From the Black-Karasinski to the Verhulst model to accommodate the unconventional Fed's policy

TL;DR

This paper modifies the Black-Karasinski interest rate model into a Verhulst logistic model to better reflect current market conditions and Fed policies, providing new integral equations and efficient solutions.

Contribution

It introduces a Verhulst model as a modification of the Black-Karasinski model, with new integral equations and closed-form solutions for bond pricing.

Findings

Verhulst model better fits current economic environment

Derived new integral equations for bond prices

Proposed efficient solution methods outperform standard techniques

Abstract

In this paper, we argue that some of the most popular short-term interest models have to be revisited and modified to reflect current market conditions better. In particular, we propose a modification of the popular Black-Karasinski model, which is widely used by practitioners for modeling interest rates, credit, and commodities. Our adjustment gives rise to the stochastic Verhulst model, which is well-known in the population dynamics and epidemiology as a logistic model. We demonstrate that the Verhulst model's dynamics are well suited to the current economic environment and the Fed's actions. Besides, we derive new integral equations for the zero-coupon bond prices for both the BK and Verhulst models. For the BK model for small maturities up to 2 years, we solve the corresponding integral equation by using the reduced differential transform method. For the Verhulst integral equation, under some mild assumptions, we find the closed-form solution. Numerical examples show that computationally our approach is significantly more efficient than the standard finite difference method.