On the consistency of finite difference approximations of the Black Scholes equation on nonuniform grids

Transcription

1 INVOLVE :49 On the consistency of finite difference approimations of the Black Scholes equation on nonuniform grids Myles D. Baker and Daniel D. Sheng Communicated by Johnny Henderson The Black Scholes equation has been used for modeling option pricing etensively. When the volatility of financial markets creates irregularities, the model equation is difficult to solve numerically; for this reason nonuniform grids are often used for greater accuracy. This paper studies the numerical consistency of popular eplicit, implicit and leapfrog finite difference schemes for solving the Black Scholes equation when nonuniform meshes are utilized. Mathematical tools including Taylor epansions are used throughout our analysis. The consistency ensures the basic reliability of the finite difference schemes based on choices of temporal and variable spatial derivative approimations. Truncation error terms are derived and discussed, and numerical eperiments using C, C++ and Matlab are given to illustrate our discussions. We show that, though orders of accuracy are lower compared with their peers on uniform grids, nonuniform algorithms are easy to implement and use for turbulent financial markets.. An introduction of the differential equation and nonuniform grids Let f = f, t be the value of an option in the option market. In [Brandimarte 6; Wilmott et al. 995], for eample, we find the linearized Black Scholes equation B f, t = f t, t + α f f, t + β, t γ f, t =, t, - where t is time, is the asset price or space variable, and B f is the so-called Black Scholes operator, epressed in terms of partial derivatives. The constants MSC: primary 65G5, 65L, 65N6, 65N5; secondary 65L7, 65L8, 65D5. Keywords: finite difference approimation, difference algorithm, eplicity, implicity, consistency, accuracy, matri equations. This collaborated research was supported by a Undergraduate Research and Scholarly Achievement Grant no. 3-8-URSA from Baylor University. 479

2 48 MYLES D. BAKER AND DANIEL D. SHENG α, β and γ are nonnegative; they represent important parameters in economic and financial calculations. Let T > be a sufficiently large number. We consider a rectangular space-time domain D where, t T for -. We further adopt an initial option value distribution f, = sinaπ,, - as well as the Dirichlet boundary conditions f, t = sinbπt, f, t = sinaπt +, t >, -3 where a and b are constants. Since conventional difference approimations are consistent to the first derivative [Jain and Sheng 7; Urban et al. 4], we may adopt a uniform grid in the temporal direction, while maintaining nonuniform discretization in the -direction. Let τ > be the temporal step size used, and h, h, h,..., h n the spatial step sizes, where in general and h i = h i+, i {,,,..., n }, n h k =. k= Thus, a two-dimensional nonuniform grid D h,τ = { i, t j : i = i + h i, i =,,..., n + ; =, n+ = } is a discrete set over the domain D. Any P i, j = i, t j D h,τ is called a grid point of D h,τ. It is an internal grid point if i =, n+ and j =, a boundary point if i {, n + } and an initial point if j =. In Figure we show a particular two-dimensional nonuniform grid. In the design of nonuniform grids, we define the smoothness ratios r k = h k h k, k =,,..., n, and we avoid etreme r k values since they may cause nonphysical oscillations of the numerical solutions inconsistent with the assumption that - adheres to the principles of geometric Brownian motion [Sheng 8; Wilmott et al. 995, 3.5, pp. 4 43]. For concreteness and simplicity, we will fi the parameter values throughout our investigations. α =, β =, γ = -4

3 FINITE DIFFERENCE APPROXIMATIONS OF THE BLACK SCHOLES EQUATION 48 t Τ j t J { t j Τ j { f 4, t j f i, t j f i, t J t j t j f, t j t j 3 f, t j 3 t 3 f, t 3 t Τ Τ { t { f, t 3 4 i i h h h i h Figure. An illustration of the nonuniform grid D h,τ. A temporal step τ =.5 is used. Particular spatial steps h =.5, h =.8, h =.7, h 3 =.6, h 4 =., h 5 =., h 6 =., h 7 =., h 8 =.8, h 9 =.9, h =.8, h =., h =.7, h 3 =. are used.. The eplicit scheme An eplicit scheme is an algorithm which can be eecuted readily, or straightforwardly, without using a system of nonlinear solvers [Atkinson and Han 4; Sheng 8; Smith 985]. In our case, unknowns can be evaluated by first finding the initial values - and boundary values to the left and right -3, and then using those values to arrive at the targeted f i, j. We note that the scheme is being taken along a nonuniform grid rather than a uniform grid. In order to find values in the net temporal level, a recursive formula needs to be implemented. The application of computer software is very helpful in computing these f values, where else we would never have been able to see comple numerical results.

