Volatility Level Dependence and Linear-Rational Term Structure Models

ABSTRACT

We outline a subclass of linear-rational term structure models, based on CEV dynamics. A tractable and arbitrage-free term structure results from the linear-rational aspect of the model, independently of the term-structure volatility dynamics that follow from the CEV specification. This specification is devised to capture a flexible degree of volatility-level dependence, i.e., the degree to which yield-curve volatility depends on yield levels. We estimate the model based on a panel of South African swap rates, and extract the degree of volatility-level dependence inherent in the time series of rates, without interference from the shape of the swap curve. The CEV exponent parameters are found to be essential for matching the low degree of volatility-level dependence that tends to be observed in the high interest-rate environments of emerging markets.

KEYWORDS:

• Interest-rate volatility
• volatility-level dependence
• term structure models
• linear-rational models

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are available from Bloomberg. Restrictions apply to the availability of these data, which were used under license for this study.

Supplementary material

Supplemental data for this article can be accessed on the publisher’s website.

Notes

1. Potential arbitrages are of two kinds: based on trading various points on the yield curve, and based on additional claims dependent on the yield curve.

2. Realizations, unlike expectations, are associated with sampling error, and exert significantly less sway on, say, a likelihood function. See Duffee and Stanton (2012) for a detailed study of this issue; they focus on the market price of risk, which relates expectation-based cross-sectional behavior to realization-based time-series behavior.

3. This assumption is only necessary if ${\gamma }_i<1/2$. In this case, with the assumption, Andersen and Piterbarg (2007) derive the stationary distribution of ${\left(X_t\right)}_{t\ge 0}$.

4. See, e.g., Lindsay and Brecher (2012) for proof that a driftless CEV processes (with exponents no larger than one) are martingales. The mean reversion that we include further stabilizes the process. Andersen and Piterbarg (2007) derive the density of a mean-reverting CEV process, which can integrated to show the finiteness of process variance, and indeed to deduce the effect of the mean-reversion parameters.

5. Criticism includes Bliss and Smith (1998), who find the key result to be overstated on the basis of overlooked policy regime changes, and Treepongkaruna and Gray (2003), who find the results to vary greatly from market to market.

6. The data were sourced from Bloomberg, specifically their composition of quotes from the relevant banks and brokers.

7. While our data is available at a daily frequency, we prefer weekly observations because this avoids any inhomogeneity caused by weekends (i.e., the change from Friday to Monday is not necessarily comparable to other daily changes). This weekly approach is taken by, e.g., Filipović, Larsson, and Trolle (2017) and Backwell (2021). Daily interest-rate data has been used for maximum-likelihood estimation (see, e.g., Filipović and Trolle (2013)), but one should be cautious in emerging markets, where the data may be not be perfectly liquid, and may show some staleness when viewed at a daily frequency.

8. This is because (1), which flows into (14) below, does not explicitly account for the possibility of default.

9. See Jakarasi, Labuschagne, and Mahomed (2015) for an approach to this specific problem.

10. The procedure should strictly speaking be described as a quasi-maximum likelihood, as the state values that feed into the volatility of the transition, as per (9), are not known, and are replaced by the filtered estimates. As discussed in §B of Filipović et al. (n. d.), this approximation is not significant.

11. Note that the CEV exponents give an indication of the level-dependence exhibited by the state variables, but not an indication of how important the state variables are for fitting the term structure. Recall from Section 2.1 that the CEV exponents do not enter the bond pricing function. For instance, our second state variable in the CEV model has a virtually constant volatility (given the extremely low exponent estimate) but this does not bear on its role in fitting the term structure.

12. If the log-likelihood increased by just $5$, we would still be able to reject the null hypothesis that the LRCEV model brings no meaningful improvement relative to the LRSQ, at a $99\mathrm{\%}$ level of significance. If the increase were $7$, this improves to a $99.9\mathrm{\%}$ level. Our observed log-likelihood increase is well beyond this, leading to a clear test result. Note that one must multiply the log-likelihood change by two in order to have a chi-squared-distributed statistic given the null hypothesis, with two degrees of freedom (as two additional parameters are included in the LRCEV model).

13. Note that Section 2 applies regardless of how $\alpha$ is specified; model implementation is based around (7), whether $\alpha$ is specified directly or, as in our case, as some function of the other parameters.

14. This does mean that negative rates, as low as one negative percent, are theoretically possible. They are highly improbable however, with the bulk of rate distributions sitting significantly away from zero, given the relatively high interest rates considered.

15. Note that this is distinct from the average value of the metric, which is much more heavily swayed by the spikes in the unshifted LRCEV model.