  We propose a modification of the classical Black-Derman-Toy (BDT) interest
rate tree model, which includes the possibility of a jump with small
probability at each step to a practically zero interest rate. The corresponding
BDT algorithms are consequently modified to calibrate the tree containing the
zero interest rate scenarios. This modification is motivated by the recent
2008-2009 crisis in the United States and it quantifies the risk of a future
crises in bond prices and derivatives. The proposed model is useful to price
derivatives. This exercise also provides a tool to calibrate the probability of
this event. A comparison of option prices and implied volatilities on US
Treasury bonds computed with both the proposed and the classical tree model is
provided, in six different scenarios along the different periods comprising the
years 2002-2017.