4 48 MYLES D. BAKER AND DANIEL D. SHENG Denote the function value f i, t j by f i, j. From the structure of D h,τ, we know that for any internal grid point P i, j we have i = h + h + + h i, i {,,..., n}; t j = jτ, j >. According to [Atkinson and Han 4, 4., p. 37; Jain and Sheng 7], at P i, j we have f f i, j+ f i, j, - t i, j τ f i,, f i, j = D,+ f i, j D, f i, j j h i + h i / = h i h i + h i f i+, j /h i + /h i h i + h i / f i, j + h i h i + h i f i, j = h i h i + h i f i+, j f i, j + h i h i h i h i + h i f i, j - f f i+, j f i, j. -3 i, j h i + h i Original concepts of the finite differences - -3 can be traced back to calculations of variations in L p spaces and beyond [Fonseca and Leoni 7,..]. Recall our choices α =, β =, γ =. If we substitute - -3 into - and remove the error terms, we acquire an eplicit finite difference equation at P i, j : B h,τ f i, j = f i, j+ f i, j τ f i, j h i + h i h i f i, j + + f i+, j =, h i h i h i + h i h i where B h,τ can be viewed as a discretized Black Scholes operator. The above can be reformulated to the recursive formula [ f i, j+ = f i, j + σ i f i, j + h i + h i f i, j + f i+, j ], h i h i h i h i τ σ i =, h i + h i or f i, j+ = σ i f i, j + + τ h i h i h i σ i = τ h i + h i, f i, j σ i + h i f i+, j, -4 which runs for the temporal level inde j from to J, as far as Jτ T. The numerical solution can thus be derived from the recursive relation -4 together

5 FINITE DIFFERENCE APPROXIMATIONS OF THE BLACK SCHOLES EQUATION 483 with the initial and boundary conditions -, -3. This eplicit finite difference scheme -4, -, -3 is originally presented in [Sheng 8]. To analyze -4 we need the following definitions: Definition. Order of accuracy. Consider a discretized Black Scholes operator B h,τ, where < h, τ <. Assume that the function f, t is sufficiently smooth no cusps or discontinuities of partial derivatives up to a desired order over D. If ma B f i, t j B h,τ f i, j = Oh p + τ q, -5 i,t j D h,τ where h = ma{h, h,..., h N }, we say that the difference scheme defined by B h,τ is an order- p, q scheme for solving the Black Scholes equation -. We also say that B h,τ is an order-r scheme, where r = min{p, q}. A difference method is practically meaningful when both p and q are positive. Definition.. A difference method B h,τ is consistent if it has order r >. There is always an error associated with a finite difference approimation. If we use Taylor s Theorem to epand certain known finite difference approimations along difference schemes, such as -4, we can empirically prove that these approimations may or may not be useful when applied to the Black Scholes equation. For this purpose, we need the following notion: Definition.3. The truncation error function of the difference scheme is defined as err f i, j = B f i, t j B h,τ f i, j, i, t j D h,τ. Theorem.4. For the eplicit finite difference scheme -4 we have the truncation error estimate err E f i, j = Oh + τ, where h = ma{h, h,..., h N }. Therefore the eplicit scheme -4 is of first order. If the local spatial grid is uniform, that is, h i = h i = h >, the scheme -4 is locally of second order in space: err E f i, j = Oh + τ. Proof. We use the so-called forward, central and modified central difference operators, defined respectively by t f = f t i, j+ f t i, j τ i, j+ τ i, j, δ f = f i+, j f i, j h i+, j h i, j,, f = f f h i, j +h i, j /, for i X, j T. In the last epression is defined like t, and the backward spatial difference operator is similar, but with all spatial indices decreased by. In this notation we obtain from -4

