We propose a new monotone finite-difference scheme for the second-order local approximation on a nonuniform grid that approximates the Dirichlet initial boundary value problem (IBVP) for the quasi-linear convection-diffusion equation with unbounded nonlinearity, namely, for the Gamma equation obtained by transformation of the nonlinear Black-Scholes equation into a quasilinear parabolic equation. Using the difference maximum principle, a two-sided estimate and an a priori estimate in the c-norm are obtained for the solution of the difference schemes that approximate this equation.
Mathematics and Computer Science | Mathematics Doi: 10.31276/VJSTE.61(4).03-08 Finite-difference method for the Gamma equation on non-uniform grids Le Minh Hieu1*, Truong Thi Hieu Hanh1, Dang Ngoc Hoang Thanh2 University of Economics, The University of Danang Hue College of Industry Received August 2019; accepted 11 November 2019 Abstract: Introduction We propose a new monotone finite-difference scheme for the second-order local approximation on a nonuniform grid that approximates the Dirichlet initial boundary value problem (IBVP) for the quasi-linear convection-diffusion equation with unbounded nonlinearity, namely, for the Gamma equation obtained by transformation of the nonlinear Black-Scholes equation into a quasilinear parabolic equation Using the difference maximum principle, a two-sided estimate and an a priori estimate in the c-norm are obtained for the solution of the difference schemes that approximate this equation In the theory of difference schemes [1-3], the maximum principle is used to study the stability and convergence of a difference solution in the uniform norm Computational methods that satisfy the maximum principle are usually called monotone [1, 2] The monotone schemes play an critical role in computational practice They make it possible to obtain a numerical solution without oscillations even in the case of non-smooth solutions [4] Keywords: Gamma equation, maximum principle, monotone finite-difference scheme, non-uniform grid, quasi-linear parabolic equation, scientific computing, two-side estimates Classification number: 1.1 When constructing monotone difference schemes, it is desirable to preserve the second order approximation with respect to the spatial variable Such schemes are constructed for parabolic and hyperbolic equations in the presence of lower derivatives For example, a nonconservative scheme of second order approximation for linear parabolic equations of general form on uniform grids is given in [1, 2] When solving two-dimensional partial differential equations in the free domain, we need to construct a difference scheme on a non-uniform grid We must first confirm that a nonuniform grid is more general than a uniform grid While one can easily convert a non-uniform grid to uniform grid, the inverse transformation is not so straightforward, and it cannot preserve the conservation properties [5] For the nonlinear Black-Scholes equation, it is helpful to implement the grid to the payoff of the option, because the price of an option may be more sensitive in a precise area [6] In this case, the uniform grid is not appropriate In the case of nonuniform grids for equations in mathematical physics with variable coefficients without lower derivatives, a scheme was obtained in [7] for which the conditions of the maximum principle are fulfilled without relations on the coefficients and parameters of the grid (unconditional monotonicity) In [8], the unconditionally monotone and economical schemes of second order approximation were constructed on a non-uniform grid for non-stationary multidimensional convection-diffusion problems In the present work, the previously obtained results are *Corresponding author: Email: hieulm@due.edu.vn DECEMBER 2019 • Vol.61 Number Vietnam Journal of Science, Technology and Engineering ( ) is *based on + the appropriate choice of the perturbed 10] The construction of such schemes Jandacka-Sevcovic model model [9]) or [9]) or (the Frey (the mo ( Jandacka-Sevcovic ) ) ) and a priori ( two-sided coefficient, similar to In[1, 2, 8] the difference maximum(principle, more sensitive in a precise appropriate case of ofUsing precisearea area[6] [6].InInthis thiscase, case,thetheuniform uniformgrid gridisisnotnot appropriate Inthethe case ( as) respectively, as respectively, ( ) more sensitivegrids in afor precise area [6] In this case, physics the uniform isare notcoefficients appropriate Ininthelower case of estimates obtained the normal for* solving+ difference schemes that approximate the equations ininmathematical with variable without non-uniform for equations mathematical physics withgrid variable coefficients without lower (t Note (that ) is a parameter that depends( on each (concrete model, for example, )) is the volatili where ) ) , ( ( ) ) ( non-uniform forwas equations in inmathematical