The forward-path method for pricing multi-asset American-style options under general diffusion processes

The heat conduction equation is a combination of first-order and second-order differential equations. Solving first-order differential equations is necessary to examine temperature as a function of time. Meanwhile, solving second-order differential equations is needed to examine temperature as a function of space. The heat flux equation is based on Fourier's law, which shows that temperature is a function of time and space. Understanding heat conduction can be improved by building a heat propagation model on the Solar ERK dryer plate. Analysis of heat propagation on the drying plate used the Finite Difference Approach (FDA) method with explicit and implicit schemes. With an explicit scheme, the FDA method calculates the temperature (T) at a point on the spatial derivative term, when T is at time t, while the implicit scheme calculates T at a point on the space derivative term when T is at time t+Δt. Heat propagation at each time change was analyzed by developing a program using the MATLAB 17 application. The results of the analysis show that there are differences in heat propagation between the explicit and implicit schemes. The convergence and stability of calculations in explicit schemes are unstable, causing problems at the time step. Meanwhile, the implicit scheme is carried out simultaneously on all nodes so that convergence and stability are easily maintained, and there are no time-step limitations.

This paper develops a numerical procedure to accelerate the convergence of the Favre-averaged non-linear harmonic (FNLH) method. The scheme provides a unified mathematical framework for solving the sparse linear systems formed by the mean flow and the time-linearized harmonic flows of FNLH in an explicit or implicit fashion. The approach explores the similarity of the sparse linear systems of FNLH and leads to a memory-efficient procedure, so that its memory consumption does not depend on the number of harmonics to compute. The proposed method has been implemented in the industrial computational fluid dynamics solver Hydra. Three test cases are used to conduct a comparative study of explicit and implicit schemes in terms of convergence, computational efficiency, and memory consumption. Comparisons show that the implicit scheme yields better convergence than the explicit scheme and is also roughly 7–10 times more computationally efficient than the explicit scheme with four levels of multigrid. Furthermore, the implicit scheme consumes only approximately 50% of the memory required by the explicit scheme with four levels of multigrid. Compared with the full-annulus unsteady Reynolds-averaged Navier–Stokes simulations, the implicit scheme produces comparable results to URANS with computational time and memory consumption that are two orders of magnitude smaller.

The locally implicit leapfrog scheme has been proposed as one of the signal/power integrity (SI/PI) circuit simulation techniques of an arbitrary shaped power distribution network (PDN) modeled by triangular meshes. This method can perform transient analysis several times faster than the explicit leapfrog scheme by employing a locally implicit scheme instead of the globally explicit leapfrog scheme. In this paper, we describe the nonlinear circuit simulation technique of the locally implicit leapfrog scheme for fast simulation of the PDN modeled by triangle meshes. First, the modeling method by using the triangular meshes and the formulation of the locally implicit leapfrog scheme. Next, we describe the nonlinear circuit simulation technique of the locally implicit leapfrog scheme. Finally, some numerical results are shown, and it is confirmed that the proposed technique is useful and efficient for the fast simulation of the PDN including nonlinear circuit modeled by triangle meshes.

A numerical solution for a coupled hyperbolic thermoelasticity problem of the Lord‐Shulman type is presented. An infinite layer with free boundaries that are kept at a constant temperature is irradiated by a short laser impulse. In this case the problem can be studied in the one‐dimensional formulation. An impact of the laser impulse on a medium is modeled by an internal heat source. The laser impulse intensity decays with a distance from an irradiated surface according to the Bouguer law. The layer is in an unperturbed state at the initial moment of time. The material properties of copper are taken to be the thermo‐mechanical properties of the layer.Two approaches for the boundary value problem are considered: the finite‐difference method (FDM) and the finite‐volume method (FVM). In the first case the system of two second‐order PDE is investigated, in the second case a system of integral balance equations and an integral thermal conductivity equation are composed. To integrate these equations both the explicit and implicit schemes are used. The obtained results are compared with each other and with an analytical solution. A numerical error is estimated for different time and space step sizes. The features inherent in various ways of integration of the coupled hyperbolic thermoelastic problem with the thermal excitation are found.

A modified Euler-Maruyama (EM) scheme was constructed for a class of multi-term fractional nonlinear stochastic differential equations with weak singularity kernels, and the strong convergence of this modified EM scheme was proved. Specifically, according to the sufficient condition for stochastic integral decomposition, the multi-term fractional stochastic differential equation was equivalently transformed into the stochastic Volterra integral equation, and then the corresponding modified EM scheme and its strong convergence were derived and proved, respectively. The order of strong convergence is α_m-α_m-1, where α_i is the index of fractional derivative satisfying 0<α₁<…<α_m-1<α_m<1. Finally, numerical experiments verify the correctness of the theoretical results.

We present a quasi-analytical method for pricing multi-dimensional American options based on interpolating two arbitrage bounds, along the lines of Johnson in J Financ Quant Anal 18(1):141–148 (1983). Our method allows for the close examination of the interpolation parameter on a rigorous theoretical footing instead of empirical regression. The method can be adapted to general diffusion processes as long as quick and accurate pricing methods exist for the corresponding European and perpetual American options. The American option price is shown to be approximately equal to an interpolation of two European option prices with the interpolation weight proportional to a perpetual American option. In the Black-Scholes model, our method achieves the same efficiency as the quadratic approximation of Barone-Adesi and Whaley in J Financ 42:301–320 (1987), with our method being generally more accurate for out-of-the-money and long-maturity options. When applied to Heston’s stochastic volatility model, our method is shown to be extremely efficient and fairly accurate.

