# Normal Black-Scholes model for swaptions isn't working properly

I just wrote two functions in Matlab which calculates the swaption prices based on the Lognormal model and on the Normal model, although I have the idea that the Normal model is wrong because the swaption price is (I think) too high.

Hereby the Lognormal function in Matlab:

 function [Receiver, Payer] = BlackSwaptionModel(K,S,Bvol,Time,Reonia,TenorSwap)
d1 = (log(S/K) + 1/2*Bvol^2*Time) / (Bvol*sqrt(Time));
d2 = d1 - (Bvol * sqrt(Time));

Receiver = ((1-1/(1+S)^(TenorSwap)) / S) * exp(-Reonia*Time) * (S*normcdf(d1) - K*normcdf(d2)); %Value receiver swaption Black Model
Payer = ((1-1/(1+S)^(TenorSwap)) / S) * exp(-Reonia*Time) * (K*normcdf(-d2) - S*normcdf(-d1)); % Value payer swaption Black Model
end


Plus the Normal model function in Matlab:

function [Receiver, Payer] = NormalSwaptionModel(K,S,Nvol,Time,Reonia,TenorSwap)

d1 = (S-K) / (Nvol * sqrt(Time));
d2 = -(S-K) / (Nvol * sqrt(Time));

Receiver = Nvol * sqrt(Time) * (d1*normcdf(d1) + normpdf(d1)) * ((1-1/(1+S)^(TenorSwap)) / S) * exp(-Reonia*Time);

Payer = Nvol * sqrt(Time) * (d2*normcdf(d2) + normpdf(d2)) * ((1-1/(1+S)^(TenorSwap)) / S) * exp(-Reonia*Time);

end


Could anybody see what's going wrong here. Thanks.

• what are your input parameters? Feb 25, 2016 at 15:14
• And where did you get your forumlas? Feb 25, 2016 at 16:51
• Input parameterrs are: S = 0.02, K=0.02, Reonia=0.01, Time=1, Bvol=0.20, Nvol=0.20,Tenorswap=10. Now I know that when using these parameters the two formula's won't yield the same value but I think there is something wrong with the NormalSwaptionModel because the Nvol needs to be very small (which doesn't make sense) will these formula's give the same value for the swaptions. Feb 25, 2016 at 17:25
• The vols do not have the same order of magnitude. To get an idea, at the money, a 30% lognormal vol can correspond to a 0.60% normal vol. Normal vols are usually quoted in bps = 0.01%.
– AFK
Feb 25, 2016 at 19:38
• Hereby the link of the article: milliman.com/insight/2015/… Feb 26, 2016 at 7:56