This article's main aim is to study the concepts of the generalized neighborhood and generalized quasi-neighborhood in fuzzy bitopological spaces. It also introduces fundamental theorems for determining the relationships between them.... more
This article's main aim is to study the concepts of the generalized neighborhood and generalized quasi-neighborhood in fuzzy bitopological spaces. It also introduces fundamental theorems for determining the relationships between them. Additionally, some significant examples were examined to demonstrate the significance of the interconnections, some theorems were also introduced to study some main properties of neighborhood structures. Finally, we also studied the concepts of closure, interior, and each of their critical theories and properties by generalized neighborhood systems in fuzzy bitopological spaces.
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This study aims to construct some new Milne-type integral inequalities for functions whose modulus of the local fractional derivatives is convex on the fractal set. To that end, we develop a novel generalized integral identity involving... more
This study aims to construct some new Milne-type integral inequalities for functions whose modulus of the local fractional derivatives is convex on the fractal set. To that end, we develop a novel generalized integral identity involving first-order generalized derivatives. Finally, as applications, some error estimates for the Milne-type quadrature formula and new inequalities for the generalized arithmetic and p-Logarithmic means are derived. This paper’s findings represent a significant improvement over previously published results. The paper’s ideas and formidable tools may inspire and motivate further research in this worthy and fascinating field.
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In this work, the exact and approximate solution for generalized linear, nonlinear, and coupled systems of fractional singular M-dimensional pseudo-hyperbolic equations are examined by using the multi-dimensional Laplace Adomian... more
In this work, the exact and approximate solution for generalized linear, nonlinear, and coupled systems of fractional singular M-dimensional pseudo-hyperbolic equations are examined by using the multi-dimensional Laplace Adomian decomposition method (M-DLADM). In particular, some two-dimensional illustrative examples are provided to confirm the efficiency and accuracy of the present method.
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In this present work, we perform a numerical analysis of the value of the European style options as well as a sensitivity analysis for the option price with respect to some parameters of the model when the underlying price process is... more
In this present work, we perform a numerical analysis of the value of the European style options as well as a sensitivity analysis for the option price with respect to some parameters of the model when the underlying price process is driven by a fractional Lévy process. The option price is given by a deterministic representation by means of a real valued function satisfying some fractional PDE. The numerical scheme of the fractional PDE is obtained by means of a weighted and shifted Grunwald approximation.
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In mathematics, distance and similarity are known as dual concepts. However, the concept of similarity is interpreted as fuzzy similarity or T-equivalence relation, where T is a triangular norm (t-norm in brief), when we discuss a fuzzy... more
In mathematics, distance and similarity are known as dual concepts. However, the concept of similarity is interpreted as fuzzy similarity or T-equivalence relation, where T is a triangular norm (t-norm in brief), when we discuss a fuzzy environment. Dealing with multi-polarity in practical examples with fuzzy data leadsus to introduce a new concept called m-polar T-equivalence relations based on a finitely multivalued t-norm T, and to study the metric behavior of such relations. First, we study the new operators including the m-polar triangular norm T and conorm S as well as m-polar implication I and m-polar negation N, acting on the Cartesian product of [0,1]m-times.Then, using the m-polar negations N, we provide a method to construct a new type of metric spaces, called m-polar S-pseudo-ultrametric, from the m-polar T-equivalences, and reciprocally for constructing m-polar T-equivalences based on the m-polar S-pseudo-ultrametrics. Finally, the link between fuzzy graphs and m-polar ...
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This paper outlines a comprehensive study of the fluid-flow in the presence of heat and mass transfer. The governing non-linear ODE are solved by means of the homotopy perturbation method. A comparison of the present solution is also made... more
This paper outlines a comprehensive study of the fluid-flow in the presence of heat and mass transfer. The governing non-linear ODE are solved by means of the homotopy perturbation method. A comparison of the present solution is also made with the existing solution and excellent agreement is observed. The implementation of homotopy perturbation method proved to be extremely effective and highly suitable. The solution procedure explicitly elucidates the remarkable accuracy of the proposed algorithm.
