Pricing an Arithmetic Average Reset Option Using the Green Function Method

Numerical solving of the generalized Black-Scholes differential equation using Laguerre neural network

Propagation of a short pulse through human breast tissues is studied by numerically solving the diffuse equation. Different numerical methods, such as Monte Carlo and finite difference time domain, have been used to study the short pulse laser propagation through biological tissues. In this paper, we use boundary integral method to study the laser pulse–tissue interaction. The diffuse equation is used to examine the propagation of laser into biological tissues, and boundary integral method is used to alter this equation to the integral form and the result is solved by using boundary element method. To verify the precision of a boundary element method code, we compared the obtained results with those obtained by finite difference time domain method. In addition, the effects of different optical parameters of breast tissues, i.e., reduced scattering and absorption coefficients, on time evolution of a diffusely reflected pulse are studied.

Option contracts can be valued by using the Black-Scholes equation, a partial differential equation with initial conditions. An exact solution for European style options is known. The computation time and the error need to be minimized simultaneously. In this paper, the authors have solved the Black-Scholes equation by employing a reasonably accurate implicit method. Options with known analytic solutions have been evaluated. Furthermore, an overall second order accurate space and time discretization has been accomplished in this paper.

Black Scholes(B-S) model is one of the popular methods of calculating the option prices. It consists of the closed form solution of the linear Black Scholes equation. Researchers have also found the numerical solutions of the Black Scholes equation which is much more intuitive compared to its closed form solution. In this paper, we have compared the analytical solution with the numerical solution of linear Black-Scholes equation governing option pricing. Finite Difference methods (FDMs) are used to discretize B-S equation base on three different schemes namely, Explicit, Implicit and the Crank Nicholson scheme. We have also tried to found out whether this solution are validate to compute the price the options of the higher strike prices such as those of BANKNIFTY options. It is also showed that numerical solutions are independent of number of grids.

The green's function method in quantum statistics

This thesis focuses on the modification of the classical Black-Scholes equation by introducing fractional derivatives instead of the usual integer order derivatives. The project has two main parts: (1) develop efficient numerical methods, such as finite difference methods and a space-time spectral method, for the resulting fractional partial differential equations, and (2) employ an estimation of the associated parameter of the time-fractional Black-Scholes equation and explore the underlying meaning of fractional-order derivatives under different economic conditions. This study aims to provide a general framework for the application of fractional-order option pricing models to deal with real option pricing problems.

An analytic and numerical solution with spectral Green's function method for transport equation in spherical geometry

This paper presents the numerical valuation of the two-asset step-down equitylinked securities (ELS) option by using the operator-splitting method (OSM). The ELS is one of the most popular financial options. The value of ELS option can be modeled by a modified Black-Scholes partial differential equation. However, regardless of whether there is a closedform solution, it is difficult and not efficient to evaluate the solution because such a solution would be represented by multiple integrations. Thus, a fast and accurate numerical algorithm is needed to value the price of the ELS option. This paper uses a finite difference method to discretize the governing equation and applies the OSM to solve the resulting discrete equations. The OSM is very robust and accurate in evaluating finite difference discretizations. We provide a detailed numerical algorithm and computational results showing the performance of the method for two underlying asset option pricing problems such as cash-or-nothing and stepdown ELS. Final option value of two-asset step-down ELS is obtained by a weighted average value using probability which is estimated by performing a MC simulation.

The deregulation of the electricity market, the appearance of competence and new risks, has made to emerge in the electricity sector a new activity as it can be the trading. This new activity is due to the necessity to ensure a supply price and transfer the risks to the speculators of the market. Also, it has been introduced more competence in the retail market and the companies integrated vertically have been fractionated. Whether we compare the electric markets with the traditional financial markets, the latters are less complicated and more mature, considering that the electric power has a more complex behavior in relation to the transmission, meteorology, market coupling and stocking. Thus, the final objective of this work is to analyze and quantify the risk that a big industrial company assumes when negotiating with electricity prices, and in this way, enhance the industrial companies the decision-making for getting a profit adjusted to the risk (RORAC). For that, different tools are going to be used, such as the Montecarlo method, the VaR and the Black-Scholes equation. The losses that can be obtained have been calculated when you make a hedge by a futures portfolio or and an options portfolio, and it has been decided which one is more adequate taking into account the economic loss and the quantity of energy that can be hedged.

In order to price some exotic options in the presence of transaction costs, the new type of reset option is considered. By incorporating the random stopping property of the American option, we propose some pricing theorems. The specific reset option pricing is within the framework of portfolio optimization problems. Utility functions are defined so as to derive the utility indifference prices for the reset option buyers. Detailed numerical procedures are proposed for computing reset option prices and the corresponding optimal trading strategies. We also elaborate on the practical procedures of implementing our numerical algorithms and some direction of the future work.

