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


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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 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 6.3/6.4: Black-Scholes in R
Last updated August 2016

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