Monotone Finite-Difference Schemes With Second Order Approximation Based on Regularization Approach for the Dirichlet Boundary Problem of the Gamma Equation

We investigate the initial boundary value problem for the Gamma equation transformed from the nonlinear Black-Scholes equation for pricing option to a quasilinear parabolic equation of second derivative. Furthermore, two-side estimates for the exact solution are also provided. By using regularization principle, the unconditionally monotone second order approximation finite-difference scheme on uniform and nonuniform grids is generalized, in that the maximum principle is satisfied without depending on relations of the coefficients and grid parameters. By using the difference maximum principle, we acquired two-side estimates for difference solution for the arbitrary non-sign-constant input data. Finally, we also provide a proof for a priori estimate. It can be confirmed that the two-side estimates for difference solution are completely consistent with the differential problem. Otherwise, the maximal and minimal values of the difference solution is independent from the diffusion and convection coefficients.


I. INTRODUCTION
Over the last decades, not only financial engineers but also mathematicians have paid special attention to the valuation of derivative financial instruments. Indeed, since being introduced by Fischer Black and Myron Scholes in 1973, the Black-Scholes model based on partial differential equation has been widely employed in modern mathematical finance and become a common-sense approach for pricing options as well as many other financial securities [25]. This mathematical model was derived from the principle that yielding profits from making portfolios of both short and long positions in options as well as their underlying stocks should not be possible, if option prices are rightly priced in the market [15]. These scholars indicated that a European option's value on a stock, whose price or the log return of underlying price is supposed to follow a geometric Brownian motion with constant volatility and drift, is determined by a second-order parabolic The associate editor coordinating the review of this manuscript and approving it for publication was Jenny Mahoney. equation concerning time and stock price. Nevertheless, the assumptions of Black-Scholes equation based on perfectly liquid market are so idealistic in comparison with the high illiquidity recently. There are many numerical methods used for studying properties of typical non-linear Black-Scholes equations [1]- [12], [14], [16], [17].
Not only in mathematical physics, but also in economics, there is a need to solve partial differential equations containing lower derivatives. For example, in financial mathematics, it is of interest to study the Gamma equation obtained by a transformation of the nonlinear Black-Scholes equation to a quasilinear parabolic equation [26], [27], [30], [42]. The approximate solution of the Gamma equation is the main goal of this study.
We consider the following quasilinear parabolic equation, which is called the Gamma equation [26], [27] ∂u ∂t = ∂ 2 β (u) ∂x 2 + ∂β (u) ∂x + c ∂u ∂x , u = u (x, t) , The present paper will focus on some models related to nonlinear Black-Scholes equations for European option whose volatility relies upon various factors like the stock price, the option price, the time as well as its derivatives due to the presence of transaction cost. The option's behaviour would be disclosed by higher derivative of its price which is mentioned as the Greeks in the financial literature. Not only giving a good approximation for the pricing option, reliable numerical methods are also essential for its derivatives because of the relevance of the Greeks to quantitative analysis.
In order to find the difference solution of problem (1)-(2), we must restrict it to a finite spatial interval x ∈ (−L, L), with L > 0 being a sufficiently large number. Since S = Ee x , we limit S ∈ (0, +∞) the interval by interval S ∈ Ee −L , Ee L . In practical calculations, we can choose L ≈ 1.5 to contain important values of S. Thus, instead of (2), we consider the Gamma equation (1) with Dirichlet boundary conditions at x = ±L [26], i.e., Because the Gamma equation has no exact solutions (analytical solutions), to assess efficiency of the proposed difference scheme and to maintain the equality of the Gamma equation, we must add a residual term f (x, t) to the right-hand side of (1). Thus, we consider equation (1) in the form (13).
With no loss of generality, we construct a differential scheme for quasi-linear parabolic equation with the initial condition of u 0 (x). Depending on the specific model, the initial condition will have different forms. In this work, we have introduced two above models (Jandacka-Sevcovic and Frey) and the initial condition has the form of a Delta function. In order to get a suitable initial condition for computation, we consider the regularization of u 0 (x) = δ (x), given by the function 27]. In order to solve the Black-Scholes equation, as well as other nonlinear partial differential equations (PDEs), there are some effective methods such as the finite difference methods, the Galerkin method [54], the Monte-Carlo method [55], reduced basis methods [53], etc. Nowadays, with many achievements of computing techniques, novel methods utilizing the machine learning technique such as the deep Galerkin method [52] are effective for solving nonlinear PDEs. The reduced basis methods and machine-learning-based methods are especially effective in the case of grids containing a huge number of nodes. For this case, methods such as the finite difference methods and the Galerkin method usually require a lot of memory resources. However, the finite difference methods have an advantage: easy to process parallelly. This is effective to be implemented on high-performance computing systems. In this article, we only focus on finite difference methods to develop new finite difference schemes and analyze their two-sided estimates.
