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