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On the Numerical Solution of Black-Scholes Equation

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

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