The maximum principle has attracted a lot of attention in the theory of difference schemes [39]. In particular, this principle is used to study the stability and convergence of a difference solution on a uniform norm. Computational methods satisfying the maximum principle are called to be monotone. Monotone schemes play a vital role in experiments, because the considered discrete problems are usually well-posed [24]. Furthermore they provide a numerical solution not being oscillating even if solutions are non-smooth [37].
It must be noticed that we can acquire lower estimates of the solutions of differential-difference problems or of twosided estimates problems. This is very important to investigate theoretical properties of the approximate methods with unbounded nonlinearities, in that it is necessary to prove that numerical solution lays in a neighbourhood of the exact solution. In the case of linear problems, the estimates is to find the range of values of the desired solution in terms of the problem input data (the coefficients and right-hand side of the equation as well as the initial and boundary conditions). In the case of nonlinearity, the estimates provide an approach to confirm the nonnegativity of the exact solution. This feature is very important in physical problems, as well as to find conditions of the input data to let the problem being parabolic or elliptic, for example, investigating the Gamma equation in financial mathematics to model pricing of options.
Finding nontrivial solutions estimates of the initialboundary value problems based on a special method is provided by Ladyzhenskaya [28] (see also the monograph [29, p. 22]), whereby we can introduce a parameterdependent change of variables and then minimize or maximize some functions depending on this parameter; the resulting extremal values will provide the corresponding estimates for the numerical solution. Naturally, it is neccessary to acquire such estimates for the approximate solution. The theory of finite-difference schemes [39, p. 229] includes the technique, well developed for linear problems on the grid satisfying the maximum principle, which provides two-sided estimates for the approximate solution. The accuracy of the acquired estimates of the solutions of finite-difference problems is not higher [19] than one of the corresponding estimates of the solutions of differential problems [29, p. 22]. There is an interesting work [18], [20], in that two-sided estimates for the finite element method is used for approximating the Dirichlet boundary problem for the linear parabolic equation in the discrete and continuous cases.
On uniform grids [35], for the case of the one-dimensional convection-diffusion equation, there is the simplest approximations of first derivatives leading to upwind difference schemes. Such schemes are unconditionally monotone. However, they have only the first order of approximation. Moreover, the convective term can be approximated using central difference schemes. In the case of the second order approximation, the monotonicity property of the schemes is satisfied based on constraints of the spatial grid step. Unconditionally monotone second-order accurate difference schemes for convection-diffusion problems on uniform grids were constructed by applying the regularization principle (see [39], [44]).
In problems of mathematical physics, it is very important to increase the order of accuracy of approximate methods without enlarging the standard stencil. In applied multidimensional problems with singularities in domains of complex geometry, mathematical simulation frequently relies on nonuniform grids. Nevertheless, in the transition from uniform to nonuniform grids, the order of the local approximation error is usually reduced. For example, approximating the second derivative on the usual three-point stencil (see [39]) ensures only the first order of accuracy in the uniform norm and in the grid norm of L 2 : , h i is the step size of the non-uniform grid. The second order approximation accuracy of the corresponding difference schemes on nonuniform grids can be proved only by applying a negative norm.
To increase the order approximation accuracy of a difference method, we must approximate the original differential equation not at grid nodes, but rather at some intermediate points of the computational domain. Specifically, it turns out that the usual approximation of the second difference derivative preserves the second order with respect to the point This simple idea was further developed by A.A. Samarskii, P.N. Vabishchevich, and P.P. Matus. For example, various high-order approximation schemes for a second-order ordinary differential equation and one-dimensional parabolic and hyperbolic equations were constructed in [41]. On an arbitrary rectangular nonuniform grid, monotone conservative schemes of second-order local approximation for the multidimensional Poisson equation were designed in [38], [40]. High-order accurate schemes for various equations of mathematical physics were constructed and investigated in [11]. The construction and study of high-order accurate monotone difference schemes on arbitrary nonuniform grids for stationary and nonstationary convection-diffusion equations were considered in [31], [43], [47], [51].
In the present paper, the Gamma equation is considered, on the basis of the technique from [29], two-sided estimates are obtained for its exact solution. The acquired results are generalized to the construction of monotone finite-difference schemes of second-order of local approximation on uniform and non-uniform grids for a given equation. The construction of such schemes is based on the appropriate choice of the perturbed coefficient, similarly to [39]. Using the difference maximum principle, two-sided and a priori estimates are obtained in the C-norm for the difference solution. It is interesting to note that the provided two-sided estimates of the difference solution are completely consistent with the estimates of exact solution of differential problem. Monotone schemes are very effective for well-posed problem. They also provide a numerical solution not being oscillating even if solutions are non-smooth. Moreover, two-side estimates not only provide a manner to prove the nonnegativity of the exact solution, but it is also helpful to find out sufficient conditions based on the input data if the nonlinear problem is parabolic. Consequently, a priori estimate of the approximate solution in the grid norm C depending on the initial and boundary conditions only is proved.