Simulations that are meant to determine the steady-state distribution of a diffusible solute such as oxygen in tissues have typically used finite difference methods to solve the diffusion equation. Finite difference methods require a tissue mesh with enough points to resolve oxygen gradients near and between discrete blood vessels. The large number of points that are typically required can make these calculations very slow. In this paper, we investigate a numerical method known as the Green's function method which is not bound by the same constraint. The Green's function method is expected to yield an accurate oxygen distribution more quickly by requiring fewer mesh points. Both methods were applied to calculate the steady state oxygen distribution in a model simulation region. When the Green's function calculation used meshes with 1/2, 1/4 and, 1/8 of the resolution required for the finite-difference mesh, there was good agreement with the finite difference calculation in all cases. When the volume of the domain was increased 8-fold the Green's function method was able to calculate the O2 field in 22 minutes, whereas the finite difference calculation is expected to take approximately 1 week. The number of steps required for the Green's function calculation increases quadratically with the number of points in the tissue mesh. As a result, small meshes are calculated very quickly using Green's functions, while for larger mesh sizes this method experiences a significant decrease in efficiency.

Printed circuits in bounded media encompass a wide range of practical structures such as discontinuities in waveguides, planar circuits embedded in shielded multilayered media or even two-dimensional printed periodic structures. The Electromagnetic (EM) modeling of printed circuits in layered bounded media is performed via an Integral Equation (IE) technique. Green's functions (GFs) are specially defined to satisfy both the Boundary Conditions (BCs) imposed by the layered media and by the transverse boundary enclosing the entire structure. Finally, a system of IEs on the equivalent sources can be solved numerically by means of the Method of Moments (MoM). Each of the problems enumerated above has already been solved by other authors using IE-MoM techniques. Nevertheless, our formulation introduces a unified approach applicable to all the aforementioned problems. Due to the symmetry presented by a bounded layered media, the GF problem can be reduced into a two-dimensional transverse boundary problem and a one-dimensional transmission line problem in the normal direction. Both problems can be treated independently. This thesis proposes and fully develops an efficient technique that encompasses different laterally bounded multilayered problems with a seamless transition between them. The method is based on a modal representation of the transverse boundary problem and on the expansion of the equivalent surface currents by zero-curl & constant-charge Basis Functions (BFs). It offers a unified and versatile approach that, on one hand eliminates redundancy in the formulation and on the other hand simplifies each particular problem to the evaluation of constant coefficients or basic line integrals. Analytical solutions can be found for the combination of linear subsectional basis functions in rectangular and circular Perfect Electric Conductor (PEC) boundaries as well as for periodic lattices. This thesis then solves the problem of transmission line model in the longitudinal direction by proposing an efficient algorithm that guarantees numerical stability under a variety of known critical conditions where other already known formulations fail. In addition, it introduces alternate equivalent expressions of this formulation that allow new interpretations of the problem. Due to its practical interest, the method is applied for the EM modeling of multilayered boxed printed circuits. This motivated the implementation of a dedicated software tool for the efficient analysis of these topologies including losses. Extensive numerical experiments have been carried out to assess the validity of the aforementioned theory and some properties of test-structures (losses, mesh, etc).

Black-Scholes equation of dual currency exchange option pricing is a typical multi-asset option pricing model. It is important to research numerical solution of this equation. In this paper, accelerated additive operator splitting (AOS) differential algorithm is used to transform the multi-dimensional Black-Scholes equation into equivalent one-dimensional equations. Then ’explicit-implicit’ and ’implicit-explicit’ schemes will be constructed. These schemes proved to be stable and convergent unconditionally and they have second-order accuracy. Finally, the numerical example shows the effectiveness of the accelerated AOS difference algorithm. It illustrates that accelerated AOS algorithm effectively avoids the computational complexity of the high-dimensional Black-scholes equation, and dramatically improves the computational speed. Therefore, it is applicable for multi-asset option pricing.

This paper reviews the past research on option pricing, including the Bachelier option pricing formula, Black-Scholes model, and the Cox-Ross-Rubinstein (CRR) binomial option pricing model. Then some practical calculations are performed on the Black-Scholes model and CRR binomial model and the differences between the results of these two models are analyzed for investors to choose the appropriate model for option prices. From the practical calculation results, it is verified that the Black-Scholes model can give out the analytical solutions of option prices and decrease the calculation of solution, while it can only be used for European options with stock prices regarded as a continuous process. For the CRR binomial model, although it can only give numerical solutions and requires lots of calculation because of too many branches, but it can solve the American option prices with early exercises and the European option prices with stock prices regarded as a discrete process.

We have calculated the Debye-Waller factor (DWF) of Cu from a model that was used successfully in earlier calculations of anharmonicity by Cowley and Shukla. The present calculation has been carried out using quasiharmonic theory, the lowest-order (λ2) anharmonic perturbation theory, and a Green's function (GF) method which sums an infinite series of the λ2−type anharmonic terms. The static approximation ω → 0 in the cubic contribution to the self-energy of the GF, introduced in the earlier work on the DWF by Shukla and Hubschle is further justified by showing that in the high-temperature limit the exact results for the λ2 anharmonic contributions (cubic and quartic) to the Helmholtz free energy are given in this approximation. Results for the DWF are also obtained for a modified version of the Morse potential with λ2 perturbation theory (PT) and the GF method. The GF results are in excellent agreement with the experimental Mossbauer and X-ray data in the entire temperature range, 300 K  T  120...