CONSTRUCTION OF GREEN S FUNCTIONS FOR THE BLACK-SCHOLES EQUATION. 1. Introduction The well-known function, in financial mathematics [3, 4, 6],

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1 Electronic Journal of Differential Equations, Vol , No. 53, pp. 4. ISSN: URL: or ftp ejde.math.txstate.edu login: ftp CONSTRUCTION OF GREEN S FUNCTIONS FOR THE BLACK-SCHOLES EQUATION MAX Y. MELNIKOV, YURI A. MELNIKOV Abstract. A technique is proposed for the construction of Green s functions for terminal-boundary value problems of the Black-Scholes equation. The technique permits an application to a variety of problems that vary by boundary conditions imposed. This is possible by extension of an approach that was earlier developed for partial differential equations in applied mechanics. The technique is based on the method of integral Laplace transform and the method of variation of parameters. It provides closed form analytic representations for the constructed Green s functions.. Introduction The well-known function, in financial mathematics [3, 4, 6], GS, t; S exp rt t = S[πσ T t] exp [lns/ S + r σ /T t] / σ T t. is referred to as the Green s function of the backward in time parabolic partial differential equation vs, t + σ S vs, t vs, t t S + rs rvs, t = 0. S which is called the Black-Scholes equation []. To be more specific mathematically, we note that. represents the Green s function for the homogeneous terminalboundary value problem corresponding to vs, T = fs.3 v0, t < and v, t <..4 This problem was posed for the Black-Scholes equation in the quarter-plane Ω = 0 < S < T > t > of the S, t-plane. In the above setting, v = vs, t is the price of the derivative product, fs is the pay-off function of a given derivative problem at the expiration time T, with S and t being the share price of the underlying asset and time, respectively. The parameters σ and r > 0 represent the volatility of the underlying asset and the risk-free interest rate, respectively. The variable S 0, in. plays the role of a source point. 000 Mathematics Subject Classification. 35K0, 58J35. Key words and phrases. Black-Scholes equation; Green s function. c 007 Texas State University - San Marcos. Submitted August 8, 007. Published November 4, 007.

2 M. Y. MELNIKOV, Y. A. MELNIKOV EJDE-007/53 A special comment is required as to the symbolism used in specifying the boundary conditions in.4. Both the end-points of the domain for the independent variable S represent the so-called singular points [5] to the Black-Scholes equation, in which case the corresponding boundary conditions cannot formally assign certain values to the solution of the governing differential equation. Instead, the conditions in.4 imply that the solution that we are looking for has to be bounded as the variable S approaches both zero and infinity. The function in. represents the only Green s function for. that is available in financial mathematics for decades. This study proposes a new approach that enables one to construct Green s functions to the Black-Scholes equation not only for the boundary conditions in.4 but also for a variety of others. The approach flows out from a technique proposed earlier [] for boundary value problems in applied mechanics. It is not based on the classical formalism for the diffusion equation as in [3, 4]. Instead, the emphasis is made on the parabolic single-parameter partial differential equation forward in time ux, τ τ = ux, τ ux, τ x + c cux, τ.5 x which is traditionally obtained [3, 6] from. by introducing new independent variables x = ln S and τ = σ T t.6 and setting ux, τ = vs, t. The parameter c in.5 is defined in terms of r and σ of the Black-Scholes equation as c = r/σ. To illustrate the effectiveness of our approach, a validation example is considered in the next section where we derive the Green s function of.. After the approach is validated, it is used, in the following sections, to tackle some other terminalboundary value problems for the Black-Scholes equation. New Green s functions are obtained none of which have earlier been presented in literature.. A validation example By introducing new variables x and τ in compliance with the relations in.6, the terminal-boundary value problem of.-.4 transforms to the following initial-boundary value problem ux, 0 = fexp x. u, τ <, u, τ <. for.5 on the half-plane < x < 0 < τ <. Applying the Laplace transform Ux; s = L{ux, τ} = 0 exp sτux, τdτ to the problem in.5,. and., one arrives at the boundary value problem d Ux; s dux; s dx + c s + cux; s = fexp x,.3 dx U; s <, U ; s <.4

