Backbone of Displaced Diffusion model

 Our model Black76 does not work in negative interest rate market.　One of popular alternative of in negative interest rate market is displaced diffusion model first introduced by Rubinstein(1983).
 We can implement displaced diffusion model just by replacing forward for (forward + displacement) in Black76 formula.

Here are the backbones of displaced diffusion model for some displacement parameters.
All the three graphs represents the same prices for each model, bu they looks differently.

 y axis displays log-normal volatility.

 

y axis displays 50bp displaced log-normal volatility.

 y axis displays 100bp displaced log-normal volatility.

Log-normal volatility cannot be calculated in negative forward region, since the log-normal model(Black76) does not work when forward is negative.

If you use displaced diffusion model and some finite displacement parameter, you can go into negative forward region. When the displacement parameter equals 0%, displaced diffusion model corresponds with Black76 model.

 When the displacement parameter is Xbp, the backbone of displaced diffusion model displayed in Xbp displaced log-normal volatility is flat. If you display the backbone in Ybp(X > Y) displaced log-normal volatility, the shape will be downward sloping and vice versa.