6 484 MYLES D. BAKER AND DANIEL D. SHENG err E f i, j = f t t f i, j + α f, f i, j + β f δ f i, j. -6 We have see [Jain and Sheng 7] for details f t t f i, j = τ f tt i, t j 6 τ f ttt i, t j +, f, f i, j = 3 h i h i f i, t j h i h ih i + hi f 4 i, t j, f δ f i, j = h h i + h i i f i, t j hi f i, t j + = h i h i f i, t j + 6 h i h ih i +h i f i, t j +. Thus the lowest-order terms are linear in ma{h i, h i+ } and τ j ; but when h i = h i+ = h, the linear contributions in h drop out. 3. The implicit scheme Instead of the forward finite difference -, we may consider the backward difference formula f = f i, j f i, j + Oτ. 3- t i, j τ Substitution of 3-, -, and -3 into - yields f i, j f i, j = f i, j τ h i + h i h i + h i h i f i, j + f i+, j, h i + h i h i which is significantly different from the eplicit scheme -4. For later convenience we replace the inde j by j + and j by j. Our difference equation then becomes f i, j+ f i, j τ = f i, j+ h i + h i h i which can further be written as σ i τ f i, j+ h i h i h i + f i, j+ + f i+, j+, h i h i h i + h i h i f i, j+ +σ i + h i f i+, j+ = f i, j, 3- where σ i is the same as defined before. The implicit finite difference algorithm 3-, together with conditions -, -3, is studied in [Sheng 8]. Equation 3- cannot be solved independently without collaboration between the rest of the equations at the temporal level j +. We note that there are n internal

7 FINITE DIFFERENCE APPROXIMATIONS OF THE BLACK SCHOLES EQUATION 485 grid points. Thus we have n difference equations at each of the temporal levels, and we get a linear system of size n, which can be epressed in the matri form as M f j+ = g j+, j {,,..., J}, 3-3 where M is a tridiagonal matri with nontrivial elements For the vectors we have m i,i+ = σ i + /h i, i =,,..., n, m i,i = τ/h i h i, i =,,..., n, m i,i = σ i /h i, i =, 3,..., n. f j+ i = f i, j+, i =,,..., n, g j+ = f, j σ /h f, j+, g j+ i = f i, j, i =, 3,..., n, g j+ n = f n, j + σ n + /h n f n+, j+, where the values of f, j+ and f n+, j+ are given by the condition -3. The tridiagonal system of linear equations 3-3 can be solved conveniently by using a special subroutine in C and Matlab; see [Jain and Sheng 7; Pratap 999; Sheng 8]. Theorem 3.. For the implicit finite difference scheme 3- or 3-3 we have the truncation error estimate err I f i, j = Oh + τ, where h = ma{h, h,..., h N }. Therefore the implicit scheme is of first order. If the local spatial grid region is uniform, that is, h i = h i = h >, the scheme is locally of second order in space: Proof. We have err I f i, j = Oh + τ. err I f i, j = f t t f i, j + α f, f i, j + β f δ f i, j. 3-4 The α and β terms are as in the proof of Theorem.4, while for the remaining term we have from [Jain and Sheng 7], for instance. f t t f i, j = τ f tt i, t j 6 τ f ttt i, t j +. Substitution into 3-4 and consideration of the special case h i = h i yields the result.

8 486 MYLES D. BAKER AND DANIEL D. SHENG 4. The leapfrog scheme In this much more sophisticated approach, we first replace - by the central difference formula f = f i, j+ f i, j + Oτ. 4- t i, j τ We immediately notice the increase in the order of approimation. Now, instead of -, -3 for the spatial derivatives, we consider the average formulas f i, j = = D,+ f i, j D, f i, j h i + h i / h i h i + h i f i+, j + f i+, j+ + + D,+ f i, j+ D, f i, j+ +Oh h i + h i / f i, j + f i, j+ h i h i h i h i + h i f i, j + f i, j+ + Oh, 4- f = fi+, j f i, j + f i+, j+ f i, j+ + Oh, 4-3 i, j h i + h i h i + h i where h = ma k {h k }. The order of approimation of the spatial derivatives is still due to the nonuniform grid. Substitution of into - yields f i, j+ f i, j τ = f i, j + f i, j+ + f i, j + f i, j+ h i + h i h i h i h i Let τ σ i =. h i + h i Then the difference equation can be rearranged as f i, j+ = f i, j + σ i f i, j + f i, j+ h i which leads to σ i τ f i, j+ + h i h i h i = σ i h i f i, j + f i+, j + f i+, j+. h i + h i h i + τ h i h i f i, j + f i, j+ σ i + h i f i+, j + f i+, j+, f i, j+ σ i + h i f i+, j+ τ h i h i + f i, j + σ i + h i f i+, j. 4-4