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The generalized Kuramoto–Sivashinsky equation is investigated using the modified Kudryashov method for the new exact solutions. The modified Kudryashov method converts the given nonlinear partial differential equation to algebraic... more
The generalized Kuramoto–Sivashinsky equation is investigated using the modified Kudryashov method for the new exact solutions. The modified Kudryashov method converts the given nonlinear partial differential equation to algebraic equations, as a result of various steps, which upon solving the so-obtained equation systems yields the analytical solution. By this way, various exact solutions including complex structures are found, and their behavior is drawn in the 2D plane by Maple to compare the uniqueness and wave traveling of the solutions.
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Derived from Lorenz-Haken equations, this paper presents a new 4D chaotic laser system with three equilibria and only two quadratic nonlinearities. Dynamics analysis, including stability of symmetric equilibria and the existence of... more
Derived from Lorenz-Haken equations, this paper presents a new 4D chaotic laser system with three equilibria and only two quadratic nonlinearities. Dynamics analysis, including stability of symmetric equilibria and the existence of coexisting multiple Hopf bifurcations on these equilibria, are investigated, and the complex coexisting behaviors of two and three attractors of stable point and chaotic are numerically revealed. Moreover, a conducted research on the complexity of the laser system reveals that the complexity of the system time series can locate and determine the parameters and initial values that show coexisting attractors. To investigate how much a chaotic system with multistability behavior is suitable for cryptographic applications, we generate a pseudo-random number generator (PRNG) based on the complexity results of the laser system. The randomness test results show that the generated PRNG from the multistability regions fail to pass most of the statistical tests.
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We consider subclasses of functions with bounded turning for normalized analytic functions in the unit disk. The geometric representation is introduced, some subordination relations are suggested, and the upper bound of the pre-Schwarzian... more
We consider subclasses of functions with bounded turning for normalized analytic functions in the unit disk. The geometric representation is introduced, some subordination relations are suggested, and the upper bound of the pre-Schwarzian norm for these functions is computed. Moreover, by employing Jack's lemma, we obtain a convex class in the class of functions of bounded turning and relations with other classes are posed.
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This paper gives a robust pseudospectral scheme for solving a class of nonlinear optimal control problems (OCPs) governed by differential inclusions. The basic idea includes two major stages. At the first stage, we linearize the nonlinear... more
This paper gives a robust pseudospectral scheme for solving a class of nonlinear optimal control problems (OCPs) governed by differential inclusions. The basic idea includes two major stages. At the first stage, we linearize the nonlinear dynamical system by an interesting technique which is called linear combination property of intervals. After this stage, the linearized dynamical system is transformed into a multi domain dynamical system via computational interval partitioning. Moreover, the integral form of this multidomain dynamical system is considered. Collocating these constraints at the Legendre Gauss Lobatto (LGL) points together with using the Legendre Gauss Lobatto quadrature rule for approximating the involved integrals enables us to transform the basic OCPs into the associated nonlinear programming problems (NLPs). In all parts of this procedure, the associated control and state functions are approximated by piecewise constants and piecewise polynomials, respectively. A...
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The fractional complex transformation is used to transform nonlinear partial differential equations to nonlinear ordinary differential equations. The improved ()-expansion method is suggested to solve the space and time fractional foam... more
The fractional complex transformation is used to transform nonlinear partial differential equations to nonlinear ordinary differential equations. The improved ()-expansion method is suggested to solve the space and time fractional foam drainage and KdV equations. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.
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The goal of this research article is to introduce a sequence of α–Stancu–Schurer–Kantorovich operators. We calculate moments and central moments and find the order of approximation with the aid of modulus of continuity. A... more
The goal of this research article is to introduce a sequence of α–Stancu–Schurer–Kantorovich operators. We calculate moments and central moments and find the order of approximation with the aid of modulus of continuity. A Voronovskaja-type approximation result is also proven. Next, error analysis and convergence of the operators for certain functions are presented numerically and graphically. Furthermore, two-dimensional α–Stancu–Schurer–Kantorovich operators are constructed and their rate of convergence, graphical representation of approximation and numerical error estimates are presented.