3 EJDE-007/53 CONSTRUCTION OF GREEN S FUNCTIONS 3 for the Laplace transform Ux; s of ux, τ. Note that.3 is a linear nonhomogeneous ordinary differential equation with constant coefficients since s is just a parameter and Ux; s is treated as a single-variable function of x. To find a fundamental set of solutions to the homogeneous equation corresponding to.3, consider its characteristic equation whose roots are k + c k s + c = 0 k = α + ω, k = α ω where ω = s + β /, while the parameters α and β are defined in terms of c as α = c, β = + c.5 This yields two linearly independent particular solutions to the homogeneous equation corresponding to.3 as U x; s = expα + ωx, with their linear combination U x; s = expα ωx Ux; s = Ax; s expα + ωx + Bx; s expα ωx.6 representing, according to the method of variation of parameters, the general solution to.3. Following the procedure of this method, one arrives at the well-posed system expα + ωx expα ωx α + ω expα + ωx α ω expα ωx A x; s B = x; s 0 fexp x of linear algebraic equations in the derivatives with respect to x of the coefficients Ax; s and Bx; s of the linear combination in.6. The solution of the above system is obtained as A x; s = exp α + ωx fexp x, B x; s = exp α ωx fexp x Upon integration, the coefficients Ax; s and Bx; s are found in the form Ax; s = Bx; s = exp α + ωξfexp ξdξ + Ms, exp α ωξfexp ξdξ + Ns Substitution of these in.6 yields the general solution to.3 in the form Ux; s = exp αx ξ[exp ωξ x exp ωx ξ]fexp ξdξ + Ms expα + ωx + Ns expα ωx.7 The constants of integration Ms and Ns can be obtained upon satisfying the boundary conditions of.4. Omitting details, we have Ns = 0, Ms = exp α + ωξfexp ξdξ

4 4 M. Y. MELNIKOV, Y. A. MELNIKOV EJDE-007/53 Upon substituting these in.7, one obtains the solution to the boundary value problem in.3 and.4 in the form Ux; s = + exp αx ξ [exp ωξ x exp ωx ξ]fexp ξdξ exp αx ξ exp ωx ξfexp ξdξ which can be rewritten in a compact single-integral form as Ux; s = exp αx ξ exp ω x ξ fexp ξdξ.8 The solution ux, τ to the initial-boundary value problem stated by.5,. and. can be obtained from Ux; s with the aid of the inverse Laplace transform. In doing so, we keep in mind that the parameter ω has earlier been introduced in terms of the parameter s of the Laplace transform as ω = s + β /. This yields ux, τ = L {Ux; s} = = exp αx ξl { exp s + β/ x ξ s + β / }fexp ξdξ exp αx ξ exp βτ πτ / exp x ξ fexp ξdξ.9 To obtain the solution vs, t to the setting in.-.4, we make the backward substitutions in compliance with the relations of.6. This implies that the variables x, τ and ξ ought to be replaced with S, t and S, respectively as x = ln S, τ = σ T t, ξ = ln S The differential of the variable of integration ξ in.9 converts to the form dξ = S d S while the interval of integration, in.9 transforms, according to the relation ξ = ln S, to the interval [0, with respect to S. With all this in mind, one arrives at the solution to the terminal-boundary value problem in.-.4 as vs, t = 0 σ S[πT t] exp / α lns/ S β σ [lns/ S] T t σ f Sd S T t.0 revealing the Green s function to the problem in.-.4 in the form GS, t; S = S[πσ T t] exp / α lns/ S β σ [lns/ S] T t σ T t. It is not evident that the above representation for GS, t; S and the one in. are identical. To verify their identity, we express α and β in. in terms of the original parameters σ and r of the Black-Scholes equation as α = σ / r σ, β = r + σ / σ