II. MAXIMUM PRINCIPLE FOR DIFFERENCE SCHEMES WITH VARIABLE SIGN INPUT DATA
Suppose that in the n-dimensional Euclidean space a finite number of points of the grid is given and denoted by h . For each point x ∈ h we connect one and only one stencil M(x) -a subset of h , holding this point. We call a set which is called to be the canonical form of the finitedifference scheme [39, p. 226]. Must notice that M (x) could be an empty set, for example, for Dirichlet boundary condition. A system of linear algebraic equations obtained by discretizing the boundary problem is often called difference scheme. For the grid h , we consider its arbitrary subset ω h and we denote For example, let ω h be a set of the internal nodes to approximate the Poisson equation. In this case ω h = h . According to [36], [39], the point x is called to be a grid boundary node.
In the case of Dirichlet boundary condition is posed, we have where γ is a collection of boundary nodes. We must notice that the grid may not hold boundary nodes in the case of approximating boundary conditions of second/third kind, i.e. all grid nodes will become internal nodes. In this case, we suppose that the usual positivity conditions for the coefficients is fulfilled The conditions will guarantee the unique solvability, monotonicity and stability for the difference scheme (5) in the uniform norm for small pertubation of input data. Now, we formulate the basic results before establishing two-sided estimates for the numerical solution at non-sign-definite input data of the problem F(x). Lemma 1 ( [34], [36]): Assume that conditions (6)-(7) that the coefficients are positive are satisfied. Then the maximum and minimum values of the solution of the finite-difference scheme (5) belong to the range of the input data Similar to the scalar case we introduce the canonical form of the vector-difference schemes Here the matrices The point x is called to be a grid boundary node, on condition that the Dirichlet condition is posed where γ is a collection of the boundary nodes, Definition 1: Vector-difference scheme (10)- (11) is called monotone if its solution satisfies the conditions: ii (x) = 0, a (2) ij (x) = −a ij (x) , i = j, i, j = 1, m. Then we write vector equation (10) in the form We will assume that the positivity conditions for matrix coefficients is satisfied (i.e. all elements of matrix are positive [23]) (12) and define Lemma 2: ( [33], [36]) Suppose that the positivity conditions for matrix coefficients (12) are fulfilled. Then the maximal and minimal values of the solution of the 45122 VOLUME 8, 2020 vector-difference scheme (10)- (11) belong to value interval of the input data Corollary 2 ( [33]): Let conditions of Lemma 2 be fulfilled. Then vector-difference schemes (10)- (11) is monotone and for her the following estimate holds

III. STATEMENT OF THE PROBLEM AND TWO-SIDED ESTIMATE OF THE EXACT SOLUTION
Without loss of generality in a rectangleQ T = {(x, t) : l 1 x l 2 , 0 t T } we consider the following initial boundary value problem for a quasilinear parabolic equation (the so called Gamma equation), that is a generalization of Jandacka and Sevcovic model in [26] ∂u ∂t = with boundary conditions and initial conditions Equation (13) can be written as with coefficients We assume that parabolicity condition of equation (16) on the solution [22] is satisfied wherē We assume in what follows that there exists a unique solution of problem (13)- (15) and all coefficients in Eq. (16) and the desired function have continuous bounded derivatives of order that is required as the presentation proceeds. Let Using the technique from [29], we prove two-sided estimates for the exact solution of problem (13)-(15). (13)-(15) at any point (x, t 1 ) ∈Q T the following two-sided estimates are valid:

Theorem 1: Let condition (18) be satisfied. Then for solution u(x, t) of problem
where

IV. UNCONDITIONALLY MONOTONE FINITE-DIFFERENCE SCHEME OF SECOND-ORDER APPROXIMATION ON UNIFORM GRIDS
In this section we construct a new finite difference scheme on uniform grid in space and in time for the problem (13)- (15). Some important properties for the solution of this schemes are studied, namely, order of approximation and monotonicity of scheme, two-sided estimate and a priori estimate on C-norm for difference solution.
A. DIFFERENCE SCHEME Using the principle of regularization [39] on a regular uniform grid in space and timeω =ω h ×ω τ , wherē we approximate equation (13) with a difference scheme of the form where

B. APPROXIMATION ERROR
The difference scheme (20) has the following approximation error where It follows from (21) that Hence, order of spatial approximation of the difference scheme (20) is two and for temporal approximation -one.

C. MONOTONICITY, TWO-SIDED AND A PRIORI ESTIMATES
We write the difference scheme (20) in the canonical form (5) with coefficients defined as follows The scheme (22)- (23) is monotone if the positivity conditions of the coefficients (6)- (7) are satisfied [39], i.e. if Let us prove that y n i ∈D u for all i, n. We take an auxiliary grid function z(x, t) = z n i = y n i e −λt n , λ = 0. The function z(x, t) satisfies the finite-difference equation We introduce the coefficients D n ( * )i as follows: for all λτ > 0. Take an arbitrary t n ∈ ω τ . The following three cases are possible for the function z(x, t).
1. max ω tn z(x, t) is nonpositive (i.e., z(x, t) 0, (x, t) ∈ ω t n ); 2. max ω tn z(x, t) is located on the base t = 0 or on the boundary (i.e., the inequality z(x, t) max ω tn e −λt {µ 1 (t), µ 2 (t), u 0 (x)}, (x, t) ∈ ω t n , holds); 3. A positive maximum is attained at some interior point Obviously, for n = 0 we have y 0 i = u 0,i ∈D u for all i = 1, N − 1. Assume that, for any arbitrary n, the inclusion y n i ∈ D u is also true. We need prove that y n+1 i ∈D u is true. From this assumption we have A n ( * )i > 0, B n ( * )i > 0, C n ( * )i > 0. According to Lemma 1 on the base of the estimate (8) for arbitrary t = t n ∈ ω τ and all i = 0, 1, . . . , N , we have Then, in all cases (1.)-(3.), the function z(x, t) satisfies the estimate , t t n . 45124 VOLUME 8, 2020 Next we consider the case where a positive maximum is attained at some interior point (x, t n+1 ), then once again with the help of Lemma 1 we obtain where m n+1 In a similar way, we obtain the lower bound where m n+1 Since τ e λτ (1 + τ q) − 1 1 λ + q for all λ, τ > 0, we see that the estimates (19) and (24) i.e., y n+1 i ∈ [m 1 , m 2 ], i = 1, N − 1. In this sense, the finite-difference estimates inherit the properties of the differential problem. Moreover, because positivity conditions for the coefficients (6)-(7) are fulfilled, the difference scheme (20) is monotone for all values of h and τ (i.e. the difference scheme is unconditionally monotone).
Thus, we have proved the following theorem.

Theorem 2: Suppose that the conditions (18) are fulfilled. Then the finite-difference scheme (20) is unconditionally monotone (without constraints on the steps τ and h) and for its solution y ∈D u the above two-sided estimates (24)-(25) hold at any point (x, t) ∈ ω.
On the base of the maximum principle in a standard way [36] we acquire the a priori estimate in the C-norm: Theorem 3: Let the condition (18) be fulfilled. Then for the solution of the difference problem (20) the following a priori estimate holds

Remark 1: It is interesting to note that the maximal and minimal values of the difference solution do not depend on the diffusion coefficient k(u) and the convection coefficient r(u).
Remark 2: For the case of c = q(x) = f (x, t) ≡ 0, equation (16) can be written as (24)- (25) are fully consistent with the estimates of exact solution of differential problem (19).

