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

Copyright 2016 Jon Danielsson. Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at. http://www.apache.org/licenses/LICENSE-2.0. Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.

##### Listing 6.1: Black-Scholes function in R Last edited: 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 6.2: Black-Scholes function in Matlab Last edited: 2011

function  res= bs(K,P,r,sigma,T)
d1 = (log(P./K)+(r+(sigma^2)/2)*T)./(sigma*sqrt(T));
d2 = d1 - sigma*sqrt(T);
res.Call = P.*normcdf(d1,0,1)-K.*exp(-r*T).*normcdf(d2,0,1);
res.Put = K.*exp(-r*T).*normcdf(-d2,0,1)-P.*normcdf(-d1,0,1);
res.Delta.Call = normcdf(d1,0,1);
res.Delta.Put = res.Delta.Call -1;
res.Gamma = normpdf(d1,0,1)./(P*sigma*sqrt(T));
end


##### Listing 6.3: Black-Scholes in R Last edited: August 2016

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

##### Listing 6.4: Black-Scholes in Matlab Last edited: 2011

>> f=bs(90,100,0.05,0.2,0.5)
f =
Call: 13.4985
Put: 1.2764
Delta: [1x1 struct]
Gamma: 0.0172
>> f.Delta
ans =
Call: 0.8395
Put: -0.1605