physics withequation variable coefficients without ( above derivatives, a grids scheme obtained thetheconditions ofofthethe maximum are ( ) scheme was obtained in[7] [7]forforwhich which conditions maximumprinciple principlelower are Jandacka-Sevcovic* model [9]) or cost measure, (the Frey model [11]), is which be writt transaction thecanrisk pre | Mathematics derivatives, a scheme wasand obtained [7] for which theparameters conditions ofof principle are ((( ))) Mathematics Science * ( )++ relations onon Computer the and thethemaximum grid fulfilled without relations theincoefficients coefficients and parameters ofthe grid(unconditional (unconditional respectively, as ( )) ( )) , is volatili where ( , is the volatility of the where ( measuring the market liquidity Using the change ( ) Problem settingtheand two-sided estimate of the solution * exact+ that fulfilled withoutIn [8], relations on the coefficients and parameters grid (unconditional Note ((that ))) is a parameter concrete model, for example, (t the monotone and economical ofofsecond monotonicity) [8], the unconditionally unconditionally monotone and economicalofschemes schemes secondorder order ( )+ , depends on each * transaction ( is the ) and ( , substituting cost measure, is premium the risk pre (( transaction )) measure, risk me ( cost ) ) ( ( ) monotonicity) In [8], the unconditionally monotone and economical schemes of second order We consider following quasilinear parabolic which is called the Gamma Jandacka-Sevcovic model [9]) or (the (1)–(2) Frey model [11]), which can be writt(to forfornon-stationary multidimensional ( measuring ) approximation were were constructed constructed onon aa non-uniform non-uniformgrid grid non-stationarythe multidimensional we obtain problem Then the function measuring the market liquidity Using the change ( )market Note thatequation, depends on each model, for example, the liquidity Using the change of indepen (that ) ≥ is (t Note that is0 aaisparameter parameter on measure, each concrete concrete model, forthe example, generalized to the construction of monotone difference asset, M the cost C ≥ is ( transaction ) that depends approximation were constructed on a non-uniform grid for non-stationary multidimensional respectively, as ( ) equation [9, 10]: convection-diffusion problems convection-diffusion problems corresponding models will also become [9, 10]: ( ) ( ( ) ( ) , , and substituting , , and substituting Jandacka-Sevcovic model [9]) or (the Frey model [11]), which can be writt ( )) , is the volatility of the underlying asset, is where ( Notepremium that is ameasure, parameter that onaeach concrete model, for example, schemes of second-order local approximation on non- risk anddepends ρ ≥ is parameter measuring Jandacka-Sevcovic model [9]) (the Frey model [11]), which can be writt(t convection-diffusion problems Note is a parameter that or depends on each concrete model, for the example, (tt In the present work, the obtained arearegeneralized totothetheconstruction of of that work, the previously previously obtainedresults results generalized construction we problem (1)–(2) Then we cost obtain problem Then the function (is )a and respectively, ) obtain (model transaction measure, is(1)–(2) the risk premium measure, and whichfunction (as )[9]) uniform spatial grids for the Gamma equation for the second the market liquidity Using the change of independent Jandacka-Sevcovic or (the Frey model [11]), can beparame writt respectively, as In the present work, the previously obtained results are generalized to the construction of ( ) ( ) Jandacka-Sevcovic model [9]) or x will Frey [11]), which[9,can be writt schemes ofofsecond-order corresponding models will 10]: monotone difference difference schemes second-order local approximationonmathematics onnon-uniform non-uniformspatial spatialgrids grids corresponding models become 10]: (x as=market )ln(S/E), derivative of the option price local in approximation financial , also t (the = (of T-τ, t model ∈)also (0,[9,become T) and variables where measuring the liquidity Using ∈ the variables respectively, ( ) (1) (( ) ), wh )independent ( change ) ( ( ) monotone difference schemes of second-order local approximation on non-uniform spatial grids respectively, as ( )) for the Gamma equation for the second derivative of the option price in financial mathematics [9, ) ( is(3)the the underlying asset, is (( , t) (= SV ) ,andinsubstituting for the secondofderivative of the option is price in financial mathematicswhere [9, , [9, 10] Theequation construction such schemes forvolatility the(two)ofabove inmodels, (3) for the two ( pricebased ) financial (on the) substituting ( ) u(x, ( ) ) SS (2) above mode for equation the schemes second derivative option [9,obtain ) ( ( ) construction ofofforsuch isis based thethe appropriate thethe transaction cost measure, is the risk premium