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


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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 July 2020

using Distributions;
##If using function_name(;) to define a function,
##you need to supply argument names when calling the functions
function bs(; X = 1, P = 1, r = 0.05, sigma = 1, T = 1)
#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 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)