Comparing Algebraic and Numerical Solutions of Classical Diffusion Process Equations in Computational Financial Mathematics

We revise the interrelations between the classical Black Scholes equation, the diffusion equation and Burgers equation. Some of the algebraic properties the diffusion equation shows are elaborated and qualitatively presented. The related numerical elementary recipes are briefly elucidated in context of the diffusion equation. The quality of the approximations to the exact solutions is compared throughout the visualizations. The article mainly is based on the pedagogical style of the presentations to the Novacella Easter School 2000 on Financial Mathematics.


INTRODUCTION
The Black Scholes equation has played and is playing a strong role in mathematical modelling of financial markets.It connects the understanding of option prizing with diffusion models and thus yields a fascinating interaction between phenomena in nature with phenomena in society.For a detailed description concerning the Black Scholes equation, see [10].For general reference, see [1-4, 6-9, 11].Discretizations of this equation have also been investigated in detail, and in principle it seems that there is nothing really new about this celebrated equation.Its symmetry properties however are still a remarkable mathe- matical fact, in detail with respect to its transfor- mation properties into the classical heat or diffusion equation.
In this article, we prosecute the strategy of reducing solutions of the Black-Scholes equation to solutions of the classical diffusion equation as described in detail in [10,11].We give some explicit examples in which we compare the quality of exact algebraic solutions to the classical diffusion equation with solutions obtained by the standard numerical recipes as proposed in [10].As a consequence, we obtain the main result that has been elucidated in great detail on the Novacella Easter School 2000: The conventional recipes proposed in [10,11] yield a fine qualitative accuracy.As for the standard error analysis, we recommend the reader to refer to [5].
The Easter School in Novacella, Italy, was an attempt to communicate questions in the area of discrete dynamics in nature and society to students in mathematical finance, mathematics, to students in computer science and students in physics and to encourage them to scientifically interact.
The homepage of the Novacella Easter School 2000 is www-ml.ma.tum.de/bilder/2000/ostersem_ 4 00 The organization of this article shall now be as follows: In the second chapter we revise the interrelation between Black-Scholes equation, diffusion equa- tion and Burgers equation.
In the third and the fourth chapter, algebraic solutions and numerical approximations are devel- oped.They are compared at the end of the fourth chapter.
The authors would like to mention in particular that the impressive graphic results were also partially made possible by the help of Tobias Lasser who had addressed the topic of solution strategies to ordinary differential equations during the Novacella Easter School.

EXAMPLES OF EQUATION TRANSFORMS
The connection between the Black Scholes equa- tion and the diffusion equation belongs to the standard procedures being taught in computa- tional finance.For the convenience of the reader, we are going to state it right here at the very beginning of the article.The Blaek-Scholes equation is given by where C refes to a call, S denoting the underlying asset, the time.0" refers to the volatility of the underlying asset, r means the interest rate.
Let us show that also another partial differential equation can be reduced to the diffusion equation by standard methods: The partial differential equation OU OU 02U 0--7 + u 0-Ox 2 (7) is nonlinear and usually referred to by the name Burgers equation.It is remarkable that one can linearize this equation in the following way.
Assume that v(x, t) is a purely positive function for which the expression 0 oxln(v(x,t)) (8) is well defined the function for any (x,t)E R2.Defining From a classification viewpoint, the diffusion resp.
heat equation is a second order linear partial differential equation which is parabolic, one obtains first by evaluating (7 v Ox We now differentiate the function u with respect to t, x and obtain Now, taking the second derivative of u with respect to x, we receive 02it 4 (OV Combining these expressions in the sense of (7), one recognizes the following: The function u we have introduced by v yields a solution to Burgers Thus, if the solution v is a positive solution to the heat resp.diffusion Eq. ( 6) for any (x, t)E 2, the function u, defined by (9) provides a solution to Burgers equation.
W have now got some motivation to focus on comparing solutions to the diffusion equa- tion which, as we have seen, is closely related to the Black-Scholes equation and to Burgers equation.
0t/ 02bt (15) Ot Ox 2 In a conventional sense, linearity is understood as usual, i.e., along with the solutions Ul (x, t) and u2 (x, t) for the equation, all linear combinations Cl Ul (x, t)+ c2 u2 (x, t) solve the equation as well.The solutions u (x, t) (u: x ) model the time dependent temperature in a long, fully insulated bar which is assumed to be one-dimensional.
One obtains the most simple solutions by combining polynomial functions in x and as follows:  o GRAPH 3 u(x, t) (1/2)x q-t.
Concerning the solution structure, "more diffi- cult" solutions exist as well, for instance u(x, t) ce n=l This can easily be checked by using induction methods: The case k-1 is obviously clear, therefore we address kk: As the partial derivative of a function is lin- ear, y=lcne nx+nt gives a solution because k-1 n--1 c,enx+"t satisfies the initial condition.As for cke x+t compare (19).Starting the summa- tion at a lower negative index doesn't cause any problems.
We can construct a different solution fulfilling the following boundary conditions: u(x, O) f(x) O < x < Tr u(0, t) 0 t>0 u(Tr, t)-O t>0 Obviously further solutions are obtained by changing sin into cos, sinh or cosh.Due to the linearity of the heat resp.diffusion equation, the sum again gives a solution: b/(X, t) ZCn sin (nx)e -nzt (Cn E ) (22) n=l GRAPH 7 u(x,t)=sin ((1/2)x)e -O/4)t with =k, k E [0, 10].
The factor c is necessary to ensure that the integral f_+ c(s)ds equals which reflects the fact that the probability is conserved.
By direct calculation one can verify that the density function with parameter # is already a As usual, we replace the partial derivatives by difference quotients where we introduce first order difference quotients and second order difference quotients separately: The approximation is performed as follows: &is assumed to be sufficiently small The evaluation for (Ou/Ox) (x, r) goes in an analogous way.