Event-driven sorting algorithm-based Monte Carlo method with neighbour merging method for solving aerosol dynamics

In this paper, a multiscale approach with partially explicit time discretization is proposed. The idea is to use a partially explicit time scheme, considering a filtration problem in a fractured medium, where the implicit scheme is used for nodes whose subdomains contain fractures, and the explicit scheme is used for all others. In this way, it is possible to use a time step that is independent of the diffusion coefficient for fractures. Numerical results demonstrating high accuracy of calculations are presented.

A Closed-Form Solution to American Options under General Diffusions

Split-step forward methods for stochastic differential equations

This work presents the development of an explicit/implicit particle finite element method (PFEM) for the 2D modeling of metal cutting processes. The purpose is to study the efficiency of implicit and explicit time integration schemes in terms of precision, accuracy and computing time. The formulation for implicit and explicit time marching schemes is developed, and a detailed study on the explicit solution steps is presented. The PFEM remeshing procedures for insertion and removal of particles have been improved to model the multiple scales of time and/or space of the solution. The detection and treatment of the rigid tool contact are presented for both, implicit and explicit schemes. The performance of explicit/implicit integration is studied with a set of different two-dimensional orthogonal cutting tests of AISI 4340 steel at cutting speeds ranging from 1 m/s up to 30 m/s. It was shown that if the correct selection of the time integration scheme is made, the computing time can decrease up to 40 times. It allows us to affirm that the computing time of the PFEM simulations can be excessive due to the used time marching scheme independently of the meshing process. As a practical result, a set of recommendations to select the time integration schemes for a given cutting speed are given. This is intended to minimize one of the negative constraints pointed out by the industry when using metal cutting simulators.

Positivity preserving schemes for the fractional Klein-Kramers equation with boundaries

This study compares the efficiency of 3-D transient electromagnetic forward modeling schemes on the multi-resolution grid for various modeling scenarios. We developed time-domain finite-difference modeling based on the explicit scheme earlier. In this work, we additionally implement 3-D transient electromagnetic forward modeling using the backward Euler implicit scheme. The iterative solver is used for solving the system of equations and requires a proper initial guess that has significant effect on the convergence. The standard approach usually employs the solution of a previous time step as an initial guess, which might be too conservative. Instead, we test various initial guesses based on the linear extrapolation or linear combination of the solutions from several previous steps. We build up the implicit scheme forward modeling on the multi-resolution grid, which allows for the adjustment of the horizontal resolution with depth, hence improving the performance of the forward operator. Synthetic examples show the implicit scheme forward modeling using the linearly combined initial guess estimate on the multi-resolution grid additionally reduces the run time compared to the standard initial guess approach. The result of comparison between the implicit scheme developed here with the previously developed explicit scheme shows that the explicit scheme modeling is more efficient for more conductive background models often found in environmental studies. However, the implicit scheme modeling is more suitable for the simulation with highly resistive background models, usually occurring in mineral exploration scenarios. Thus, the inverse problem can be solved using more efficient forward solution depending on the modeling setup and background resistivity.

Many fluctuation phenomena, in physics and other fields, can be modeled byFokker–Planck or stochastic differential equations whose coefficients, associated with driftand diffusion components, may be estimated directly from the observed time series. Itscorrect characterization is crucial to determine the system quantifiers. However, due to thefinite sampling rates of real data, the empirical estimates may significantly differ from theirtrue functional forms. In the literature, low-order corrections, or even no corrections, havebeen applied to the finite-time estimates. A frequent outcome consists of lineardrift and quadratic diffusion coefficients. For this case, exact corrections havebeen recently found, from Itô–Taylor expansions. Nevertheless, model validationconstitutes a necessary step before determining and applying the appropriatecorrections. Here, we exploit the consequences of the exact theoretical results obtainedfor the linear–quadratic model. In particular, we discuss whether the observedfinite-time estimates are actually a manifestation of that model. The relevanceof this analysis is put into evidence by its application to two contrasting realdata examples in which finite-time linear drift and quadratic diffusion coefficientsare observed. In one case the linear–quadratic model is readily rejected while inthe other, although the model constitutes a very good approximation, low-ordercorrections are inappropriate. These examples give warning signs about the properinterpretation of finite-time analysis even in more general diffusion processes.

An increased cooperation between universities and industries is seen by many politicians as an ultimate remedy against economic under-development by fostering regional growth in a world in which the know-how content of products is ever increasing. Furthermore, this know-how is increasingly separable from the goods and, thus, can be developed into an Intellectual Property Right (IPR). As we well know, technological development was historically often linked to the machinery in which this specific technology was used, and enterprises tried to keep implicit knowledge within their firms, protecting it against diffusion into economic space. The more that know-how became public, for instance through the development within publicly financed research units, such as universities, the more a general diffusion process became possible. It was in the nineteenth century that nation-states started to found their universities of technology, their royal research institutes, etc. in order to promote technological development. The necessity to protect generally accessible know-how that was privately developed became of strategic importance and, thus, international regulations on trade marks, patents, their licensing, etc. developed.