Python and Julia Chapter 6. Analytical Value-at-Risk for Options and Bonds

# Chapter 6. Analytical Value-at-Risk for Options and Bonds

### Python and Julia

Copyright 2011 - 2023 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: www.gnu.org/licenses.

##### Black-Scholes function in Python
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}

##### Black-Scholes function in Julia
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


##### Black-Scholes in Python
f = bs(X = 90, P = 100, r = 0.05, sigma = 0.2, T = 0.5)
print(f)

##### Black-Scholes in Julia
f = bs(X = 90, P = 100, r = 0.05, sigma = 0.2, T = 0.5)


##### Financial Risk Forecasting
Market risk forecasting with R, Julia, Python and Matlab. Code, lecture slides, implementation notes, seminar assignments and questions.