Derivatives of Second Order
Reducing the second order derivatives to the given derivatives of first order yields 02U(x,r) lim (OU(x+6x, r) OU(x,r)) There are three given methods for approximating the first derivative.
In total we obtain 27 possibilities of approx- imating the second derivatives in terms of finite differences to the u-functions In the outlined context, the symmetric central differences are often preferred due to their invariance with respect to the transformations x -+ -x bzw.7---7-: 02U (X, 7) , 4.2.Considering the Errors As a consequence of the respective Taylor expan- sion and the additional assumption that u(x, 7") allows to calculate its lowest derivatives, we can derive the following equations that give a first insight into the error behavior: On the one hand, we obtain OU (X, 7")(67")q-R2 u(x, , + 6-) u(x, -) + -2 This means that the error term of the forward resp.backward difference goes to zero in a linear way as 6- 0. Adding the forward and backward difference, we can derive the following approximations from (24): 0 (x, ) (x + x, ) (x x, ) Thus the given central differences yield a quadratic convergence in the stated sense.The stated facts open the possibility of determining exact error bounds.

Evaluating the Difference Scheme
The discretization is performed by choosing an equidistant lattice with mesh lenghts x resp.GRAPH 21 u(x, t) v'] sin (nx)e-"2t.
Andreas Ruffing gratefully appreciates the finan- cial support of the Stifterverband within the award "Reformfakult/it Zentrum Mathematik der Tech- nischen Universit/it Miinchen" to organize the Easter School 2000 in Novacella.He likes to thank Claudia Kliippelberg for arising his interest in several analytical topics of financial mathe- matics.This work was co-sponsored by the Stadtsparkasse Miinchen, the Nova Kreditna Banka Maribor and by Feilmeier and Junker.
The financial support by the Ministry of Science and Technology of the Slovenian Government is acknowledged with special thanks.We like to thank in great detail the director of the project "Summerschools" at the Center for Mathematical Sciences at Technical University of Munich, Rupert Lasser.With his help, the Easter School 2000 in Novacella became possible.A lot of thanks are given to Peter Gritzmann for encoura- ging the organizer of the Easter School at an essential state of the school's origin and for supporting the project as dean of the Center for Mathematical Sciences.The great moral support by Marko Robnik and by Christian Kredler in thinking in an avantgarde way of modern science is acknowledged in particular.The positive attitude of Wolfgang A. Herrmann, President of the Technische Universitt Miinchen and of Alto Brachner, Director of Natural Sciences at Deutsches Museum, Miinchen towards the Summerschool Project of the "Reformfakultt Zentrum Mathematik" is very highly appreciated.

1 .
Stating the Difference Scheme