9 FINITE DIFFERENCE APPROXIMATIONS OF THE BLACK SCHOLES EQUATION 487 Like3-, the system of linear equations 4-4 can be put into matri form: P f j+ = Q f j + s j+, 4-5 where P, Q are n n tridiagonal matrices and s j+ can be determined via the boundary condition -3. Therefore 4-5 can be readily solved by computers [Atkinson and Han 4; Sheng 8; Smith 985]. The leapfrog scheme 4-4, or 4-5, is implicit since it must be solved as a linear system. A peculiarity of the leapfrog method is that if we start from the initial value at t =, we can only obtain the numerical solution of - -3 on even temporal levels. To compute numerical solutions on odd temporal levels, we need solution values on the first temporal level, which can be generated by using one step via either the eplicit method or implicit method. Theorem 4.. For the leapfrog implicit finite difference scheme 4-4 or 4-5 we have the truncation error estimate: err L f i, j = Oh + τ, where h = ma{h, h,..., h N }. Therefore the leapfrog scheme is of first order in space and second order in time. If the local spatial grid region is uniform, that is, h i = h i = h >, the leapfrog scheme becomes a second order method locally: err L f i, j = Oh + τ. Proof. For the leapfrog scheme 4-4 or 4-5, err L f i, j = f t δ t f i, j + α f i, j, f i, j +, f i, j+ + β f i, j δ f i, j + δ f i, j Again, according to [Jain and Sheng 7; Sheng 8], we have f t δ t f i, j = 6 τ f ttt i, t j τ 4 f t 5 i, t j. 4-7 Note that, since the grid distribution in space is irregular, 4-7 cannot be etended for estimating the difference f i, j δ f i, j + δ f i, j+. Instead, employing the epansion δ f i, j = f i, j + h i h i f i, j + 6 h i h ih i +h i f i, j +

10 488 MYLES D. BAKER AND DANIEL D. SHENG and performing simplifications, we obtain f i, j δ f i, j + δ f i, j+ = 4 h i h i f i, j + f i, j+ h i h ih i + h i f i, j + f i, j+ + Oh For the α term in 4-7 we will assume for simplicity that the grids { i } remain the same on different temporal levels. Using the epansion, f i, j = 3 h i h i 3 f i, j + h i h ih i + h i f 4 i, j + and simplifying, we obtain f i, j, f i, j +, f i, j+ = 6 h i h i f i, j + 3 f i, j + f i, j+ 4 h i + h i f 4 i, j + 6 f 4 i, j + f i, j+ + 4 h ih i f 4 i, j + f 4 i, j+ + Oh Substituting into 4-6 we get for err L f i, j the epression [ 6 τ f ttt i, t j + α 6 h i h i f i, j + 3 f i, j + f i, j+ + 4 h i + h i f 4 i, j + 6 f 4 i, j + f 4 i, j+ + 4 h ] ih i f 4 i, j + f 4 i, j+ [ + β 4 h i h i f i, j + f i, j+ + h i h ih i + hi ] f i, j + f i, j+ + Oh 3 + τ 4, which is generally of first order in space and second order in time, but becomes of second order in both time and space when h i = h i. Since a leapfrog scheme spans three temporal levels, the computation using 4-4 or 4-5 can be started initially. One strategy is to use an implicit or an eplicit scheme for calculating the numerical solution at the first temporal level, that is, when j =. Then, by using the numerical solutions at temporal levels and, a leapfrog scheme can generate solutions at higher temporal levels. However, this treatment may reduce the overall order of accuracy, given that an eplicit or implicit method is used to generate the solution on the first temporal level. The computer club members had several fruitful discussions on the issue. At the end, we realized that such a computation is not necessary. Let τ be halved. Why not collect numerical solutions on the even number of temporal levels only?

11 FINITE DIFFERENCE APPROXIMATIONS OF THE BLACK SCHOLES EQUATION 489 This will guarantee our overall accuracy. We may introduce imaginary middle temporal grid points [Atkinson and Han 4; Sheng 8; Smith 985] in the analysis, which may help to retain the overall second order accuracy in time t. We will report details elsewhere. 5. Simulation results In our numerical eperiments, we consider the following simplified Black Scholes initial-boundary value problem [Brandimarte 6]: f t, t = f f, t, t,, t, 5- f, = sin.π,, 5- f, t =.5 sin7πt, f, t = sin.πt, t >. 5-3 For the sake of simplicity, we will only provide numerical results from the eplicit scheme -4. Results from the other two schemes are similar. Our nonuniform grid region is designed as follows: Let N = and use C and Matlab programs to generate N random numbers, < < 3 < < N, on the interval,. Denote =, N+ =. We have the set of nonuniform spatial step sizes: h i = i i, i =,,..., N +. Now, set τ = /. Thus a nonuniform two-dimensional grid region is completed. In Figure, we show the spatial step sizes across the interval [, ]. We also show the ratio distribution of the neighboring spatial step sizes, h i+ /h i, i =,,..., N, random mesh step distribution radio of random mesh steps size of steps ratio of steps number of steps number of steps Figure. Left: distribution of the random spatial step sizes. Right: ratio function value distribution across the interval [, ].