Then, as construction of monotone difference schemes for it, we do not need to use the regularization principle. Remark 3: The estimates obtained in
Remark 4: For the case of the Frey model [21] β(u) = u/(1 − ρu) 2 , ρ > 0 from (17) we obtain the coefficient k(u) of the form k(u) = (1 + ρu)/(1 − ρu) 3 . Then, by virtue of (18), equation (13) will be parabolic if k(u) > 0, ∀u ∈D u , i.e., if With the help of the estimates (19) we can choice such input data that conditions (26) are satisfied. Then for solution of the difference scheme (20), which approximates problem (13)- (15), conditions (26) As the coefficient k (u) = (1 + u) /(1 − u) 3 is not defined at u = 1, then it is not defined for the initial function u 0 (x) = sin x at x = x * = π/2. So we build uniform grid with step h = π/ (2N + 1) in order to x i = x * . The approximate solution of the problem (27) at t = 1, obtained by the difference scheme (20), is shown on Fig. 1. Fig. 1

is not mathematically correct because the solution of (27) is not defined for such choice of the initial data.
Therefore it is important to construct grid domain so that an extremal point is a grid node.

V. MONOTONE FINITE-DIFFERENCE SCHEME OF SECOND-ORDER APPROXIMATION ON NON-UNIFORM SPATIAL GRIDS
In this section, a new monotone difference schemes for the convection-diffusion problem for Gamma equation (13)-(15) will be constructed and analysed on non-uniform grids in space. The construction of such schemes is based on the appropriate choice of the perturbed coefficient (regularization principle), similar to [31], [39]. Using the difference maximum principle, two-sided estimate and a priori estimates in the C-normal are obtained for the solution of the difference scheme that approximates the above equation.

A. DIFFERENCE SCHEME
We introduce an arbitrary non-uniform spatial grid Taking into account the identity and using standard notation [39] h we construct a new difference scheme for a quasilinear parabolic equation (16) on a non-uniform grid ω =ω h × ω τ where ,

B. APPROXIMATION ERROR
Let us prove that the scheme (29) approximates the problem (13)- (15) with the second order with respect to the non-calculated pointx i = x i +h i (in the case of a uniform gridx i = x i ). To do this, we focus on the relationship [40] vxx when the condition of variable in space weight factors is fulfilled β k , β k+1 By virtue of (30) In view of (31) we obtain

From (30)-(33) it follows that
Using the Taylor series expansion we conclude that Since Finally, from (34)- (35), (37) we find out that the approximation error is of second order in space Hence, order of spatial approximation of the difference scheme (29) is two and order of temporal approximation is one.

C. MONOTONICITY, TWO-SIDED AND A PRIORI ESTIMATES
We write the difference scheme (29) in the canonical form (22)-(23) with coefficients defined as follows Let the following inequality be fulfilled: Theorem 4: Let the conditions (38) be met. Then the finite-difference scheme (29) is monotone, its solution belongs to the value interval of exact solution y ∈D u and the following two-sided estimates hold at any where m n 1 = sup λ>0 min 0, min Proof: To prove the upper bound (39), we consider an auxiliary function z = z (x, t n ) = y (x, t n ) e −λt n , where λ > 0 is a parameter. Let x 0 , t 0 be the maximum point of z in the grid domain ω t τ and z 0 = z x 0 , t 0 . There are only the following possibilities: 1. The maximum z 0 is non-positive; 2. The point x 0 , t 0 is on the boundary of the grid domain ω t τ ; 3. The maximum z 0 is positive, and the point x 0 , t 0 is the interior point of the grid domain ω t τ . In case (3.) at the maximum point x 0 , t 0 the following relations are fulfilled: Now we need to find a condition so that y n i ∈D u for all i, n. If n = 0, it is obvious that y 0 i = u 0 (x i ) ∈D u . Assume that, for any arbitrary n, y n i ∈D u is also true for all i. From this assumption for the case ofh > 0, kxx > 0 (we do not consider trivial cases ofh = 0 and kxx = 0) we get the concrete values of the weights The expressions k (β 1 β 2 ) (y), −β 4 kxx (y), −τdβ 6 are converted to the form It follows that A > 0. Discarding the positive term in the expression for the coefficient B with b + i (y n ), we obtain Since h /h + < 1, then 1 +h/h + k + + 1 −h/h + k ≥ 2k 1 . On the other hand, we have ) .

