On the Numerical Solution of Black-Scholes Equation - Article Example

Numerical Solution of Black-Scholes equation Introduction Black-Scholes model is used to describe behaviour of derivative instruments like call and put option. The model yields a partial differential equation called the Black-scholes equation which can reduced to the form of the famous heat equation. Partial difference equations can be solved using either Finite difference methods or Finite element methods. Both these methods make use of a discrete mesh and the values are calculated at the mesh points by iteration using known values.

This paper discusses an alternative method which is meshless and is based on Radial Basis functions(RBF’s).It discusses the results obtained after using different sets of RBF’s namely ThinPlateSpline(TPS),Multiquadric(MQ),Cubic and Gaussian 2 Black-Scholes equation A derivative is a financial instrument whose value is derived from one or more underlying commodities. An option is a financial instrument that gives an individual the right to buy or sell an asset, at some time in the future. Options are traded on a number of exchanges throughout the world, the first of which was the Chicago Board Options Exchange (CBOE), which started in 1971.

The price V(S,t) of the derivative or the option depends on the price of the underlying S and time t. V(S,t) satisfies the Black Scholes partial differential equation. (1) Where r is the interest free interst rate and ? is the volatility of the underlying. The right to buy the underlying in the future for an agreed upon price , called the Exercise price(E) ,with in a date called the expiry date, is called a Call Option.. Similarly the right to sell the underlying for the Exercise price before the expiry date is the Put Opttion.

The time to expiry is the expiry time denoted by T. The pay-off equations for the options gives the boundary conditions for the Black-Scholes equation.The variable t can take values between 0 and T while S can take values from 0 to ? . If V(S,t) is the option price with S ? [0,?) and t ? [0,T] then the boundary condtions for the Black-Scholes equation are V(S,T)=max(E-S,0) for the Put option and V(S,T)=max(S-E,0) for the Call Option which are the pay-off’s of the options on the Exercise date.

3 Radial Basis function 3.1 Definitions: A radial basis is a continous spline of the form where rj is the Euclidean norm or distance .The most common RBF’s are Cubic: Gaussian: Thin Plate Spline(TPS): Multiquadric(MQ): 3.2 Expanding V(S,t) : Approximation of a function can be written as a linear combination of the basis Functions .Here V(S,t) can be represented approximately as a linear combination of any of the four set of basis functions as, (2) Where N is the number of data points and ?’s are the coefficients to be determined .and ?’s are the basis functions.

4 Solving the Black-Scholes Equation 4.1 Discretizing Black-Scholes in time using the ? method (3) ?=0.5 for Crak-Nicholson and ?t is the time step 4.2 Solving the Black-Scholes equation Using the notation Vn =V(S,tn ) for the value of the option at the time step V(S,t)=Vn V(S,t+?t)=Vn+1 And putting ?t(1-?) = ? ??t = ? and rearranging Equation (3) becomes (4) Defining operators and Substituting for V from equation (2) equation(4) becomes (5) Which is system of linear equations which is to be solved at each time step n?

t with known values of to get A system of linear equations can be solved using Gauss-Jordan elimination with partial pivoting.Once the ?j values are known V(S,t) can be calculated as done by the authors of this paper 5 Results The authors of the paper takes for comparison purposes a European put option with Exercise price E=10 ,time to expiry T=.5 years .The price of the underlying is expected to vary between 0 and 30.The interest rate r is 5% and the volatility is 20%. Black-Scholes equation is solved with the four different sets of Basis functions and the option prices are calculated for different values of the underlying in the given interval and tabulated .

The expected error in the results is calculated using the equation Delta (the rate of change of the option with respect to the cost of the underlying) also is tabulated for different basis functions.The error values for different number of nodes is also tabulated .From the tables it can be easily concluded that TPS and MQ RBF’s are superior to Gaussian and Cubic. 6 Summary and Conclusions The authors have successfully solved the Black-Scholes equation by using RBF’s. This method does not require discretization of the derivatives and is similar to the spectral method of solving differential equations.

The paper also shows that of all RBF’s TPS and MQ generate better results than the other RBF’s. The table of results also show that the method is superior to mesh dependent methods . References [1] Choi S.and Marcozzi M.D.,A Numerical approach to American Currency Option Valuation,Journal of Derivatives ,19,2001 [2] Buhmann M.D.,Radial Basis Functions ,Acta Numerica (pp1-38)2000 [3] Alonso Pena Option Pricing with Radial Basis Functions :Tutorial