12 49 MYLES D. BAKER AND DANIEL D. SHENG since the values are important in adaptive computations [Jain and Sheng 7]. Note the large peak value of the ratio function near i = 8, eemplifying the socalled nonsmoothness phenomenon of random spatial grids [Jain and Sheng 7; Sheng 8]. Because the step sizes are random, each numerical eperiment with the same Black Scholes problem yields a different Figure. We choose the smooth test function f, t = sinπ e.t, as in [Atkinson and Han 4; Jain and Sheng 7]. Its partial derivatives are readily calculated. Figure 3 shows illustrative plots of the finite difference approimations for f and its derivatives. The D, formula is used in approimating the second derivative since repeated use of conventional first order differences does not yield a consistent approimation of the second derivative [Fonseca and Leoni 7; Jain and Sheng 7]. For ease of comparison, all function surfaces are plotted from the same viewpoint. eperiment eplicit scheme.8.5 derivative ft testing function f eperiment eplicit scheme eperiment eplicit scheme.5 t t eperiment eplicit scheme 4 3 derivative f derivative f t.5 t Figure 3. Clockwise from top left: graphs of f, t, f t, t, f, t similar to f t, t and f, t. Second derivatives use the D, formula.

13 49 FINITE DIFFERENCE APPROXIMATIONS OF THE BLACK SCHOLES EQUATION eperiment eplicit scheme eperiment eplicit scheme relative error relative error t t Figure 4. Left: three-dimensional distribution of the relative numerical error. Right: projection of the graph onto the X Z plane. Consider the eplicit scheme -4. To check the numerical truncation error of the algorithm, we submit the function f, t into -6 and evaluate the outcome on all internal nonuniform grid points. The relative error is adopted for a more reasonable evaluation of the errors [Atkinson and Han 4]. Figure 4 gives the error distribution over the nonuniform grids of the domain, t. In the first figure, we show a three-dimensional relative error distribution. The second figure represents the numerical error projected onto the X Z plane so that we can view more clearly the oscillatory features of the error pattern probably due to the random spatial steps used. The overall numerical error is oscillatory but small. The maimal relative error appears to be approimately.45, that is, 4.5%, which is satisfactory. It is interesting that the error pattern does not seem to be similar to the mesh pattern given in Figure, though they must be related. A more in-depth numerical analysis may be required to reveal their internal connections. The numerical eperiments for estimating the order of convergence is more comple and tricky too, since the feature of two-dimensional problems and the nonuniform grids used. Our eperiments are based on the following stages: For a given testing function f, t, let the numerical truncation error function be err i, j. Since h i = Oτ, we may assume that err i, j Mτ p, 5-4 where M is a positive constant, for all valid indees i, j. Halve both the spatial and temporal step sizes. This is easy to achieve in time but relatively tricky in space. For the sake of simplicity, we may halve each of the h i generated, although this yields pairs of identical step sizes.

14 49 MYLES D. BAKER AND DANIEL D. SHENG eperiment eplicit scheme 4 4 computed order p computed order p t Figure 5. Left: three-dimensional distribution of the order function p. Right: contour map of the function p over the region, t. 3 Repeat the computation on the refined grid region. Suppose that the new numerical truncation error function is err i, j, where the indees are taken only on grid points where err i, j is defined. Thus, we have err i, j Mτ/ p 5-5 for all such indees i, j. 4 From 5-4, 5-5 we deduce that err i, j err p, which offers an estimate i, j p ln ln err i, j err. 5-6 i, j Evidently, such a p is also a function of i, j. Therefore an average value of such p values would provide a more reasonable estimate of the order for the underlying numerical method. The process may be repeated by halving the step sizes again. However, note that 5-6 provides only an estimate which can be used as a reference. Figure 5 demonstrates a solution surface of the p values obtained via 5-6. It is interesting to notice that ma i, j p = 3.435, min i, j p =.7357, average p = 8.3. The numerical results seem to be much higher than the linear truncation error predicted by Theorem.4 in the situation. Most of our randomly chosen -grids have demonstrated a similar conclusion. However, since the testing function is artificially chosen and the grid points in the -direction are randomly chosen, we cannot conclude that the actual truncation order is much better than predicted. The