It follows that
From the last inequality we conclude that B n i > 0 at τ ≥ It is similar, we can find out conditions for other cases.
Therefore, the inequality (38) also ansures that the positivity of the coefficients (6)-(7) is fulfilled (i.e. the difference scheme (29) is monotone). According to Lemma 1 on the base of the estimate (8) for arbitrary t = t n ∈ ω τ and all i = 0, 1, . . . , N , we have Therefore, in all above cases the function z will satisfy the following estimate and for the original function y, we acquire the upper bound (39). Analogous arguments for the minimum point hold the lower estimate (39). As we see that the estimates (19) and ( show that difference estimates are consistent with the estimates for the solution of the differential problem.
Basing on the maximum principle in a standard way [36], we can acquire the a priori estimate in the C-norm: Theorem 5: Let the condition (38) be fulfilled. Then for the solution of the difference problem (29) the following a priori estimate holds Proof: According to Corollary 1, we can conclude that y n+1 Since the variable weighting factors β 1 , β 2 ≥ 0 are non-negative, then Substituting this estimate in the last inequality, we find the chain of relations Given that from (41) we obtain the required relation (40).

D. NUMERICAL TEST
We consider the problem (13)-(17) with a given input data l 1 = 0, l 2 = π, T = 1, k (u) = u 2 , r (u) = u, q (x) = 1 + x 2 and the exact solution u (x, t) = e t (sin x + 1). The starting non-uniform spatial nodes are shown in Table 1. To increase the number of the nodes, we use the following law where r ∈ [0; 0, 25) -is a random variable following the normal distribution. For illustration of the efficiency of the new algorithms on non-uniform grids the error of the method in maximum form |y (x, t) − u (x, t)| , 45128 VOLUME 8, 2020  of difference scheme (29) approximating the given problem are shown in Table 2.
The numerical experiments illustrate the increasing of the accuracy of the new scheme and for scheme (29) the order of accuracy O h 2 + τ is reached on coarse grids.

VI. ANOTHER APPROACH FOR CONSTRUCTION OF MONOTONE SECOND-ORDER FINITE-DIFFERENCE SCHEMES ON NON-UNIFORM GRIDS
To simplify evaluation, we consider the linear Gamma equation with the convective transport being non-divergent [44] We can construct similar schemes by using the identities (28) and ku = 0.5 (ku) + ku − k u . Suppose that b (x) = r (x) /k (x) and L 1 v = v + bv we rewrite the equation (42) in the form We change the differential operator L 1 on the gridω h by the following second order difference operator L 1h : On the gridω h we change the differential operator L by L h The variable in space weightsβ 3 ,β 4 , β 5 , β 6 are defined by fulfilment of the condition (32) in the following waȳ

Therefore, the difference scheme
can be used for approximating differential problem (42) on arbitrary non-uniform grid with the second order approximation. For uniform grid R = 0.5h |r| /k, the difference scheme (45) will degenerate to be the well-known monotone scheme of second-order approximation [39]. The difference scheme (45) can be rewritten as the following canonical form (5) with coefficients It is obvious that Hence, for arbitrary non-uniform grid refinement, the coefficients (46) of the difference scheme (45) will fulfill the conditions (6)-(7) (unconditional monotonicity). By Lemma 1 we acquire two-sided estimates for the difference scheme solution (45) for i = 0, N y i ≥ min µ 1 , µ 2 , min Moreover, according to Corollary 1, the difference scheme (45) is stable by right-hand side and by the boundary conditions. For the solution, we have the following a priori estimate Substituting y = z+u in the equation (45), we get the problem for the method error It is obvious that for the problem (48) all conditions (6)-(7) of the maximum principle are fulfilled, from (47) we can find out that z C ≤ ψ C ≤ ch 2 , i.e. the difference scheme (45) will converge to the exact solution with the order convergence being two.
It is similar, we can use formulas (33), (34), (43), (44) to construct the usual six-point stencil of monotone secondorder approximation difference scheme for the Gamma equation (13)-(15) on the non-uniform grid ω =ω h × ω τ with the VOLUME 8, 2020 support of the change g (x, u) = r (x) /k (u) and the operator where It is easy to show, that the difference scheme (49) approximates the original differential problem with second order on arbitrary non-uniform spatial grid, and the inequality (38) guarantees the fulfilment of the positivity condition for the coefficients (6)-(7) (i.e., the difference scheme (49) is monotone), and for a difference solution the two-sided estimates of the form (39) are valid.