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The basic aim of this study is to include nonnegative real parameters to allow for approximation findings of the Stancu variant of Phillips operators. We concentrate on the uniform modulus of smoothness in a simple manner before moving on... more
The basic aim of this study is to include nonnegative real parameters to allow for approximation findings of the Stancu variant of Phillips operators. We concentrate on the uniform modulus of smoothness in a simple manner before moving on to the approximation in weighted Korovkin’s space. Our study’s goals and outcomes are to fully develop the uniformly approximated findings of Phillips operators. We determine the order of convergence in terms of Lipschitz maximal function and Peetre’s K-functional. In addition, the Voronovskaja-type theorem is also proved.
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Abstract. The Gamma function)(xΓ and the associated Gamma functions) ( ±Γ x are defined as distributions and neutrix product ++−Γ xnxx rs AD)() ( is evaluated. J.G. van der Corput developed the neutrix calculus having noticed that, in... more
Abstract. The Gamma function)(xΓ and the associated Gamma functions) ( ±Γ x are defined as distributions and neutrix product ++−Γ xnxx rs AD)() ( is evaluated. J.G. van der Corput developed the neutrix calculus having noticed that, in study of the asymptotic behaviour of integrals, functions of certain type could be neglected. This idea was also used by Fisher (see [3]) in order to define the neutrix product of the distributions. The neutrix product of distributions generalizes the definition of the product of distributions by Gelfand and Shilov and applicable to broader class of distributions. In the following, we let N be the neutrix, see van der Corput [1], having domain},..,2,1 { "nN = ′ and range the real numbers, with negligible functions finite linear sums of the functions "AA,2,1,0:,1 => − rnnnnn rr λλ and all functions which converge to zero in the normal sense as n tends to infinity. We now let)(xρ be any infinitely differentiable function having the following...
In this article, we introduce Stancu type generalization of Baskakov–Durrmeyer operators by using inverse Pólya–Eggenberger distribution. We discuss some basic results and approximation properties. Moreover, we study the statistical... more
In this article, we introduce Stancu type generalization of Baskakov–Durrmeyer operators by using inverse Pólya–Eggenberger distribution. We discuss some basic results and approximation properties. Moreover, we study the statistical convergence for these operators.
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In this paper, by making use of a certain family of fractional derivative operators in the complex domain, we introduce and investigate a new subclass P?,?(k,?,?) of analytic and univalent functions in the open unit disk U. In particular,... more
In this paper, by making use of a certain family of fractional derivative operators in the complex domain, we introduce and investigate a new subclass P?,?(k,?,?) of analytic and univalent functions in the open unit disk U. In particular, for functions in the class P?,?(k,?,?), we derive sufficient coefficient inequalities and coefficient estimates, distortion theorems involving the above-mentioned fractional derivative operators, and the radii of starlikeness and convexity. In addition, some applications of functions in the class P?,?(k,?,?) are also pointed out.
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In this study, we first present a time-fractional L$\hat{e}$vy diffusion equation of the exponential option pricing models of European option pricing and the risk-neutral parameter. Then, we modify a particular L$\hat{e}$vy-time... more
In this study, we first present a time-fractional L$\hat{e}$vy diffusion equation of the exponential option pricing models of European option pricing and the risk-neutral parameter. Then, we modify a particular L$\hat{e}$vy-time fractional diffusion equation of European-style options. Introduce a more general model from the models based on the L$\hat{e}$vy-time fractional diffusion equation and review some recent findings regarding of the Europe option pricing of risk-neutral free.
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We consider a model of predator-prey interaction at fractional-order where the predation obeys the ratio-dependent functional response and the prey is linearly harvested. For the proposed model, we show the existence, uniqueness,... more
We consider a model of predator-prey interaction at fractional-order where the predation obeys the ratio-dependent functional response and the prey is linearly harvested. For the proposed model, we show the existence, uniqueness, non-negativity as well as the boundedness of the solutions. Conditions for the existence of all possible equilibrium points and their stability criteria, both locally and globally, are also investigated. The local stability conditions are derived using the Magtinon's theorem, while the global stability is proven by formulating an appropriate Lyapunov function. The occurance of Hopf bifurcation around the interior point is also shown analytically. At the end, we implement the Predictor-Corrector scheme to perform some numerical simulations.