15 FINITE DIFFERENCE APPROXIMATIONS OF THE BLACK SCHOLES EQUATION 493 eperiment results only indicate a strong possibility that our numerical scheme may behave better than anticipated. 6. Conclusion Financial confidence is of the utmost importance when making investments, and in this paper we analyze the consistency of eplicit, implicit and leapfrog finite difference schemes for solving Black Scholes partial differential equation. We show the potential these numerical methods have for making accurate predictions of the option values when nonuniform discretizations are necessary. By using the, difference formula, we prove that all difference methods developed are consistent. While the eplicit and implicit schemes provide a truncation error of the order Oτ +h, the leapfrog scheme offers a higher order Oτ +h. More precise error estimates for the three numerical methods are delivered in the corollaries for closer comparisons. Numerical eperiments based on the eplicit finite difference scheme have demonstrated a satisfactory result, indicating their good potential use in real-world implementations. The orders of approimation can be further improved, but, the use of more sophisticated finite difference formulas may lead to complicated numerical schemes which are either difficult to use or difficult to analyze. The benefit of order improvement may be limited in actual financial computations, though its theory in numerical investigations is always meaningful. The eploration of better, higherorder numerical schemes for solving the Black Scholes equation is one of the goals in our forthcoming study. There are many interesting problems to be eplored for the finite difference schemes implemented. A particularly relevant issue is numerical stability in the von Neumann sense [Atkinson and Han 4; Sheng 8; Smith 985]. This concern focuses on the question: once a tiny error is introduced during computations, will it affect significantly further option values of f? If such a consequence is unavoidable, then is there a strategy to reduce the damage? We prefer to leave the answers to our forthcoming investigations. We also encourage the reader to eplore any possible solutions. Acknowledgments The authors are grateful to Dr. Johnny Henderson and Dr. Lance Littlejohn, professors of mathematics at Baylor University, and Mr. Andrew D. Sheng, undergraduate student of computer science at Carnegie Mellon University, for reading our manuscripts, constructing and eperimenting many programs via C, C++ and Matlab languages, and for sharing important research notes and thoughts. Our

16 494 MYLES D. BAKER AND DANIEL D. SHENG special thanks go to Baylor University for its Undergraduate Research and Scholarly Achievement URSA grant, which enabled this collaborative research. We also thank our project advisor, Dr. Qin Sheng, professor of mathematics at Baylor University, for suggesting this line of undergraduate research and for the encouragement and numerous discussions throughout the project. Last, but not least, we appreciate our reviewers for their comments and suggestions that helped to improve the contents of this paper. References [Atkinson and Han 4] K. Atkinson and W. Han, Elementary numerical analysis, 3rd ed., Wiley, Somerset, NJ, 4. Zbl [Brandimarte 6] P. Brandimarte, Numerical methods in finance and economics: A MATLABbased introduction, nd ed., Wiley, Hoboken, NJ, 6. MR 7d:9 Zbl 9.9 [Fonseca and Leoni 7] I. Fonseca and G. Leoni, Modern methods in the calculus of variations: L p spaces, Springer, New York, 7. MR 8m:49 Zbl [Jain and Sheng 7] B. Jain and A. D. Sheng, An eploration of the approimation of derivative functions via finite differences, Rose-Hulman Undergraduate Math Journal 8 7, [Pratap 999] R. Pratap, Getting started with MatLab 5, Oford University Press, Oford and New York, 999. [Sheng 8] A. D. Sheng, Optimized finite difference approimations on non-uniform grids with applications: A midterm project report to the URSA program, preprint, Baylor University, 8, Available at [Smith 985] G. D. Smith, Numerical solution of partial differential equations: Finite difference methods, 3rd ed., Oford App. Math. and Computing Science Series 3, Oford Univ. Press, New York, 985. MR 87c:65 Zbl [Urban et al. 4] P. Urban, J. Owen, D. Martin, R. Haese, S. Haese, and M. Bruce, Mathematics for the international student : Mathematics HL core, Haese & Harris Publications, Adelaide Airport, Australia, 4. [Wilmott et al. 995] P. Wilmott, S. Howison, and J. Dewynne, The mathematics of financial derivatives: A student introduction, Cambridge University Press, Cambridge, 995. MR 96h:98 Zbl Received: Revised: Accepted: mylesdanielbaker@gmail.com danieldsheng@gmail.com Department of Mathematics, Baylor University, Waco, TX , United States Westwood High School, Austin, TX 7875, United States