VII. IMPLICATIONS
A great attention of many scholars to mathematical models utilised for pricing complex financial instruments, in recent years, has suggested the interesting research area of financial mathematics in which Black-Scholes model (BS) is a prominent example. Indeed, there is a large number of studies on this equation in the solution and application context with the employment of diverse numerical schemes like explicit difference schemes [13], finite difference scheme [45], multivariate padé approximation scheme [46] and Cauchy Euler method [32]. Nevertheless, applications of monotone finite difference scheme of second-order approximation for pricing models in finance are still very few. Therefore, to fill this research gap in mathematical financial, this paper adopts this kind of scheme for Gramma equation which is transformed from the non-linear BS equation for option price in an effort to derive the approximation solution of the nonlinear parabolic equations (for the second derivative of the option price). Specifically, in the problems of unlimited nonlinearity (or nonlinearity of unlimited growth), the basic properties of the differential problem may not be preserved when converting from a problem for a differential equation to the corresponding one for a difference equation. For instance, the approximation solution's values may not belong in a certain neighborhood of the exact solution's values.
Hence, two-side estimates for difference solution need to be proved to attain the approximation solution that belongs to some neighborhood of the exact solution (i.e. difference solution y ∈D u , then condition (18) is satisfied, and it follows that equation (16) is uniformly parabolic). Based on the technique of O.A Ladyzhenskava, the findings of this paper show that a two-side estimate for the difference solution, which is completely consistent with a similar solution estimate of the differential problem, was obtained. The empirical results are expected to be generalised to construct monotone finite difference schemes for two-dimensional quasilinear parabolic BS equations. These issues will be clarified in future work. In addition, this application of this method to solve BS pricing models can be reproduced for other mathematical models in finance such as Kolmogorov equations, Vasicek pricing equations.

VIII. BLACK-SCHOLES EQUATION FROM THE PERSPECTIVE OF ELECTRICAL AND ELECTRONICS ENGINEERING
The proposed difference schemes can be used for solving the Black-Scholes equation (by supporting the Gamma equation). The Black-Scholes equation plays an important role not only in financial engineering but also in electrical and electronics engineering. In this section, we discuss on two well-known equations that have many important applications in electrical and electronics engineering: the heat equation and the Schrodinger equation. They can be transformed into the Black-Scholes equation. The equations are widely used in studying quantum computing, light-emitting diodes, microprocessor, radiological imaging [48], microscopy imaging [49], image restoration (e.g., based on Perona-Malik model [50]), etc. Computing the partial derivatives: ∂v ∂t = βφu + φ ∂u ∂t , ∂v ∂x = αφu + φ ∂u ∂x , Hence, we acquired the following Black-Scholes equation: x ∈ (−∞, +∞) , t ∈ 0, σ 2 2 T , v (x, 0) = V e x , T = f e x , x ∈ (−∞, +∞) .

B. TRANSFORMATION OF THE SCHRODINGER EQUATION TO THE BLACK-SCHOLES EQUATION
We consider the following Schrodinger equation: i -the imaginary unit,h = h 2π , h -the Planck const, V (x, t) -a potential of an electric field. We only consider the free Schrodinger equation, i.e., V (x, t) = 0. Then, we set: As we can see that the Black-Scholes equation, as well as the Gamma equation, has a close relation to equations of mathematical physics such as the heat equation and the Schrodinger equation. In other words, the proposed difference schemes can be applied for solving problems of the field of electrical and electronics engineering.

IX. CONCLUSION
Based on regularization principle, we developed monotone finite-difference schemes with second-order of local approximation on both uniform and nonuniform grids for the initial boundary value problem for the Gamma equation. Moreover, two-side estimates of the difference solution are presented. Such estimates not only provide a manner to prove the non-negativity of the exact solution, but it is also helpful to find out sufficient conditions based on the input data if the nonlinear problem is parabolic. Consequently, a priori estimate of the approximate solution in the grid norm C depending on the initial and boundary conditions only is proved.