5 EJDE-007/53 CONSTRUCTION OF GREEN S FUNCTIONS 5 and then rewrite. as GS, t; S = σ S[πσ T t] exp / r / σ lns/ S r + σ / [lns/ S] σ T t σ T t. Multiplying the above expression by the product of the two factors exp rt t exp rt t which is identically equal to one, we leave the first of these factors the negative exponent in its current form, combine the second factor with the existing exponential term in. and rewrite subsequently the latter as GS, t; S = exp rt t S[πσ T t] exp r σ / / σ lns/ S + rt t r + σ / [lns/ S] σ T t σ T t Combining the second and the third additive terms in the argument of the extended exponential function, we reduce the latter to the form exp r σ / σ lns/ S r σ / [lns/ S] σ T t σ T t which can immediately be transformed into exp [lns/ S] + r σ /T t lns/ S + r σ / T t σ T t It is evident that the numerator in the argument of this exponential function represents a complete square, reducing the above to exp [lns/ S + r σ /T t] σ T t Thus, the representation in. is, indeed, identical to that of.. This implies that the function that we came up with in. does really represent the Green s function to the terminal-boundary value problem in.-.4. In other words, our approach is proven productive, and in the next section we bring a convincing justification of its successful applicability to other terminal-boundary value problems for the Black-Scholes equation. 3. Other Green s functions Two particular terminal-boundary value problems with different boundary conditions imposed are considered as an illustration to the assertion made in the previous section.

6 6 M. Y. MELNIKOV, Y. A. MELNIKOV EJDE-007/ Dirichlet boundary conditions. As the first example, consider a terminalboundary value problem stated for. in the semi-infinite strip Ω = S < S < S T > t > of the S, t -plane. Let the terminal condition be given by.3, while the Dirichlet boundary conditions vs, t = 0, vs, t = 0 3. are imposed on the edges S = S and S = S of Ω. Note that the above setting for the Black-Scholes equation sounds quite practical for the financial engineering, whereas its Green s function is not yet available in literature. By the transformations of.6, the setting in.,.3 and 3. converts to the following initial-boundary value problem ux, 0 = fexp x, 3. ua, τ = 0, ub, τ = for.5 on the semi-infinite strip a < x < b 0 < τ < in the x, τ-plane, on which the region Ω maps by the change of variables introduced in.6. The parameters a and b are determined in terms of S and S as a = ln S, b = ln S The Laplace transform applied to the setting in.5, 3. and 3.3 converts the latter into the boundary value problem d Ux; s dux; s dx + c s + cux; s = fexp x, 3.4 dx Ua; s = 0, Ub; s = for the Laplace transform Ux; s of ux, τ. In compliance with the method of variation of parameters, the general solution to 3.4 is found, in this case, as exp αx ξ Ux, s = [exp ωξ x exp ωx ξ]fexp ξdξ a + Ms expα + ωx + Ns expα ωx 3.6 where the parameter ω is defined as ω = s + β /. Satisfying the boundary conditions of 3.5 yields the system of linear algebraic equations expα + ωa expα ωa Ms 0 = expα + ωb expα ωb Ns Ψs in Ms and Ns. Here Ψs = Solving the above system, we obtain a [expα ωb ξ expα + ωb ξ]fexp ξdξ expα ωa exp αb ξ Ms = a [exp ωa b exp ωb a] [exp ωξ b exp ωb ξ]fexp ξdξ

7 EJDE-007/53 CONSTRUCTION OF GREEN S FUNCTIONS 7 and expα + ωa exp αb ξ Ns = a [exp ωa b exp ωb a] [exp ωξ b exp ωb ξ]fexp ξdξ Upon substituting these in 3.6, the latter reads Ux, s = a exp αx ξ [exp ωξ x exp ωx ξ]fexp ξdξ exp αx ξ[exp ωx a exp ωa x] + a [exp ωa b exp ωb a] [exp ωξ b exp ωb ξ]fexp ξdξ which can be expressed in a single-integral form as exp αx ξ Ux; s = a [exp ωa b exp ωb a] { exp ω[x + ξ a + b] + exp ω[a + b x + ξ] } exp ω[ x ξ + a b] exp ω[b a x ξ ] fexp ξdξ Transforming the bracket factor in the denominator as exp ωa b exp ωb a = exp ωb a[ exp a b] we rewrite the above representation for Ux; s as exp αx ξ Ux; s = a exp ωb a[ exp a b] {exp ω[x + ξ a + b] + exp ω[a + b x + ξ] exp ω[ x ξ + a b] exp ω[b a x ξ ]}fexp ξdξ 3.7 The inverse Laplace transform of Ux; s is problematic if the latter is kept in its current form. Therefore, we adjust it first by representing the factor [ exp a b] in the integrand of 3.7 as a geometric series, exp a b = exp nωa b whose common ratio exp a b represents a negative exponential function a < b and is, therefore, less than one. This transforms 3.7 to exp αx ξ Ux; s = {exp ω[ x ξ n + b a] a n=0 n=0 + exp ω[n + a b x ξ ] exp ω[na b b + x + ξ] exp ω[na b + a x + ξ]}fexp ξdξ and the inverse Laplace transform of the above can be accomplished in the term-byterm manner. This yields the solution ux, τ to the initial-boundary value problem