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In this article, we consider the three dimensional $\alpha$-fractional nonlinear delay differential system of the form \begin{align*}... more
In this article, we consider the three dimensional $\alpha$-fractional nonlinear delay differential system of the form \begin{align*} D^{\alpha}\left(u(t)\right)&=p(t)g\left(v(\sigma(t))\right),\\D^{\alpha}\left(v(t)\right)&=-q(t)h\left(w(t))\right),\\D^{\alpha}\left(w(t)\right)&=r(t)f\left(u(\tau(t))\right),~ t \geq t_0, \end{align*} where $0 < \alpha \leq 1$, $D^{\alpha}$ denotes the Katugampola fractional derivative of order $\alpha$. We have established some new oscillation criteria of solutions of differential system by using generalized Riccati transformation and inequality technique. The obtained results are illustrated with suitable examples.
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In this work, we introduce fractional Mellin transform of order α; 0 < α ≤ 1 on a functionwhich belongs to the Lizorkin space. Further, some properties and applications of fractionalMellin transform are given.
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The composition of the distribution δ (s) (x) and an infinitely differentiable function f(x) having a simple zero at the point x=x 0 is defined by Gel’fand Shilov by the equation δ (s) (f(x))=1 |f ' (x 0 )|1 f ' (x)1 dx s δ(x-x 0... more
The composition of the distribution δ (s) (x) and an infinitely differentiable function f(x) having a simple zero at the point x=x 0 is defined by Gel’fand Shilov by the equation δ (s) (f(x))=1 |f ' (x 0 )|1 f ' (x)1 dx s δ(x-x 0 ). It is shown how this definition can be extended to functions f(x) which are not necessarily infinitely differentiable or not having simple zeros at the point x=x 0 , by defining δ (s) (f(x)) as the limit or neutrix limit of the sequence {δ n (s) (f(x))}, where {δ n (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). A number of examples are given.
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LetFbe a distribution inD'and letfbe a locally summable function. The compositionF(f(x))ofFandfis said to exist and be equal to the distributionh(x)if the limit of the sequence{Fn(f(x))}is equal toh(x),... more
LetFbe a distribution inD'and letfbe a locally summable function. The compositionF(f(x))ofFandfis said to exist and be equal to the distributionh(x)if the limit of the sequence{Fn(f(x))}is equal toh(x), whereFn(x)=F(x)*δn(x)forn=1,2,…and{δn(x)}is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the compositionδ(s)[(sinh-1x+)r]does not exists. In this study, it is proved that the neutrix compositionδ(s)[(sinh-1x+)r]exists and is given byδ(s)[(sinh-1x+)r]=∑k=0sr+r-1∑i=0k(ki)((-1)krcs,k,i/2k+1k!)δ(k)(x), fors=0,1,2,…andr=1,2,…, wherecs,k,i=(-1)ss![(k-2i+1)rs-1+(k-2i-1)rs+r-1]/(2(rs+r-1)!). Further results are also proved.
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A topological spaceXis said to be almost Lindelöf if for every open cover{Uα:α∈Δ}ofXthere exists a countable subset{αn:n∈ℕ}⊆Δsuch thatX=∪n∈ℕCl(Uαn). In this paper we study the effect of mappings and some decompositions of continuity on... more
A topological spaceXis said to be almost Lindelöf if for every open cover{Uα:α∈Δ}ofXthere exists a countable subset{αn:n∈ℕ}⊆Δsuch thatX=∪n∈ℕCl(Uαn). In this paper we study the effect of mappings and some decompositions of continuity on almost Lindelöf spaces. The main result is that a image of an almost Lindelöf space is almost Lindelöf.
