Stable approximation of unbounded matrix operators for the simultaneous inversion of source terms and initial values in time-fractional Black–Scholes equation

Previous article Next article Determination of an Unknown Heat Source from Overspecified Boundary DataJ. R. CannonJ. R. Cannonhttps://doi.org/10.1137/0705024PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. R. Cannon, Determination of an unknown coefficient in a parabolic differential equation, Duke Math. J., 30 (1963), 313–323 10.1215/S0012-7094-63-03033-3 MR0157121 (28:358) 0117.06901 CrossrefISIGoogle Scholar[2] J. R. Cannon, Determination of certain parameters in heat conduction problems, J. Math. Anal. Appl., 8 (1964), 188–201 10.1016/0022-247X(64)90061-7 MR0160047 (28:3261) 0131.32104 CrossrefGoogle Scholar[3] J. R. Cannon, Determination of the unknown coefficient $k(u)$ in the equation $\nabla \cdot k(u)\nabla u=0$ from overspecified boundary data, J. Math. Anal. Appl., 18 (1967), 112–114 10.1016/0022-247X(67)90185-0 MR0209634 (35:531) 0151.15901 CrossrefISIGoogle Scholar[4] J. R. Cannon and , D. L. Filmer, The determination of unknown parameters in analytic systems of ordinary differential equations, SIAM J. Appl. Math., 15 (1967), 799–809 10.1137/0115069 MR0218632 (36:1716) 0251.34002 LinkISIGoogle Scholar[5] J. R. Cannon, , Jim Douglas, Jr. and , B. Frank Jones, Jr., Determination of the diffusivity of an isotropic medium, Internat. J. Engrg. Sci., 1 (1963), 453–455 10.1016/0020-7225(63)90002-8 MR0160045 (28:3259) CrossrefGoogle Scholar[6] J. R. Cannon and , B. Frank Jones, Jr., Determination of the diffusivity of an anisotropic medium, Internat. J. Engrg. Sci., 1 (1963), 457–460 10.1016/0020-7225(63)90003-X MR0160046 (28:3260) CrossrefGoogle Scholar[7] J. R. Cannon and , J. H. Halton, The irrotational solution of an elliptic differential equation with an unknown coefficient, Proc. Cambridge Philos. Soc., 59 (1963), 680–682 MR0149064 (26:6560) 0117.07101 CrossrefISIGoogle Scholar[8] Jim Douglas, Jr. and , B. Frank Jones, Jr., The determination of a coefficient in a parabolic differential equation. II. Numerical approximation, J. Math. Mech., 11 (1962), 919–926 MR0153988 (27:3949) 0112.32603 ISIGoogle Scholar[9] B. Frank Jones, Jr., The determination of a coefficient in a parabolic differential equation. I. Existence and uniqueness, J. Math. Mech., 11 (1962), 907–918 MR0153987 (27:3948) 0112.32602 ISIGoogle Scholar[10] B. Frank Jones, Jr., Various methods for finding unknown coefficients in parabolic differential equations, Comm. Pure Appl. Math., 16 (1963), 33–44 MR0152760 (27:2735) 0119.08302 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Identifying a space-dependent source term in distributed order time-fractional diffusion equationsMathematical Control and Related Fields, Vol. 0, No. 0 | 1 Jan 2022 Cross Ref Identification of stationary source in the anomalous diffusion equationInverse Problems in Science and Engineering, Vol. 29, No. 13 | 21 November 2021 Cross Ref A modified quasi-reversibility method for inverse source problem of Poisson equationInverse Problems in Science and Engineering, Vol. 29, No. 12 | 22 March 2021 Cross Ref Inverse modeling of contaminant transport for pollution source identification in surface and groundwaters: a reviewGroundwater for Sustainable Development, Vol. 15 | 1 Nov 2021 Cross Ref Convergence Analysis of a Crank–Nicolson Galerkin Method for an Inverse Source Problem for Parabolic Equations with Boundary ObservationsApplied 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MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705024Article page range:pp. 275-286ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

It is well known now, that a Time Fractional Black Scholes Equation (TFBSE) with a time derivative of real order \begin{document}$ \alpha $\end{document} can be obtained to describe the price of an option, when for example the change in the underlying asset is assumed to follow a fractal transmission system. Fractional derivatives as they are called were introduced in option pricing in a bid to take advantage of their memory properties to capture both major jumps over small time periods and long range dependencies in markets. Recently new derivatives of Fractional Calculus with non local and/or non singular Kernel, have been introduced and have had substantial changes in modelling of some diffusion processes. Based on consistency and heuristic arguments, this work generalises previously obtained Time Fractional Black Scholes Equations to a new class of Time Fractional Black Scholes Equations. A numerical scheme solution is also derived. The stability of the numerical scheme is discussed, graphical simulations are produced to price a double barriers knock out call option.

In this paper we prove some new converse and saturation results for Tikhonov regularization of linear ill-posed problems Tx=y, where T is a linear operator between two Hilbert spaces.

