# 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? – Richard Feb 25 '16 at 15:14
• And where did you get your forumlas? – Bob Jansen Feb 25 '16 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. – Oamriotn Feb 25 '16 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 '16 at 19:38
• Hereby the link of the article: milliman.com/insight/2015/… – Oamriotn Feb 26 '16 at 7:56