science and Numerical Simulation, 74, 260-267.
[2] Ascione, G., & Pirozzi, E. (2021). Generalized fractional calculus for gompertz-type models. Mathematics, 9(17), 2140.
[3] Beghin, L., Cristofaro, L., & Mishura, Y. (2024). A class of processes defined in the white noise space through generalized fractional operators. Stochastic Processes and their Applications, 178, 104494.
[4] Sakthivel, R., Revathi, P., & Ren, Y. (2013). Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Analysis: Theory, Methods & Applications, 81, 70-86.