8 8 M. Y. MELNIKOV, Y. A. MELNIKOV EJDE-007/53 in.5, 3. and 3.3 in the form ux, τ = L {Ux, s} exp αx ξ exp βτ = a πτ / n=0 [ x ξ na b] + exp exp { exp [x + ξ a na b] exp [ x ξ + n + a b] [b x + ξ na b] } fexp ξdξ which converts to a more compact form by rearranging the summation in the above series. This yields ux, τ = a exp αx ξ exp βτ πτ / exp m= [b x + ξ ma b] { exp [ x ξ + ma b] } fexp ξdξ In compliance with the relations in.6, the solution vs, t to the setting in.,.3 and 3. can be attained by the backward replacement of the variables x, τ and ξ with S, t and S, respectively. Similarly to the analogous replacement that has been performed in Section with α and β replaced with the original parameters r and σ of the Black-Scholes equation, we obtain vs, t in the form vs, t = S S exp r σ / σ lns/ S r+σ / σ T t S[πσ T t] / m= { exp [lns/ S + m lns /S ] σ T t exp [lns /S S m lns /S ] } σ f Sd S T t which can be transformed, by combining the logarithmic components in the series factor. This yields vs, t = S S exp r σ / σ lns/ S r+σ / σ T t S[πσ T t] / m= { exp [lnssm m / SS σ T t exp [lnsm+ m /S SS σ T t ] ] } f Sd S 3.8

9 EJDE-007/53 CONSTRUCTION OF GREEN S FUNCTIONS 9 Thus, the kernel in the above integral, GS, t; S = exp r σ / σ lns/ S r+σ / σ T t S[πσ T t] / m= { exp [lnssm m / SS σ T t exp [lnsm+ m /S SS σ T t ] ] }, 3.9 represents the Green s function to the setting in.,.3 and 3.. The series in this representation converges at a high rate unless the term T t is very small. This implies that, in computing values of GS, t; S, an accuracy level required for applications can, in most cases, be attained by appropriately truncating the series in 3.9 to a partial sum. 3.. Mixed boundary conditions. Note that the qualitative theory of partial differential equations [5] claims that if GS, t; S is the Green s function to the setting in, say,.,.3 and 3., then the solution to this problem can be written in the integral form vs, t = S S GS, t; S f S d S 3.0 This observation determined our strategy in the development of Sections and 3. The strategy can also be applied while obtaining a Green s function to the setting in. and.3, with the following boundary conditions v0, t <, vd, t S + ϱvd, t = 0, ϱ 0 3. imposed on the boundary fragments S = 0 and S = D of the semi-infinite strip Ω = 0 < S < D T > t >. Indeed, if we manage to find the solution to the problem in.,.3 and 3. in an integral form like that in 3.0, then the kernel of the integral represents the Green s function that we are looking for. The second condition in 3. is referred to, in mathematical physics, as either mixed or Robin type. To our best knowledge, mixed boundary conditions have never been considered yet in association with the Black-Scholes equation. It is even unclear if such problem settings are timely for financial engineering. But from mathematics stand-point, they do not look unfeasible and could possibly find realistic applications in the field of finance in years to come. Upon introducing new variables x and τ as suggested in.6, one converts the setting in.,.3 and 3. to the initial-boundary value problem u, τ <, ux, 0 = fexp x 3. ub, τ x + ϱub, τ = for.5 on the quarter-plane < x < b 0 < τ <. The parameters b and ϱ in 3. are defined in terms of the initial data in the original problem as b = ln D, ϱ = Dϱ

