A Comparative Study of Black-Scholes Equation

Transcription

1 Selçuk J. Appl. Math. Vol. 10. No. 1. pp , 2009 Selçuk Journal of Applied Mathematics A Comparative Study of Black-Scholes Equation Refet Polat Department of Mathematics, Faculty of Science and Letters Yaşar University Bornova, Izmir, Turkey refet.polat@yasar.edu.tr Received: March 19, 2009 Abstract. In this paper, we analyzed Options and Black Scholes Models for the valuing and pricing of commodities. In particular, we examined the numerical solution techniques of American Option Problems. For the comparison of the results pertaining to different methods, we used classical methods utilizing chronological order. Then we compared the results of these methods and tried to determine the most efficient method. Key words: Options, American, Options, Black-Scholes Equation, Finitedifference Method Mathematics Subject Classification: 26D15. 1.Introduction The Black-Scholes Model was developed by Fisher Black and Myran Scholes in 1973 [2]. Firstly it was used to value European options that are not paying any financial profit. In the case of American options, a volatility event which will be an important research area for mathematicians occurs. BLACK SCHOLES EQUATION The Black-Scholes equation in general form can be given as (1) =0 2 where is current stock price, is option price, is volatility, is option expiration date, is risk-free rate of interest 135

2 The numerical methods [10], which are developed for solving American Options problem (1), are the following: Finite-difference method, direct discretization of PDE, implicit method, explicit method, Wilmott, Howison and Dewyne Model and Linear Complementarity Theta-weighted method. Instead of implementing the direct discretization of Black-Scholes equation, Wilmott transformed Black- Scholes equations into heat equations and came up with a solution [10,12,13]. In this study, Wilmott s results are compared with the numerical methods that are mentioned above. 2. Reduction of Black-Scholes Equation to Parabolic Equation In Wilmott s paper [12], the reduction of Black-Scholes equation given in (1) is reduced to heat equation as follows: In equation (1) Let us assign K as S s appropriate basic value i.e. (2) = Then = ( ) = 1 2 = 1 2 (3) = = = = 1 2 = µ µ 2 µ µ (4) = Ifweinsert(3)and(4)inplaceof(1) or + ( ) µ 2 =0 or +( ) =

3 (5) µ =0 2 We reach the equation (5). If we multiply this equation by ( ) = µ =0 1 = 2 2 and 2 = Rearranging the above equation becomes 2( ) 2 ; (6) +( 2 1) 1 = 2 2 Let us arrange the time variable again: (7) = ( ) For convenience, we take independent of x and t. Hence, t from equation (7); = 2 2 = 2 2 so; = = µ 2 2 is reached. If we insert this equation in (6) (8) 2 +( 2 1) 1 = is reached. In order to solve equation (8), let us apply the separation of variables method. ( ) = ( )( ). Inthiscasewereach; = ( ) 0 ( ) 2 = ( ) 00 ( ) 137

4 = ( )0 ( ) If we insert these equations in (8) ( ) ( 00 ( ) 1 ( )+( 2 1) 0 ( ) = ( ) 0 ( ) 00 ( )+( 2 1) 0 ( ) ( ) ( ) = 0 ( ) ( ) = 00 ( )+( 2 1) 0 ( ) 1 ( ) =( ) and 0 ( ) ( ) = ( ) = + 1 ( ) = 1 In order to solve 00 ( )+( 2 1) 0 ( ) ( 1 + ) ( ) =0 Then the characteristic equation is ( ) = 2 +( 2 1) ( 1 + ) =0 q 1 2 = 1 2 ( 2 1) = 1 r (1 2)+ 4 ( 1 1) ( ) = ( 2 1) + 4 ( 2 1) ( ) = ( )( ) (9) ( ) = ( 2 1) + 4 ( 2 1) ( ) The solution (9) is the semi-analytical solution for the Black-Scholes Partial Differential Equation. This solution will be compared with the numerical solutions mentioned in introduction. 138

5 3. Numerical Experiment and The Result The analysis of our results shows that among the different methods [10] used namely, Explicit, Implicit, Brennan & Schwartz First Model [3], Brennan & Schwarz Second Model [4], Courtadon [5], Wilmott Howison Dewyne Theta weighted and Linear Complementarity Theta weighted method [12] Wilmott s method is the most efficient one [10]. The table given below verifies that all obtained results are diverse due to the applied methods. These empiric results show that the stability of the equation differs from method to method; but, in our opinion, Wilmott s method may be taken as the most appropriate one. Therefore the reason that we made our comparison with respect to this solution is that the method is a semi-analytic solution. In the Table 1, it is observed that the best value is obtained with linear complementarily methods, but the complexity of the method is very high which makes it less effective. In the other numerical methods, complexity looks pretty appropriate. Nevertheless the values differ from method to method, which shows the unstability of the equation. Cpu: Pentium 4, 2.2 Table 1. Comparative results of semi-analytic method and numerical methods 139

6 References 1. C. Albenese, S. Jaimungal and D. Rubinsov, The Model of lines for Option Pricing with Jump, F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, , M. Brennan and E. Schwartz, The valuation of American put options, Journal of Finance, M. Brennan and E. Schwartz, Finite-difference methods and jump processes arising in the Pricing of contingent claims: A synthesis, Journal of Financial and Quantitative Analysis, G. Courtadon, A more accurate finite difference approximation for the valuation of Options, Journal of Financial and Quantitative Analysis, H. P Mc Kean, Appendix, A Free Boundary Problem for the Heat Equation Arising from a Problem in Mathematical Economics. Industr. Manage. Rev., 6: 32-39, Hull, John, Options, Future and other Derivative Securities, Willeys, J. J McCutcheon-W. F. Scott, An Introduction to the Mathematics of Finance, P. Van Moerbeke, On Optimal Stopping and Free Boundary Problem, Arch. Rational Mech. And. 60: , S. Sukha, Finite-Difference Methods for Pricing the American Put Options, W. Allegretto, G. Barone-Adesi and R. J. Elliot, Numerical Evaluation of the Critical Price and American Options, Europen J. Finance, 1: 69-78, Wilmott, Paul-Dewynne, Jeff-Howison, Sam, Option Pricing Mathematical Modals and Computation, Oxford Financial Press, Wilmott, Paul-Dewynne, Jeff-Howison, Sam, The Mathematics of Financial Derivatives A student Introduction,