Meshfree Approximation for Multi-Asset Options

We price multi-asset options by solving their price partial differential equations using a meshfree approach with radial basis functions under jump-diffusion and geometric Brownian motion frameworks. In the geometric Brownian motion framework, we propose an effective technique that breaks the multi-dimensional problem to multiple 3D problems. We solve the price PDEs or PIDEs with an implicit meshfree scheme using thin-plate radial basis functions. Meshfree approach is very accurate, has high order of convergence and is easily scalable and adaptable to higher dimensions and different payoff profiles. We also obtain closed form approximations for the option Greeks. We test the model on American crack spread options traded on NYMEX.Multi-asset options, radial basis function, meshfree approximation, collocation, multidimensional Lévy process, basket options, PIDE, PDE

Fine-Scale Coral Connectivity Pathways in the Florida Reef Tract: Implications for Conservation and Restoration

Connectivity between coral reefs is critical to ensure their resilience and persistence against disturbances. It is driven by ocean currents, which often have very complex patterns within reef systems. Only biophysical models that simulate both the fine-scale details of ocean currents and the life-history traits of larvae transported by these currents can help to estimate connectivity in large reef systems. Here we use the unstructured-mesh coastal ocean model SLIM that locally achieves a spatial resolution of ~100 m, 10 times finer than existing models, over the entire Florida Reef Tract (FRT). It allows us to simulate larval dispersal between the ~1,000 reefs composing the FRT. By using different connectivity measures and clustering methods, we have identified two major connectivity pathways, one originating on the westernmost end of the outer shelf and the other originating on the inner shelf, North of the Lower Keys. We introduce new connectivity indicators, based on the PageRank algorithm, to show that protection efforts should be focused on the most upstream reefs of each pathway, while reefs best suited for restoration are more evenly spread between the Lower and Upper Keys. We identify one particular reef, North of Vaca Key, that is a major stepping stone in the connectivity network. Our results are the first reef-scale connectivity estimates for the entire FRT. Such fine-scale information can provide knowledge-based decision support to allocate conservation and restoration resources optimally

Challenges and Prospects in Ocean Circulation Models

We revisit the challenges and prospects for ocean circulation models following Griffies et al. (2010). Over the past decade, ocean circulation models evolved through improved understanding, numerics, spatial discretization, grid configurations, parameterizations, data assimilation, environmental monitoring, and process-level observations and modeling. Important large scale applications over the last decade are simulations of the Southern Ocean, the Meridional Overturning Circulation and its variability, and regional sea level change. Submesoscale variability is now routinely resolved in process models and permitted in a few global models, and submesoscale effects are parameterized in most global models. The scales where nonhydrostatic effects become important are beginning to be resolved in regional and process models. Coupling to sea ice, ice shelves, and high-resolution atmospheric models has stimulated new ideas and driven improvements in numerics. Observations have provided insight into turbulence and mixing around the globe and its consequences are assessed through perturbed physics models. Relatedly, parameterizations of the mixing and overturning processes in boundary layers and the ocean interior have improved. New diagnostics being used for evaluating models alongside present and novel observations are briefly referenced. The overall goal is summarizing new developments in ocean modeling, including: how new and existing observations can be used, what modeling challenges remain, and how simulations can be used to support observations.Peer reviewe

Challenges and Prospects in Ocean Circulation Models

We revisit the challenges and prospects for ocean circulation models following Griffies et al. (2010). Over the past decade, ocean circulation models evolved through improved understanding, numerics, spatial discretization, grid configurations, parameterizations, data assimilation, environmental monitoring, and process-level observations and modeling. Important large scale applications over the last decade are simulations of the Southern Ocean, the Meridional Overturning Circulation and its variability, and regional sea level change. Submesoscale variability is now routinely resolved in process models and permitted in a few global models, and submesoscale effects are parameterized in most global models. The scales where nonhydrostatic effects become important are beginning to be resolved in regional and process models. Coupling to sea ice, ice shelves, and high-resolution atmospheric models has stimulated new ideas and driven improvements in numerics. Observations have provided insight into turbulence and mixing around the globe and its consequences are assessed through perturbed physics models. Relatedly, parameterizations of the mixing and overturning processes in boundary layers and the ocean interior have improved. New diagnostics being used for evaluating models alongside present and novel observations are briefly referenced. The overall goal is summarizing new developments in ocean modeling, including: how new and existing observations can be used, what modeling challenges remain, and how simulations can be used to support observations

On the numerical solution of space–time fractional diffusion models

A flexible numerical scheme for the discretization of the space–time fractional diffusion equation is pre- sented. The model solution is discretized in time with a pseudo-spectral expansion of Mittag–Leffler functions. For the space discretization, the proposed scheme can accommodate either low-order finite- difference and finite-element discretizations or high-order pseudo-spectral discretizations. A number of examples of numerical solutions of the space–time fractional diffusion equation are presented with various combinations of the time and space derivatives. The proposed numerical scheme is shown to be both efficient and flexible

Front dynamics in fractional-order multi-species models: Applications in epidemiology and ecology

A number of recent studies have shown that mobility patterns for both humans and biological species can be quite complex and exhibit a scale-free dynamics, characteristic of Levy flights. Levy-flight patterns have been observed in the dispersion of bank notes, in human mobility patterns derived from mobile phone data as well as in the foraging patterns of a numbers of animal species. However, current reaction-diffusion models used to describe the spread of humans and other biological species do not account for the superdiffusive effect due to Levy flights. This could result in higher spreading speeds than predicted by classical models. We have considered two-species reaction-diffusion models driven by Levy flights. That family of models is based on the Lotka-Volterra equations and has been obtained by replacing the second-order diffusion operator by a fractional-order one. Depending on the parameter values, it can be used to represent the interaction between susceptible and infective populations in an epidemics model or the interactions between ecological species competing for the same resources. Theoretical developments and numerical simulations show that fractional-order diffusion leads to an exponential acceleration of the population fronts and a power-law decay of the fronts’ leading tail. Depending on the skewness of the fractional derivative, we can derive catch-up conditions for different types of fronts. Our results confirm that second-order reaction-diffusion models are not well-suited to simulate the spatial spread of modern epidemics and biological species that follow a Levy random walk as they are inclined to underestimate the propagation speeds at which they spread

SLIM: A multi scale model of the land-sea continuum

SLIM is an unstructured-mesh hydrodynamic model that can seamlessly simulate flows from the river to the coastal ocean. It relies on the Discontinuous Galerkin finite element method to achieve unprecedented accuracy, even for very complex coastlines and bathymetry. SLIM includes the following modules to model a range of different water environments: * SLIM1D for flows in branching river networks * SLIM2D for shallow barotropic flows with or without wetting and drying * SLIM3D for more complex barotropic or baroclinic flows where the vertical structure cannot be neglected * A Lagrangian particle tracker to simulate the transport of coral larvae, seagrass propagules or plastic debris * A Eulerian transport model to simulate the dynamics of tracers such as pollutants and sediments In this talk, I will give an overview of SLIM's features and capabilities, and present some applications in the context of marine ecosystems management and water quality