Chapter 6. Analytical Value–at–Risk for Options and Bonds (in R/MATLAB)

Copyright 2011 - 2019 Jon Danielsson. This code is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This code is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. The GNU General Public License is available at:

Listing 6.1/6.2: Black-Scholes function in R
Last updated 2011

bs = function(X, P, r, sigma, T){
	d1 = (log(P/X) + (r + 0.5*sigma^2)*(T))/(sigma*sqrt(T))
	d2 = d1 - sigma*sqrt(T)
	Call = P*pnorm(d1,mean=0,sd=1)-X*exp(-r*(T))*pnorm(d2,mean=0,sd=1)
	Put = X*exp(-r*(T))*pnorm(-d2,mean=0,sd=1)-P*pnorm(-d1,mean=0,sd=1)
	Delta.Call = pnorm(d1, mean = 0, sd = 1)
	Delta.Put = Delta.Call - 1
	Gamma = dnorm(d1, mean = 0, sd = 1)/(P*sigma*sqrt(T))
Listing 6.1/6.2: Black-Scholes function in MATLAB
Last updated 2011

%% To run this code block in Jupyter notebook:
%% delete all lines above the line with file bs.m, then run
%%file bs.m
function  res = bs(K,P,r,sigma,T)
	d1 = (log(P./K)+(r+(sigma^2)/2)*T)./(sigma*sqrt(T));
	d2 = d1 - sigma*sqrt(T);
	res.Call = P.*normcdf(d1,0,1)-K.*exp(-r*T).*normcdf(d2,0,1);
	res.Put = K.*exp(-r*T).*normcdf(-d2,0,1)-P.*normcdf(-d1,0,1);
	res.Delta.Call = normcdf(d1,0,1);
	res.Delta.Put = res.Delta.Call -1;
	res.Gamma = normpdf(d1,0,1)./(P*sigma*sqrt(T));

Listing 6.3/6.4: Black-Scholes in R
Last updated August 2016

Listing 6.3/6.4: Black-Scholes in MATLAB
Last updated 2011