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We study the first-order nonhomogenous wave equation. We extend the convolution theorem into a general case with a double convolution as the nonhomogenous term. The uniqueness and continuity of the solution are proved and we provide some... more
We study the first-order nonhomogenous wave equation. We extend the convolution theorem into a general case with a double convolution as the nonhomogenous term. The uniqueness and continuity of the solution are proved and we provide some examples in order to validate our results.
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We establish exact solutions for the Schrödinger-Boussinesq Systemiut+uxx−auv=0,vtt−vxx+vxxxx−b(|u|2)xx=0, whereaandbare real constants. The (G′/G)-expansion method is used to construct exact periodic and soliton solutions of this... more
We establish exact solutions for the Schrödinger-Boussinesq Systemiut+uxx−auv=0,vtt−vxx+vxxxx−b(|u|2)xx=0, whereaandbare real constants. The (G′/G)-expansion method is used to construct exact periodic and soliton solutions of this equation. Our work is motivated by the fact that the (G′/G)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. As a result, hyperbolic function solutions and trigonometric function solutions with parameters are obtained. These solutions may be important and of significance for the explanation of some practical physical problems.
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We define the so-called box convolution product and study their properties in order to present the approximate solutions for the general coupled matrix convolution equations by using iterative methods. Furthermore, we prove that these... more
We define the so-called box convolution product and study their properties in order to present the approximate solutions for the general coupled matrix convolution equations by using iterative methods. Furthermore, we prove that these solutions consistently converge to the exact solutions and independent of the initial value.
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In the present paper we introduce some sequence spaces overn-normed spaces defined by a Musielak-Orlicz function . We also study some topological properties and prove some inclusion relations between these spaces.
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The literature has an astonishingly large number of integral formulae involving a range of special functions. In this paper, by using three Beta function formulae, we aim to establish three integral formulas whose integrands are products... more
The literature has an astonishingly large number of integral formulae involving a range of special functions. In this paper, by using three Beta function formulae, we aim to establish three integral formulas whose integrands are products of the generalized hypergeometric series p+1Fp and the integrands of the three Beta function formulae. Among the many particular instances for our formulae, several are stated clearly. Moreover, an intriguing inequality that emerges throughout the proving procedure is shown. It is worth noting that the three integral formulae shown here may be expanded further by using a variety of more generalized special functions than p+1Fp. Symmetry occurs naturally in the Beta and p+1Fp functions, which are two of the most important functions discussed in this study.
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A differential equation is a relationship between a function and its derivatives which are naturally appeared in many disciplines. They might have coefficients as constant and polynomials. There are several methods to solve a differential... more
A differential equation is a relationship between a function and its derivatives which are naturally appeared in many disciplines. They might have coefficients as constant and polynomials. There are several methods to solve a differential equation. But there is no general method to solve all the differential equations. Different problem might require different techniques. In this work after reviewing the present solution techniques we introduce some of the differential equations with distributional coefficients and make some comparisons between the solutions. Keywords: Distributions; Integral Transforms; Differential Equations with Distributional Coefficients.
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In this paper, the aim studying this topic is to extend the study of the one-dimensional fractional to the multi-dimensional fractional integral equations and their applications. The multi-dimensional Laplace transform method (M.D.L.T.M)... more
In this paper, the aim studying this topic is to extend the study of the one-dimensional fractional to the multi-dimensional fractional integral equations and their applications. The multi-dimensional Laplace transform method (M.D.L.T.M) is developed to solve multi-dimensional fractional Integrals equations. We used the one-dimensional Laplace transform for solving the fractional integral. The procedure will simply to find the Laplace transform to the equation, to solve the transform of the unknown function. Finally, find the inverse Laplace to obtain our desired solution. The result reveals that the transform method is very convenient and effective.
