# Zero Black-Derman-Toy interest rate model

## TL;DR

This paper introduces a modified Black-Derman-Toy interest rate model that incorporates rare jumps to near-zero rates, helping to better assess crisis risks and improve derivative pricing accuracy.

## Contribution

It extends the classical BDT model to include low-probability zero-interest rate jumps, enabling more realistic crisis risk modeling and calibration.

## Key findings

- The modified model captures zero-rate jumps effectively.
- Option prices and implied volatilities are better fitted during crises.
- The model provides a calibrated probability of interest rate crises.

## Abstract

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.