measure, and is a parame appropriate choice of the perturbed coefficient, similar toperturbed we problem (1)-(2) Then the function β(u) and the 10].theTheGamma construction such schemes basedofononthe appropriateinchoice choiceofofmathematics perturbed we obtain problem and) the initial (((1)–(2) ) Then (( (underlying )( () condition ) ( )( ) for )) ,, is )the volatility is where )( asset, (the (function ) ( )( ))of ))( )for volatility of( the the underlying where 10] The8] construction schemes isdifference based on the appropriate of and theandaperturbed (the [1, 2, Using maximum principle, two(x) theis the corresponding models will asset, ( ), iswh condition measuring the(models market(uwill liquidity Using change of independent variables similar toto [1, 2,2,such 8] Using the maximum principle, two-sided priori corresponding also become [9, 10]: coefficient, [1,ofthe 8].difference Using thedifference maximum principle,choice two-sided a initial priori transaction is the risk measure, and is aa parame (where )) are According the studies in [9,, 10],cost Eqs by premium transforming nonlinear , obtained the volatility the underlying asset, is where ( [9,measure, transaction cost measure, isis premium measure, parame is risk the delta coefficient, to [1,inin2,the Using thefor difference maximum two-sided priori sided aobtained priori estimates are obtained inprinciple, the to normal cand a also 10]: ( )of inthe (3)and for the twoisabove mode , (1)–(2) ( ()) ) ,and(substituting is)the the volatility offunction the underlying asset, iswh where ( market estimatesand aresimilar obtained solving difference schemes that the the8].normal normal for solving difference schemes thatapproximate approximate the become ( ) ( ) ( ) ( ) (( a parame ), measuring the liquidity Using the change of independent variables ( ) Black-Scholes equation for such that transaction cost measure, is the risk premium measure, and is ), measuring the market liquidity Using the change of independent variables wh In order to find the approximate solution of p for solving difference schemes that approximate the above ( ) ( ) estimates are obtained in the normal for solving difference schemes that approximate the ( ) for we obtain problem (1)–(2) Then the function ( ) and the initial condition transaction cost measure, is the risk premium measure, and isabove a parame above equation ( ) ( ) in (3) for the two mode , , and substituting ( ) ( ) ( ) ( ), ( ) measuring the market liquidity Using the change of independent variables wh ( ) ( ) in (3) for the two above mode , , and substituting ( ), equation spatial interval where is a su where ( ) is the delta function where ( ) is the delta function corresponding models will also become [9, 10]: above equation ( ( )),for measuring the market(1)–(2) liquidity Using the change of( independent variables wh we problem Then the function and the condition ( toIn)find (3) the two above , , (1)–(2) and interval substituting ( of ) mode for we (obtain obtain problem Then the to function ( ())) approximate and solution the)ininitial initial condition limit the by for the interval ) ( ) order find ((the solution pr (3) In order the approximate of problem (1) ( ) Problem setting and two-sided estimate of the exact solution in (3) for the two above mode , , and substituting and two-sided of the exact solution of the exact Problem setting and estimate two-sided estimate corresponding models will also become [9, 10]: ( ) ( ) ( ) for we obtain problem (1)–(2) Then the function ( ) and the initial condition corresponding models will also become [9, 10]: ( ), chooseThen to where include important ( function ), interval where is(of spatial a values sufficiently Problem setting and two-sided estimate of the exact solution ( interval )spatial )a forsu we obtain problem (1)–(2) the ( ) and the initialiscondition solution (limit ) )also ((1) ) (with (10]: ) Dirichlet (the We consider the which is iscalled thetheGamma the following following quasilinear quasilinear parabolic parabolicequation, equation, which called Gamma corresponding models will become [9, equation boundary conditions at ( ) ) ( the interval by the interval limit interval by the interval The present paper on some related to nonlinear where ( ) ismodels the delta function corresponding will also become [9, 10]: Black-Scholes equations for We consider parabolic equation, whichparabolic is calledwill the focus Gamma where δ(x) ismodels the choose delta function [9, consider 10]: the following We the quasilinear following quasilinear equation include important values include important values ofoption Thus, ((the)) approximate ))solution ( ))to (( find )) to ((factors In choose orderrelies to of the 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̅̅ + ̅̅ **( ( )( ) ) ( ) ( ) are constants chosen based on each model, and where solution of Eq (3) with q=0 and 0⩽S