# Chapter 6. Analytical Value-at-Risk for Options and Bonds (in MATLAB/Python)

Copyright 2011 - 2020 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: https://www.gnu.org/licenses/.

##### 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));
end

##### Listing 6.1/6.2: Black-Scholes function in Python Last updated June 2018

import numpy as np
from scipy import stats
def bs(X, P, r, sigma, T):
d1 = (np.log(P/X) + (r + 0.5 * sigma**2)*T)/(sigma * np.sqrt(T))
d2 = d1 - sigma * np.sqrt(T)
Call = P * stats.norm.cdf(d1) - X * np.exp(-r * T) * stats.norm.cdf(d2)
Put = X * np.exp(-r * T) * stats.norm.cdf(-d2) - P * stats.norm.cdf(-d1)
Delta_Call = stats.norm.cdf(d1)
Delta_Put = Delta_Call - 1
Gamma = stats.norm.pdf(d1) / (P * sigma * np.sqrt(T))
return {"Call": Call, "Put": Put, "Delta_Call": Delta_Call, "Delta_Put": Delta_Put, "Gamma": Gamma}


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

f=bs(90,100,0.05,0.2,0.5)

##### Listing 6.3/6.4: Black-Scholes in Python Last updated July 2020

f = bs(X = 90, P = 100, r = 0.05, sigma = 0.2, T = 0.5)
print(f)