10 0 M. Y. MELNIKOV, Y. A. MELNIKOV EJDE-007/53 Applying the Laplace transform to the problem in.5, 3. and 3.3, one arrives at the boundary value problem U; s <, dub; s + ϱub; s = 0 dx 3.4 for the equation in 3.4. Aiming at the solution to the problem in.,.3 and 3. in an integral form, we apply the method of variation of parameters to the problem in 3.4 and 3.4. This gives the general solution of 3.4 in the form Ux; s = exp αx ξ[exp ωξ x exp ωx ξ]fexp ξdξ + Ms expα + ωx + Ns expα ωx 3.5 To determine the functions Ms and Ns, we take advantage of the boundary conditions of 3.4. When x approaches negative infinity, the integral component in 3.5 vanishes, while the Ms-containing component approaches zero. Hence, for the first condition in 3.4 to hold, Ns ought to be zero Ns = because the exponential factor in the Ns-containing component in 3.5 is unbounded as x approaches negative infinity. In light of 3.6, the derivative of Ux; s reads as dux; s dx = [α ω exp ωξ x α + ω exp ωx ξ] exp αx ξfexp ξdξ + Msα + ω expα + ωx So, the second condition in 3.4 yields the following equation in Ms, [α ω exp ωξ b α + ω exp ωb ξ] exp αb ξfexp ξdξ + Msα + ω expα + ωb + ϱ exp αb ξ[exp ωξ b exp ωb ξ]fexp ξdξ + ϱms expα + ωb = 0 from which Ms is found as Ms = ϱ + α ω [ exp ωξ b exp ωb ξ] exp αξ ωbfexp ξdξ ϱ + α + ω Substituting now the above expression for Ms in 3.5 and taking into account 3.6, one obtains the solution to the boundary value problem in 3.4 and 3.4 as Ux; s = exp αx ξ[exp ωξ x exp ωx ξ]fexp ξdξ ϱ + α ω [ exp ωξ b exp ωb ξ] ϱ + α + ω exp αx ξ exp ωx bfexp ξdξ

11 EJDE-007/53 CONSTRUCTION OF GREEN S FUNCTIONS To obtain the inverse Laplace transform of Ux; s, ux, τ = L {Ux; s} which represents the solution to the initial-boundary value problem in.5, 3. and 3.3, we simplify the above expression for Ux; s. Proceeding through a tedious but quite straightforward algebra, one obtains a more compact form for Ux; s as Ux; s = [exp ω x ξ exp αx ξ fexp ξdξ ϱ + α ω exp ωx + ξ b] ϱ + α + ω 3.7 which is not, unfortunately, convenient yet for the immediate inverse Laplace transform. To facilitate the latter, we rewrite Ux; s in the equivalent form Ux; s = { ϱ + α } exp ω x ξ [ ϱ + α + ω ] exp ωx + ξ b exp αx ξ fexp ξdξ or, recalling the expression for ω in terms of the parameter s of the Laplace transform, the above reads as exp αx ξ { exp s + β / x ξ Ux; s = s + β / Φ [ Φ + s + β / ]exps + β/ x + ξ b } fexp ξdξ s + β / 3.8 where we introduced, for compactness, Φ = ϱ + α. The inverse Laplace transform of Ux; s from 3.8 represents the solution ux, τ to the initial-boundary value problem in.5, 3. and 3.3. It is found in the form { [ ux, τ = exp x ξ x + ξ b ] + exp πτ / Φ expφ τ Φx + ξ b erfc Φτ / x + ξ b } τ / exp αx ξ exp βτfexp ξdξ 3.9 where the erfc represents the complementary error function erfcϕ = π / ϕ e x dx. The solution vs, t to the terminal-boundary value problem in.,.3 and 3. can be obtained from 3.9 by making the backward substitution of the

12 M. Y. MELNIKOV, Y. A. MELNIKOV EJDE-007/53 variables in compliance with the relations of.6. This implies D vs, t = 0 S exp α ln S S β σ T t { [ [lns/ S] exp [πσ T t] / σ + exp [lns S/D ] ] T t σ T t Φ expφ σ T t/ Φ lns S/D erfc Φ [σ T t] / lns S/D }f Sd S [σ T t] / From this, one arrives at a conclusion that the kernel GS, t; S of the above integral represents the Green s function to the problem in.,.3 and 3.. After a trivial algebra, GS, t; S can be presented in the form GS, t; S = S S α σ exp β T t S { [ [lns/ S] exp [πσ T t] / σ + exp [lns S/D ] ] T t σ T t Φ S S Φ expφ D σ T t/ Φ erfc [σ T t] / lns S/D } [σ T t] / 3.0 Summarizing all the notations introduced at various stages of the present development, we can express the parameters α, β and Φ in 3.0 in terms of the original parameters σ and r of the Black-Scholes equation and the parameters D and ϱ as α = σ / r r + σ /, σ, β = Φ = Dϱ + α 3. σ 3.3. Particular cases. Note that the problem statement in.,.3 and 3. allows two particular cases that might be of interest in option pricing valuations. One of such cases occurs when the parameter ϱ is set to equal zero transforming the boundary conditions in 3. into vd, t v0, t <, = 0 3. S The Green s function for the problem in.,.3 and 3., GS, t; S = S S α σ exp β T t S { [ [lns/ S] exp [πσ T t] / σ + exp [lns S/D ] ] T t σ T t α S S D α expα σ T t/ α erfc [σ T t] / lns S/D } [σ T t] / 3.3

