How to Improve MATLAB Function for Calculating Implied Volatility?

[mts_88]

I have found the following function for calculating implied volatility in MATLAB.
When it finds a solution, the solution tends to be correct, but the function does not take into account, that the slope can be zero. If this happens, you divide by zero and the function return inf or -inf, which isn't correct at all. I have tried to take this problem into account, but the result is that the function does not converge. Does anyone have an idea how to implement this in the function?

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
% -------------------------------------------------------------------
% NEWT_IV.M
% -------------------------------------------------------------------
% The function uses Newton's method to find the root of
% the actual call price minus the Black-Scholes call price, solving
% for volatility.
% -------------------------------------------------------------------
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

% FUNCTION: newt_iv
% Input(s):
% - C: actual call price expressed in home currency
% - S: price of the underlying expressed in home currency
% - K: strike price of the underlying expressed in home currency
% - r: risk-free rate expressed as a decimal (5%=0.05)
% - T: Tenor expressed in years (6 months=0.5)
% Output(s): the approximate implied volatility using Black-Scholes
% and Newton's method. The approximated implied volatility and
% number of iterations taken to achieve an accurate result are
% displayed.
%
function ivol = newt_iv(C,S,K,r,T)

% Anonymous functions for Black-Scholes
d1 = @(S,K,r,vol,T) (log(S/K)+(r+vol^2/2)*T)/(vol*sqrt(T));
d2 = @(S,K,r,vol,T) d1(S,K,r,vol,T)-vol*sqrt(T);
bscp = @(S,K,r,vol,T) S*normcdf(d1(S,K,r,vol,T),0,1)-K*exp(-r*T)*normcdf(d2(S,K,r,vol,T),0,1);
vega = @(S,K,r,vol,T)S*normpdf(d1(S,K,r,vol,T),0,1)*sqrt(T);
% Initial guess
ivol = 0.2;

% Error tolerance
err = 0.0001;

% Set a limit to the number of iterations
i = 1; % number of iterations
ilimit = 2000000; % limit

cpdiff = err+1; % Cause the first iteration to occur (see next line)
while abs(cpdiff) > err
% Terminate the algorithm if the iteration limit has been reached
if (i > ilimit)
str = sprintf('ERROR: Implied volatility could not be found after %d iterations.', ilimit);
disp(str);
ivol = inf;
break;
% Otherwise continue the iteration
else
i = i+1;
end

% Calculate Black-Scholes call price for current implied
% volatility guess.
ccprice = bscp(S,K,r,ivol,T);

% Find the difference between calculated call price and
% actual call price.
cpdiff = ccprice - C;

% If the calculated call price minus the actual call price is not
% small enough, adjust the guess using Newton's method
if abs(cpdiff) > err
slope = vega(S,K,r,ivol,T);
y_int = cpdiff-slope*ivol;
ivol = -y_int/slope;
end
% If the calculated call price minus the actual call price is zero,
% a root as been found. Unless a root as been round, repeat
% the process using the new upper and lower bounds.
end
ivol;
end

 

[gtoguy97]

I think the best way to implement this in the function is to add a check for the value of the slope. If the slope is zero, then the implied volatility can be set to the current guess and the loop can be exited. This should allow the algorithm to converge without dividing by zero.