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

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: https://www.gnu.org/licenses/.

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

function bs(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 .* 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 Python Last updated June 2018

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

##### Listing 6.3/6.4: Black-Scholes in Julia Last updated June 2018

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