A new approach for the black-scholes model with linear and nonlinear volatilities

Since financial engineering problems are of great importance in the academic community, effective methods are still needed to analyze these models. Therefore, this article focuses mainly on capturing the discrete behavior of linear and nonlinear Black-Scholes European option pricing models. To achieve this, this article presents a combined method; a sixth order finite difference (FD6) scheme in space and a third-order strong stability preserving Runge-Kutta (SSPRK3) over time. The computed results are compared with available literature and the exact solution. The computed results revealed that the current method seems to be quite strong both quantitatively and qualitatively with minimal computational effort. Therefore, this method appears to be a very reliable alternative and flexible to implement in solving the problem while preserving the physical properties of such realistic processes. © 2019 by the authors

A novel hybrid method based cubic B-spline for one-dimensional Stefan problem with moving PCM, size-dependent thermal conductivity and periodic boundary condition

Phase change materials (PCM) are substances that release and absorb sufficient energy at phase transition to provide useful heat or cooling. Stefan problems including PCM have many applications in science and engineering. Since an exact solution for this type of Stefan problem does not exist, tracking the moving boundary and obtaining accurate temperature distribution are still challenging issues in mathematical aspects. Therefore, in this paper, a new approach based on combining the cubic B-spline and a fourth-order compact finite difference scheme is developed to capture the behavior of the one-dimensional Stefan problem including moving PCM, variable thermal conductivity and periodic boundary condition. The proposed combined method in space and the Crank-Nicolson method in time are applied to the model after the moving domain is transformed into the fixed domain by the boundary immobilization method. The comparative results have seen a good agreement for a particular case. Besides, the stability of the scheme and the effect of the model parameters on the numerical solution are analyzed. The computations reveal that the new combined method is seen to pull up the accuracy of the model solutions and presents an efficient alternative solution to the model

Hareketli sınır değer problemi olarak tahıl hidrasyonu modellerinin sayısal çözümleri üzerine

Bu tez çalışmasında, farklı sınır koşulları altında sabit ve değişken difüzyon katsayılı, bir boyutlu soya fasülyesi hidrasyon modellerinin sayısal çözümleri incelenmiştir. Bir difüzyon modeli olarak ele alınabilen soya fasülyesi hidrasyon modellerinin çözümleri sabit hacim ve değişken hacim altında incelenmiştir. Literatürde Hsu model olarak bilinen, hidrasyon boyunca hacmin değişmediğinin varsayıldığı modelin klasik sonlu fark yöntemleriyle çözümleri incelenmiş, kararlılık analizi yapılmış ve literatürdeki sonuçla karşılaştırılmıştır. Daha sonra, soya fasülyesinin yarıçapının hareketini ifade eden bir fonksiyon yardımıyla hacim değişimi göz önüne alınarak modelin sayısal çözümü yapılmış ve su alım oranları sabit hacimdeki durumla karşılaştırılmıştır. Devam eden bölümde ise hareketli sınır değer problemi olarak ele alınan soya fasülyesi hidrasyon modellerinin sayısal çözümlerine değinilmiştir. Ele alınan modeller, farklı sınır koşulları altında hem sabit difüzyon katsayılı hem de değişken difüzyon katsayılı durumlarda dördüncü ve altıncı mertebeden kompakt sonlu fark yöntemlerinin değişken uzay ızgarası (variable space grid) ve sınır sabitleme (boundary immobilization) yöntemleriyle birlikte düşünülmesiyle çözülmüştür. Elde edilen sonuçlar literatürdeki sonuçla karşılaştırılmıştır. Son olarak, sabit difüzyon katsayılı, yüzey sınırın hidrasyonun başında denge nemine ulaştığı sınır koşuluna sahip soya fasülyesi hidrasyon modeli için kısıtlı integral yönteminden faydalanarak yarıçapın hareketinin ifade edildiği bir diferansiyel denklem elde edilerek, sayısal çözümler yapılmıştır. Elde edilen sonuçlar literatürdeki diğer yöntemlerle karşılaştırılmıştır.In this thesis, numerical solutions of one dimensional soybean hydration models with constant and variable diffusion coeffcient under different boundary conditions were investigated. The solutions of soybean hydration models, which can be considered as a diffusion model, have been investigated under constant volume and variable volume. The model which is assumed to have no change in volume during hydration known as Hsu model in the literature was solved by using classical finite difference methods. Stability analysis is made and compared with the result in the literature. Then, the numerical solution of the model was solved considering the volume variation by means of a function expressing the movement of the radius of the soybean and the water intake rates were compared with the situation in the constant volume. In the following section, numerical solutions of soybean hydration models which are considered as moving boundary value problem have been mentioned. The models discussed were solved by the fourth and sixth order compact finite difference methods with variable space grid and boundary immobilization methods under both constant diffusion coefficients and variable diffusion coefficients with different boundary conditions. Stability and sensitivity analyzes of the methods were performed. Finally, soybean hydration model with constant diffusion coeffcient and boundary condition reached to equilibrium moisture at the beginning of hydration at surface were investigated. A differential equation expressing the motion of radius is obtained by using the constrained integral method and numerical solution was made. The results obtained were compared with other methods in the literature