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The rough Heston model is a form of a stochastic Volterra equation, which was proposed to model stock price volatility. It captures some important qualities that can be observed in the financial market—highly endogenous, statistical... more
The rough Heston model is a form of a stochastic Volterra equation, which was proposed to model stock price volatility. It captures some important qualities that can be observed in the financial market—highly endogenous, statistical arbitrages prevention, liquidity asymmetry, and metaorders. Unlike stochastic differential equation, the stochastic Volterra equation is extremely computationally expensive to simulate. In other words, it is difficult to compute option prices under the rough Heston model by conventional Monte Carlo simulation. In this paper, we prove that Euler’s discretization method for the stochastic Volterra equation with non-Lipschitz diffusion coefficient error[|Vt−Vtn|p] is finitely bounded by an exponential function of t. Furthermore, the weak error |error[Vt−Vtn]| and convergence for the stochastic Volterra equation are proven at the rate of O(n−H). In addition, we propose a mixed Monte Carlo method, using the control variate and multilevel methods. The numerica...
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The volatility of stock return does not follow the classical Brownian motion, but instead it follows a form that is closely related to fractional Brownian motion. Taking advantage of this information, the rough version of classical Heston... more
The volatility of stock return does not follow the classical Brownian motion, but instead it follows a form that is closely related to fractional Brownian motion. Taking advantage of this information, the rough version of classical Heston model also known as rough Heston model has been derived as the macroscopic level of microscopic Hawkes process where it acts as a high-frequency price process. Unlike the pricing of options under the classical Heston model, it is significantly harder to price options under rough Heston model due to the large computational cost needed. Previously, some studies have proposed a few approximation methods to speed up the option computation. In this study, we calibrate five different approximation methods for pricing options under rough Heston model to SPX options, namely a third-order Padé approximant, three variants of fourth-order Padé approximant, and an approximation formula made from decomposing the option price. The main purpose of this study is t...
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The rough Heston model has recently been shown to be extremely consistent with the observed empirical data in the financial market. However, the shortcoming of the model is that the conventional numerical method to compute option prices... more
The rough Heston model has recently been shown to be extremely consistent with the observed empirical data in the financial market. However, the shortcoming of the model is that the conventional numerical method to compute option prices under it requires great computational effort due to the presence of the fractional Riccati equation in its characteristic function. In this study, we contribute by providing an efficient method while still retaining the quality of the solution under varying Hurst parameter for the fractional Riccati equations in two ways. First, we show that under the Laplace–Adomian-decomposition method, the infinite series expansion of the fractional Riccati equation’s solution corresponds to the existing expansion method from previous work for at least up to the fifth order. Then, we show that the fourth-order Padé approximants can be used to construct an extremely accurate global approximation to the fractional Riccati equation in an unexpected way. The pointwise...
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Rough Heston model possesses some stylized facts that can be used to describe the stock market, i.e., markets are highly endogenous, no statistical arbitrage mechanism, liquidity asymmetry for buy and sell order, and the presence of... more
Rough Heston model possesses some stylized facts that can be used to describe the stock market, i.e., markets are highly endogenous, no statistical arbitrage mechanism, liquidity asymmetry for buy and sell order, and the presence of metaorders. This paper presents an efficient alternative to compute option prices under the rough Heston model. Through the decomposition formula of the option price under the rough Heston model, we manage to obtain an approximation formula for option prices that is simpler to compute and requires less computational effort than the Fourier inversion method. In addition, we establish finite error bounds of approximation formula of option prices under the rough Heston model for 0.1≤H<0.5 under a simple assumption. Then, the second part of the work focuses on the short-time implied volatility behavior where we use a second-order approximation on the implied volatility to match the terms of Taylor expansion of call option prices. One of the key results th...
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Rough volatility models are popularized by \cite{gatheral2018volatility}, where they have shown that the empirical volatility in the financial market is extremely consistent with rough volatility. Fractional Riccati equation as a part of... more
Rough volatility models are popularized by \cite{gatheral2018volatility}, where they have shown that the empirical volatility in the financial market is extremely consistent with rough volatility. Fractional Riccati equation as a part of computation for the characteristic function of rough Heston model is not known in explicit form as of now and therefore, we must rely on numerical methods to obtain a solution. In this paper, we give a short introduction to option pricing theory and an overview of the current advancements on the rough Heston model.