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

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 Julia Last updated July 2020

using Distributions;
function bs(; X = 1, P = 1, r = 0.05, sigma = 1, T = 1)
d1 = (log.(P/X) .+ (r .+ 0.5 .* sigma.^2).*T)./(sigma .* sqrt.(T))
d2 = d1 .- sigma * sqrt.(T)
Call = P .* cdf.(Normal(0,1), d1) .- X .* exp.(-r * T) .* cdf.(Normal(0,1), d2)
Put = X .* exp(-r .* T) .* cdf.(Normal(0,1),-d2) .- P .* cdf.(Normal(0,1), -d1)
Delta_Call = cdf.(Normal(0,1), d1)
Delta_Put = Delta_Call .- 1
Gamma = pdf.(Normal(0,1), d1) ./ (P .* sigma .* sqrt(T))
return Dict("Call" => Call, "Put" => Put, "Delta_Call" => Delta_Call, "Delta_Put" => Delta_Put, "Gamma" => Gamma)
end


##### 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 Julia Last updated July 2020

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