# Chapter 6. Analytical Value-at-Risk for Options and Bonds (in R/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 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))
return(list(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 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 R Last updated July 2020

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

##### 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)