Stochastic Volatility and Mean Drift in the Short Rate Diffusion : Sources of Steepness , Level and Curvature in the Yield Curve

The paper demonstrates that it is possible to capture all main structural features of the short-rate diffusion, i.e., the complex conditional heteroskedasticity and the autocorrelation patterns, in an intuitively appealing and fairly manageable nonlinear, three-factor model. The factors are the short rate itself, the log-volatility of the diffusion, and a time-varying, but stationary, mean level towards which the short rate reverts. The additional two factors effectively extend the (generalized) Cox-Ingersoll-Ross model to a setting characterized by slow mean drift and stochastic volatility. The mean drift may be indicative of either slowly evolving inflationary expectations or time-variation in the required real interest rate, or both. We show that the factors in straightforward fashion may be identified with forces that induce shifts in the steepness, level and curvature of the yield curve. Parameter estimation and specification testing is conducted using a novel implementation of the Efficient Method of Moments (EMM) procedure developed by Gallant and Tauchen. The method provides consistent and efficient estimation of the interest-rate diffusion and delivers powerful model diagnostics. The performance of the methodology is noteworthy. We are not aware of any alternative estimation strategy that achieves the consistency and efficiency of the EMM for this type of nonlinear stochastic volatility model, even in a discrete-time setting. The short-term interest rate establishes the price at which funds may be transferred at no risk over short horizons. It therefore provides a market rate determining the tradeoff between consumption today and tomorrow. In an intertemporal setting, the expected future path of the short rate will similarly affect the optimal consumption-investment plan of individual agents. It is thus not surprising that the short rate is a critical benchmark for the pricing of the full range of financial assets. However, the specification of an appropriate short-rate model continues to both intrigue and puzzle empirical financial economists. The most useful theoretical models are cast in terms of continuous-time diffusions, and the earlier studies assert that the short rate follows a specific parametric one-factor continuous-time model, e.g., Merton (1973), Vasicek (1977), and Cox, Ingersoll and Ross (1985), CIR. These models are particularly convenient as they afford closed form solutions for the yield curve and certain interest-rate derivative securities. Unfortunately, it has become increasingly clear that these one-factor models provide poor descriptions of the actual short-rate process. This realization has sparked research into alternative, and less restrictive, oneand multi-factor models of the short rate as well as the entire yield curve. In this paper, we aim to develop a continuous-time model for the short-rate process that is consistent with the major stylized facts and fares well when confronted with the relevant specification tests. Our study focuses exclusively on representations of the interest rate dynamics for which the short rate, r t , solves a special case of the following general stochastic differential equation (SDE): where Wt is a standard Brownian motion, and μ t andσt , the drift and diffusion functions, depend (1) drt μt ( rt ) dt σt ( rt ) dWt on the short rate itself, and possibly other state variables. 1 For example, the one-factor Chan, Karolyi, Longstaff and Sanders (1992), or CKLS, specification: is obtained from equation (1) by letting μ t (rt ) = κ (μ rt ) and σt (rt ) = σ rt. According to this (2) drt κ (μ rt ) dt σ r γ t dWt , γ ≥ 0, representation, r t reverts towards the mean level μ, with κ measuring the speed of the mean reversion. The presence of the r t γ term in the diffusion coefficient rules out negative rates and induces a form for conditional heteroskedasticity, as it implies that volatility depends on the level of the interest rate, the so-called level effect . Further imposingγ = 1⁄2 results in the one-factor CIR model, whileγ = 0 produces the Vasicek (1977) model. 1 Since the dynamics is specified directly, the models represented by equation (1) generally do not fall within the Heath, Jarrow and Morton (1992) framework. Only in special cases will models in the latter class conform to a Markovian specification, see, e.g., Ritchken and Sankarasubramanian (1995).