A Fréchet derivative-based novel approach to option pricing models in illiquid markets

Nonlinear option pricing models have been increasingly concerning in financial industries since they build more accurate values by regarding more realistic assumptions such as transaction cost, market liquidity, or uncertain volatility. This study defines a nonclassical numerical method to effectively capture the behavior of the nonlinear option pricing model in illiquid markets where the implementation of a dynamic hedging strategy affects the price of the underlying asset. Unlike the conventional numerical approaches, this study describes a numerical scheme based on the Newton iteration technique and the Fréchet derivative for linearization of the model. The linearized time-dependent PDE is then discretized by a sixth-order finite difference scheme in space and a second-order trapezoidal rule in time. The computations revealed that the current approach appears to be somewhat more effective to some extent and at the same time economical for illustrative examples compared to the existing competitors. In addition, this method helps to prevent considering the convergence issues of the Newton approach applied to the nonlinear algebraic system. © 2021 John Wiley & Sons, Ltd.There are no funders to report for this submission

Valuation of the American put option as a free boundary problem through a high-order difference scheme

Valuation of the American options encountered commonly in finance is quite difficult due to the possibility of early exercise alternatives. Since an exact solution for the American options does not exist, effective numerical methods are needed to understand the behavior of option pricing models. Therefore, in this paper, a new approach based on a high-order difference scheme is proposed to discuss the valuation of an American put option as a free boundary problem. Using a front-fixing approach that transforms the unknown free boundary (optimal stopping) into a fixed one, a sixth-order finite difference scheme (FD6) in space and a third-order strong-stability preserving Runge-Kutta (SSPRK3) in time are applied to the model converted to a nonlinear partial differential equation. The computed results revealed that the combined method is seen to attempt to pull up the capacity of the algorithm to achieve higher accuracy. It is seen that the quantitative and qualitative results produced by the method proposed with minimal computational effort are sufficiently accurate and meaningful. Therefore, this article provides some new insights about the physical characteristics of financial problems and such realistic phenomena. © 2021 Walter de Gruyter GmbH, Berlin/Boston

Characterization of thin film Li0.5La0.5Ti1−xAlxO3 electrolyte for all-solid-state Li-ion batteries

Since addition of Al in Li0.5La0.5TiO3 has enhanced ionic conductivity in bulk materials, it is important to apply this material on all solid state thin film batteries. Because some of the good ionic conductors such as Lithium Phosphorus Oxynitride (LiPON) are sensitive to oxygen and moisture and their application is limited, so amorphous Li0.5La0.5Ti1−xAlxO3 (LLTAlO) is a most promising candidate because of its stability. In this study, the crystalline LLTAlO targets were prepared changing the amount of x content by conventional solid state reactions. Using these targets, lithium lanthanum titanium oxide (LLTO) thin film electrolytes were deposited on ITO/SLG substrates by radio frequency magnetron sputtering system in Ar atmosphere. The structural and compositional properties of targets and thin films were characterized by SEM, XRD, Raman spectroscopy and XPS. It was found that all targets are crystalline while the thin films are amorphous. To understand the effect of Al doping on ionic conductivity, electrical measurements were done at room temperature by AC impedance spectroscopy forming ITO/LLTAlO/Al structure like capacitor. Highest ionic conductivity result, 0.96 × 10−6 S·cm−1, is obtained from the nominal thin film composition of Li0.5La0.5Ti1−xAlxO3 (x = 0.05) at room temperature measurements. Heat treatment is also conducted to investigate to understand its effect on ionic conductivity and the structure of the thin films. It is found that ionic conductivity enhances with annealing. Also, temperature dependent ionic conductivity measurements from 298 K to 385 K are taken in order to evaluate activation energy for Li-ion conduction.TUBITAK (114M044