13 EJDE-007/53 CONSTRUCTION OF GREEN S FUNCTIONS 3 immediately arises from that of 3.0 when the parameter Φ is replaced with α. Indeed, as it follows from 3., if ϱ = 0, then Φ = α. The second particular case of the problem statement in.,.3 and 3. occurs when the parameter ϱ approaches infinity, which transforms the boundary conditions in 3. into v0, t <, vd, t = It is difficult to directly obtain Green s function to the terminal-boundary value problem in.,.3 and 3.4 from that of 3.0. The point is that taking a limit in the latter as ϱ approaches infinity is not a trivial exercise. That is why an alternative route is suggested. We revisit 3.7 for Ux; s in the development of the previous section and, observing that ϱ + α ω lim ϱ ϱ + α + ω = lim Dϱ + α ω ϱ Dϱ + α + ω = we rewrite 3.7 for the setting in.,.3 and 3.4 as Ux; s = = exp αx ξ [exp ω x ξ exp ωx + ξ b]fexp ξdξ exp αx ξ [ exp s + β/ x ξ s + β / exps + β/ x + ξ b s + β / ]fexp ξdξ Taking the inverse Laplace transform of the above expression, we obtain ux, τ = [exp x ξ x + ξ b exp ] exp αx ξ exp βτ fexp ξdξ πτ / allowing the solution to the problem in.,.3 and 3.4 as D vs, t = S [ πσ T t ] exp α ln S S β σ / T t that can easily be simplified to vs, t = 0 [ [lns/ S] exp σ exp T t D 0 [lns S/D ] σ T t S S S α exp βσ T t/ [πσ T t] / [lns/ S] σ exp [lns S/D ] T t σ T t [ exp from which it follows that GS, t; S = S S S α exp βσ T t/ [ exp [πσ T t] / [lns/ S] σ T t exp ] f Sd S ] f Sd S [lns S/D ] ] σ T t

14 4 M. Y. MELNIKOV, Y. A. MELNIKOV EJDE-007/53 represents the closed form of the Green s function to the Black-Scholes equation satisfying the boundary conditions in 3.4. Note that the relations in 3. bring the expressions of the parameters α and β in terms of σ and r. Conclusion. Compact analytic representations are derived for Green s functions for the Black-Scholes equation. These and other Green s functions, whose compact forms can be obtained by the approach suggested in the present study, are easily accessible for both theoretical analysis and numerical work in the field. They can readily be used in solving a variety of practical problem settings in financial engineering. References [] F. Black, M. S. Scholes; The pricing of options and corporate liabilities. Journal of Political Economics, 8, [] Y. A. Melnikov, Influence Functions and Matrices New York Basel: Marcel Dekker, 998, [3] S. N. Neftci; An Introduction to the Mathematics of Financial Derivatives New York: Academic Press, 000. [4] D. Silverman; Solution of the Black-Scholes equation using the Green s function of the diffusion equation. Manuscript. Department of Physics and Astronomy. University of California. Irvine, 999, [5] V. I. Smirnov; A Course of Higher Mathematics Oxford New York: Pergamon Press, 964. [6] P. Wilmott, S. Howison, J. Dewynne; The Mathematics of Financial Derivatives: A Student Introduction Cambridge: Cambridge University Press, 995. Max Y. Melnikov Labry School of Business and Economics, Cumberland University, Lebanon, TN 37087, USA address: mmelnikov@cumberland.edu, Phone Yuri A. Melnikov Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN 373, USA address: ymelniko@mtsu.edu, Phone , Fax