This paper presents a multinomial method for option pricing when the underlying asset follows an exponential Variance Gamma (VG) process. The continuous time VG process is approximated by a continuous time process with the same first four cumulants and then discretized in time and space. This approach is particularly convenient for pricing American and Bermudan options, which can be exercised before the expiration date. Numerical computations of European and American options are presented and compared with results obtained with finite differences method and with the Black–Scholes prices.

Background: The cerebral microvasculature forms a dense network of interconnected blood vessels where flow is modulated partly by astrocytes. Increased neuronal activity stimulates astrocytes to release vasoactive substances at the endfeet, altering the diameters of connected vessels. Methods: Our study simulated the coupling between blood flow variations and vessel diameter changes driven by astrocytic activity in the rat somatosensory cortex. We developed a framework with three key components: coupling between the vasculature and synthesized astrocytic morphologies, a fluid dynamics model to compute flow in each vascular segment, and a stochastic process replicating the effect of astrocytic endfeet on vessel radii. Results: The model was validated against experimental flow values from the literature across cortical depths. We found that local vasodilation from astrocyte activity increased blood flow, especially in capillaries, exhibiting a layer-specific response in deeper cortical layers. Additionally, the highest blood flow variability occurred in capillaries, emphasizing their role in cerebral perfusion regulation. We discovered that astrocytic activity impacted blood flow dynamics in a localized, clustered manner, with most vascular segments influenced by two to three neighboring endfeet. Conclusions: These insights enhance our understanding of neurovascular coupling and guide future research on blood flow-related diseases.

The cerebral microvasculature forms a dense network of interconnected blood vessels where flow is modulated partly by astrocytes. Increased neuronal activity stimulates astrocytes to release vasoactive substances at the endfeet, altering the diameters of connected vessels. Our study simulated the coupling between blood flow variations and vessel diameter changes driven by astrocytic activity in the rat somatosensory cortex. We developed a framework with three key components: coupling between vasculature and synthesized astrocytic morphologies, a fluid dynamics model to compute flow in each vascular segment, and a stochastic process replicating the effect of astrocytic endfeet on vessel radii. The model was validated against experimental flow values from literature across cortical depths. We found that local vasodilation from astrocyte activity increased blood flow, especially in capillaries, exhibiting a layer-specific response in deeper cortical layers. Additionally, the highest blood flow variability occurred in capillaries, emphasizing their role in cerebral perfusion regulation. We discovered that astrocytic activity impacts blood flow dynamics in a localized, clustered manner, with most vascular segments influenced by two to three neighboring endfeet. These insights enhance our understanding of neurovascular coupling and guide future research on blood flow-related diseases.

This paper presents a multinomial method for option pricing when the underlying asset follows an exponential Variance Gamma process. The continuous time Variance Gamma process is approximated by a discrete time Markov chain with the same firsts four cumulants. This approach is particularly convenient for pricing American and Bermudan options, which can be exercised at any time up to expiration date. Numerical computations of European and American options are presented, and compared with results obtained with finite differences methods and with the Black Scholes model.

I present a complete numerical study of the implicit-explicit finite difference method used to solve partial integro-differential equations (PIDEs) arising in mathematical finance. The study considers the specific case of the Merton jump-diffusion model, which is used to compute prices of European call and put options. These PIDE prices are then compared with prices obtained by closed formula, Fourier inversion, and Monte Carlo methods.