A positive interest rate model with sticky barrier

Abstract

This paper proposes an efficient model for the term structure of interest rates when the interest rate takes very small values. We make the following choices: (i) we model the short-term interest rate, (ii) we assume that once the interest rate reaches zero, it stays there and we have to wait for a random time until the rate is reinitialized to a (possibly random) strictly positive value. This setting ensures that all term rates are strictly positive.

Our objective is to provide a simple method to price zero-coupon bonds. A basic statistical study of the data at hand indeed suggests a switch to a different mode of behaviour when we get to a low level of interest rates. We introduce a variable for the time already spent at 0 (during the last stay) and derive the pricing equation for the bond. We then solve this partial integro-differential equation (PIDE) on its entire domain using a finite difference method (Cranck–Nicholson scheme), a method of characteristics and a fixed point algorithm. Resulting yield curves can exhibit many different shapes, including the S shape observed on the recent Japanese market.

Keywords:

• Short-term interest rate models
• Partial integro-differential equation
• Zero-interest rate
• Finite difference methods

Acknowledgements

This research was done during the stay of YK at the Daiwa Securities Chair of Graduate School of Economics, Kyoto University in Spring 2004. SR acknowledges financial support from the Monbukagakusho. We express our gratitude for helpful comments from Rama Cont, Monique Jeanblanc, Yue Kuen Kwok, Yoshio Miyahara, Motoh Tsujimura, Nick Webber, two anonymous referees, and participants at the 2004 Daiwa International Workshop on Financial Engineering and QMF 2005 in Sydney. All remaining errors are ours, of course.

Notes

†Vašíček (1977) assumed that the short rate process r_t evolves according to the dynamics

where θ is the long-term mean, κ is the speed of convergence to the mean, σ is the volatility and W_t is a standard Brownian motion. The parameters θ, κ and σ are constant or deterministic.

†Black's idea was that the rate cannot become negative because if it does, investors can just invest in currency. The interest rate would thus be the positive part of, say, an Ornstein--Uhlenbeck process, like in the Vašíček model. The original Gaussian process is then called the shadow interest rate.

‡This is a direct consequence of the dynamics of the short rate. Under the CIR model (Cox et al. 1985) the short rate has the following dynamics:

and under the BK model (Black and Karasinki 1991):

where θ, κ, σ and W_t are defined as in (1).

†For example, at that time, the inflation rate (deflation actually) was never below -1%.

‡This is a questionable choice; it is not exactly the {\itshape short} rate, it has some counterparty risk in it (leading to probable presence of outliers, after large corporate bankruptcies for example), it is not directly related to the yields of Japanese Government Bonds (JGBs) which is the object we want to study, and---last but not least---it has a relatively short history (we only had the data since November 1989).

†Note that we did not consider the correlation between longer maturity rates and macroeconomic factors. It still seems plausible that even during zero-interest rate periods, the inflation or foreign exchange rates have a large impact on long-term interest rates, and therefore multifactor models can certainly give a more realistic evolution of the whole yield curve. However, in this paper we chose to focus on the impact of the short rates on the whole yield curve.

‡This process can be a general diffusion as long as we can employ a numerical technique similar to the one we use here.

†The assumption that P is already an equivalent martingale measure (EMM) implies certain simplifications. If we start from the real world measure, then we will need to make a change of measure to formulate our pricing equation in an EMM. Because of the presence of jumps with possibly infinite number of sizes, there might be an infinite number of EMMs. So we would have to specify a way to find one EMM, possibly discuss about the choice of the optimal EMM (e.g. minimum entropy martingale measure), and the definition of no-arbitrage in this market. We do not consider this aspect of the discussion here. For an example of selection of EMM in the case of geometric Levy processes, see Miyahara and Fujiwara (2003).

‡Note that we could choose to place the endpoint at any rate ε instead of 0 and let the short rate evolve in the domain (ε,∞). This would not change the conclusions of this argument, but it would add one more εh term in equation (8) and would make the subsequent expressions of the bond price more complicated. To keep this paper as clear and concise as possible, as well as because economically 0 seems like the only floor that every agent would agree on for the short rate, we chose to fix the endpoint at 0.

†The smaller the jump size, the faster we expect the short rate to go back to zero after going out of it. As a result, the quantity J₀ may affect the price of interest rate derivatives. See Rinaz (2006) for a detailed discussion of the impact of each model parameter on bond prices.