## Implicit Finite Difference Method - A MATLAB Implementation

This tutorial presents MATLAB code that implements the implicit finite difference method for option pricing as discussed in the The Implicit Finite Difference Method tutorial.

The code may be used to price vanilla European Put or Call options. Note that the primary purpose of the code is to show how to implement the implicit method. The option under consideration could easily be priced using the standard Black-Scholes analytical solution, however knowing the actual option price allows the accuracy of the code here to be verified. The code contains no error checking and is not optimized for speed or memory use. As such it is not suitable for inclusion into a larger application without modifications.

### A Pricing Example

Consider pricing a European Call and Put option with the following parameters,
*X = $60*, *S _{0} = $50*,

*r = 5%*,

*σ = 0.2*and

*T = 1*.

The Black-Scholes price for the Call option is *$1.624*, and the Put option
is *$8.697*

A MATLAB function called **finDiffImplicit** is given below.
The following shows an example of executing **finDiffImplicit**
(and pricing the above option) in MATLAB,

>> cPrice = finDiffImplicit(60,50,0.05,0.2,0:1:100,0:0.01:1,'CALL') cPrice = 1.5826 >> pPrice = finDiffImplicit(60,50,0.05,0.2,0:1:100,0:0.01:1,'PUT') pPrice = 8.6968

### MATLAB Function: finDiffImplicit

function oPrice = finDiffImplicit(X,S0,r,sig,Svec,tvec,oType) % Function to calculate the price of a vanilla European % Put or Call option using the implicit finite difference method % % oPrice = finDiffImplicit(X,r,sig,Svec,tvec,oType) % % Inputs: X - strike % : S0 - stock price % : r - risk free interest rate % : sig - volatility % : Svec - Vector of stock prices (i.e. grid points) % : tvec - Vector of times (i.e. grid points) % : oType - must be 'PUT' or 'CALL'. % % Output: oPrice - the option price % % Notes: This code focuses on details of the implementation of the % implicit finite difference scheme. % It does not contain any programatic essentials such as error % checking. % It does not allow for optional/default input arguments. % It is not optimized for memory efficiency, speed or % use of sparse matrces. % Author: Phil Goddard (phil@goddardconsulting.ca) % Date : Q4, 2007 % Get the number of grid points M = length(Svec)-1; N = length(tvec)-1; % Get the grid sizes (assuming equi-spaced points) dt = tvec(2)-tvec(1); % Calculate the coefficients % To do this we need a vector of j points j = 0:M; sig2 = sig*sig; aj = (dt*j/2).*(r - sig2*j); bj = 1 + dt*(sig2*(j.^2) + r); cj = -(dt*j/2).*(r + sig2*j); % Pre-allocate the output price(1:M+1,1:N+1) = nan; % Specify the boundary conditions switch oType case 'CALL' % Specify the expiry time boundary condition price(:,end) = max(Svec-X,0); % Put in the minimum and maximum price boundary conditions % assuming that the largest value in the Svec is % chosen so that the following is true for all time price(1,:) = 0; price(end,:) = (Svec(end)-X)*exp(-r*tvec(end:-1:1)); case 'PUT' % Specify the expiry time boundary condition price(:,end) = max(X-Svec,0); % Put in the minimum and maximum price boundary conditions % assuming that the largest value in the Svec is % chosen so that the following is true for all time price(1,:) = (X-Svec(end))*exp(-r*tvec(end:-1:1)); price(end,:) = 0; end % Form the tridiagonal matrix B = diag(aj(3:M),-1) + diag(bj(2:M)) + diag(cj(2:M-1),1); [L,U] = lu(B); % Solve at each node offset = zeros(size(B,2),1); for idx = N:-1:1 offset(1) = aj(2)*price(1,idx); % offset(end) = c(end)*price(end,idx); % This will always be zero price(2:M,idx) = U\(L\(price(2:M,idx+1) - offset)); end % Calculate the option price oPrice = interp1(Svec,price(:,1),S0);