[Paper Review] From the Black-Karasinski to the Verhulst model to accommodate the unconventional Fed's policy

Andrey Itkin, Alexander Lipton+1 more|arXiv (Cornell University)|Jun 21, 2020

Complex Systems and Time Series Analysis4 citations

TL;DR

This paper proposes a modified Black-Karasinski model, named the Verhulst (or MBK) model, which replaces the lognormal rate dynamics with a logistic (Verhulst) process to better capture prolonged low-interest-rate regimes under unconventional monetary policy. The model yields a closed-form solution for zero-coupon bond prices under mild assumptions and demonstrates superior computational efficiency and accuracy compared to finite difference and Monte Carlo methods, particularly for short maturities.

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.

Motivation & Objective

• To address the limitations of classical short-rate models like Black-Karasinski in capturing prolonged periods of low or negative interest rates under unconventional monetary policy.
• To develop a more tractable and structurally appropriate model that exhibits fat tails at the lower end of the rate distribution, reflecting current market conditions.
• To derive an integral equation for zero-coupon bond (ZCB) prices in the modified model and provide a closed-form solution under specific assumptions.
• To demonstrate computational superiority of the analytical approach over finite difference and Monte Carlo methods in pricing ZCBs.

Proposed method

• Replaces the mean-reverting dynamics in the Black-Karasinski model with a logistic (Verhulst) process, where the drift term is proportional to the difference between a time-dependent mean-reversion level and the exponential of the state variable.
• Introduces a transformation of the short rate as $ r_t = s(t) + R e^{z_t} $, ensuring non-negativity and allowing for a deterministic shift.
• Derives an integral equation for ZCB prices based on the characteristic function of the state variable, enabling numerical integration for general cases.
• For small maturities (up to 2 years), applies the reduced differential transform method to solve the integral equation efficiently.
• Finds a closed-form solution for the ZCB price under the assumption of time-dependent volatility $ \sigma(t) = \sqrt{\sigma_a + \sigma_b / (t + \sigma_c)} $, with $ \sigma_b < 0 $, and constant $ \kappa(t) $.
• Employs the Whittaker function and complex-valued Gamma functions in the analytical solution, with numerical integration via Simpson’s rule for evaluation.

Experimental results

Research questions

• RQ1 How can the Black-Karasinski model be modified to better reflect the current economic environment of persistently low or negative interest rates?
• RQ2 Can a stochastic process with fat tails at the lower end of the rate distribution be embedded into a tractable interest rate model?
• RQ3 Does the proposed Verhulst-based model yield a closed-form solution for zero-coupon bond prices under reasonable assumptions?
• RQ4 How does the computational performance of the analytical method compare to finite difference and Monte Carlo methods in pricing ZCBs?

Key findings

• The modified Verhulst (MBK) model successfully captures prolonged low-rate regimes by introducing a logistic mean-reversion term that produces fatter left tails than the lognormal BK model.
• For small maturities (up to 2 years), the reduced differential transform method solves the integral equation for ZCB prices with 55 ms computation time, outperforming the finite difference method (130 ms) in speed and accuracy.
• A closed-form solution for the ZCB price is derived under specific time-dependent volatility and constant mean-reversion assumptions, enabling faster and more accurate pricing.
• Numerical integration of the analytical formula yields relative errors below 0.3 bps (vs. FD) and 0.3 bps (vs. Monte Carlo) for maturities up to 2 years, increasing to 8.78 bps (vs. FD) and 36.94 bps (vs. MC) at 50 years, with higher accuracy in short-term maturities.
• The analytical method maintains high accuracy and computational efficiency, especially when using higher-order quadrature rules, and is more scalable than finite difference methods for long maturities.