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the constructive methods of exact solutions to time fractional Black–Scholes equation. By constructing one-parameter Lie symmetry transformations and their corresponding group generators, time fractional Black–Scholes equation is reduced to a fractional ordinary differential equation and some group-invariant solutions are obtained. Using the invariant subspace method, the analytical representations of two forms of exact solutions of time fractional Black–Scholes equation are given constructively, and the characteristics and differences of the two exact solutions are compared in the sense of geometric figures. In this paper, the form of the equation is generalized, and more group invariant solutions and analytical solutions in the form of separated variables are obtained.

Homotopy analysis method and its applications in the valuation of European call options with time-fractional Black-Scholes equation

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This paper tries to examine the recovery of the time-dependent implied volatility coefficient from market prices of options for the time fractional Black–Scholes equation (TFBSM) with double barriers option. We apply the linearization technique and transform the direct problem into an inverse source problem. Resultantly, we get a Volterra integral equation for the unknown linear functional, which is then solved by the regularization method. We use L 1 {L_{1}} -forward difference implicit approximation for the forward problem. Numerical results using L 1 {L_{1}} -forward difference implicit approximation ( L 1 {L_{1}} -FDIA) for the inverse problem are also discussed briefly.

A second order numerical method for the time-fractional Black–Scholes European option pricing model

‎The Black-Scholes equation is one of the most important mathematical models in option pricing theory, but this model is far from market realities and cannot show memory effect in the financial market. This paper investigates an American option based on a time-fractional Black-Scholes equation under the constant elasticity of variance (CEV) model, which parameters of interest rate and dividend yield supposed as deterministic functions of time, and the price change of the underlying asset follows a fractal transmission system. This model does not have a closed-form solution; hence, we numerically price the American option by using a compact difference scheme. Also, we compare the time-fractional Black-Scholes equation under the CEV model with its generalized Black-Scholes model as α = 1 and β = 0. Moreover, we demonstrate that the introduced difference scheme is unconditionally stable and convergent using Fourier analysis. The numerical examples illustrate the efficiency and accuracy of the introduced difference scheme.

In this paper, we consider an inverse problem to simultaneously reconstruct the source term and initial data associated with a parabolic equation based on the additional temperature data at a terminal time t = T and the temperature data on an accessible part of a boundary. The conditional stability and uniqueness of the inverse problem are established. We apply a variational regularization method to recover the source and initial value. The existence, uniqueness and stability of the minimizer of the corresponding variational problem are obtained. Taking the minimizer as a regularized solution for the inverse problem, under an a priori and an a posteriori parameter choice rule, the convergence rates of the regularized solution under a source condition are also given. Furthermore, the source condition is characterized by an optimal control approach. Finally, we use a conjugate gradient method and a stopping criterion given by Morozov's discrepancy principle to solve the variational problem. Numerical experiments are provided to demonstrate the feasibility of the method.

A compact quadratic spline collocation (QSC) method for the time-fractional Black–Scholes model governing European option pricing is presented. Firstly, after eliminating the convection term by an exponential transformation, the time-fractional Black–Scholes equation is transformed to a time-fractional sub-diffusion equation. Then applying $$L1 - 2$$ formula for the Caputo time-fractional derivative and using a collocation method based on quadratic B-spline basic functions for the space discretization, we establish a higher accuracy numerical scheme which yields $$3-\alpha $$ order convergence in time and fourth-order convergence in space. Furthermore, the uniqueness of the numerical solution and the convergence of the algorithm are investigated. Finally, numerical experiments are carried out to verify the theoretical order of accuracy and demonstrate the effectiveness of the new technique. Moreover, we also study the effect of different parameters on option price in time-fractional Black–Scholes model.

In this paper we present a numerical method for a time-fractional Black–Scholes equation, which is used for modeling the fractional structure of the financial market. The method is based on—first, discretization in time and then the weighted finite difference spatial approximation. Some properties of the spatial discretization are studied. The main difficulty (that originates from the non-local structure of the differential operator) of the algorithm is the impossibility to advance layer by layer in time. Numerical experiments are discussed.

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of the time fractional Black–Scholes equation derived by the fractional Brownian motion. Some exact solutions are obtained, the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, a new conservation theorem and a generalization of the Noether operators are developed to construct the conservation laws for the time fractional Black–Scholes equation.

In this paper, we apply a local discontinuous Galerkin (LDG) method to solve some fractional inverse problems. In fact, we determine a timedependent source term in an inverse problem of the time-fractional diffusion equation. The method is based on a finite difference scheme in time and a LDG method in space. A numerical stability theorem as well as an error estimate is provided. Finally, some numerical examples are tested to confirm theoretical results and to illustrate effectiveness of the method. It must be pointed out that proposed method generates stable and accurate numerical approximations without using any regularization methods which are necessary for other numerical methods for solving such ill-posed inverse problems.

This paper addresses the initial value inversion problem for a nonlinear time-fractional Black–Scholes equation. The existence and uniqueness of the solution to the inverse problem are established, and its ill-posed nature is analyzed. Under appropriate source conditions, a conditional stability estimate is provided. To regularize the ill-posed problem, a simplified quasi-reversibility method is introduced, and convergence rates are derived under both a priori and a posteriori parameter choice strategies. Numerical experiments, based on a finite difference discretization, demonstrate the feasibility and effectiveness of the proposed approach.