Abstract
Over the past quarter century, mathematical modeling of the behavior of the interest rate and the resulting yield curve has been a topic of considerable interest. In the continuous-time modeling of stock prices, one only need specify the diffusion term, because the assumption of risk-neutrality for pricing identifies the expected change. But this is not true for yield curve modeling. This paper explores what types of diffusion and drift terms forbid negative yields, but nevertheless allow any yield to be arbitrarily close to zero. We show that several models have these characteristics; however, they may also have other odd properties. In particular, the square root model of Cox–Ingersoll–Ross has such a solution, but only in a singular case. In other cases, bubbles will occur in bond prices leading to unusually behaved solutions. Other models, such as the CIR three-halves power model, are free of such oddities.
The author has benefited from his discussions with his colleagues. This paper is reprinted from Advances in Quantitative Analysis of Finance and Accounting, 7 (2009), pp. 219–252.
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Notes
- 1.
When pricing bonds and other fixed-income assets in the presence of interest rate uncertainty, the equivalent martingale process that allows discounting at the interest rate does not result from assuming risk-neutrality on the part of investors as it does in the Black–Scholes model. Nevertheless, the term “risk-neutral” is still commonly applied. See Cox et al. (1981) for further discussion of this matter.
- 2.
In discrete time, the yield in Equation (102.2) is \(1 + Y \equiv {\left (\hat{\mathbb{E}}[{((1 + {r}_{1})\cdots (1 + {r}_{n}))}^{-1}]\right )}^{-1/n}\). By Jensen’s inequality this is less than \(\hat{\mathbb{E}}[{((1 + {r}_{1})\cdots (1 + {r}_{n}))}^{1/n}]\). So one plus the yield to maturity is less than the expectation of the geometric average of one plus the future prevailing spot rates. Since a geometric average is never larger than the corresponding arithmetic average, the yield to maturity must be less than the risk-neutral expected spot rate prevailing in the future.
- 3.
The lower bounds for all yields also increase with κ because the asymptotic value, Y ∞ = 2κθ ∕ (κ + γ), does. See Dybvig, et al. (1996) for an analysis of the asymptotic long rate.
- 4.
See, for example, Longstaff (1992), which solves the bond-pricing problem in the CIR square-root framework with r = 0, an absorbing barrier.
- 5.
- 6.
The drift term, μ( ⋅), must satisfy mild regularity conditions. If μ is continuous and the stochastic process is not explosive so that r = ∞ is inaccessible, then the density function for r ∈ (0, ∞) will exist for all future t with a limiting steady state distribution of
$$\frac{c} {{\sigma}^{2}r}\exp \left [2{\sigma}^{-2}{\int \nolimits \nolimits}^{r}\mu (x){x}^{-1}dx\right ]$$where c is chosen to ensure it integrates to unity. The density function may of course be zero for some values.
- 7.
The function coth(x) is the hyperbolic cotangent: \(\coth x \equiv ({e}^{x} + {e}^{-x})\left /\right. ({e}^{x} - {e}^{-x})\). The hyperbolic functions are related to the standard circular functions as:
$$\begin{array}{rcl} & & \sinh x = -i\sin \mathit{ix},\cosh x =\cos \mathit{ix},\tanh x = -i\tan \mathit{ix}, \\ & & \quad \mathrm{and}\ \coth x = -i\cot \mathit{ix}.\end{array}$$ - 8.
This condition is sufficient because zero is inaccessible for the CIR process \(\mathit{dr} = k(\overline{r} - r)\mathit{dt} + \sigma \surd \mathit{rd}\omega \) if \(2k\overline{r} \geq {\sigma}^{2}\), and by assumption, the specified process with drift \(\mu (\cdot ) \geq \frac{1} {2}{\sigma}^{2}\) for small r dominates the CIR process for sufficiently small r. This model is a special case of Heston, et al. (2007).
- 9.
In Figs. 102.2 through 102.6, the parameters used, κ = − 0. 03, σ = 0. 04, are the midpoints of the estimates by PW, though they were fitting their bond pricing function not the bubble-free function.
- 10.
Pan and Wu refer to their model as a three-factor (i.e., r, κ, σ) model with a single dynamic factor, r. In fitting their model they allow the parameters to vary over time, hence adding two additional sources of risk. This “stochastic-parameter” method has been widely used in practice since being introduced to term-structure modeling by Black et al. (1990). As Pan and Wu point out, this is inconsistent with their derivation, which assumes the parameters to be constant. Were the parameters actually varying, then bond prices would not be given by Equation (102.5) or (102.10). A true multifactor model giving results similar to PW would be a special case of the Longstaff and Schwartz (1992) multifactor extension to the CIR model with the constants in the drift terms set to zero. That is, \({\mathit{ds}}_{i} = -{\kappa}_{i}{s}_{i}\mathit{dt} + {\sigma}_{i}{\sqrt{s}}_{i}d{\omega}_{i}\) and r = s 1 + s 2 + s 3. The zero-coupon yield to maturity in this model is Y ({ s}, τ) = τ− 1 ∑B i (τ)s i . This Longstaff–Schwartz model is an immediate counterexample to PW’s claim that their formula is unique, but as in the PW model each of the state variables can be trapped at zero, and the interest rate becomes trapped at zero once all three state variables are so trapped.
- 11.
The average lag in the exponential average is δ ∫0 ∞ se − δs ds = δ− 1. An exponentially smoothed average is the continuous-time equivalent of a discrete-time geometrically smoothed average x t = (1 − η) ∑ηs r t − s . Geometrically smoothed averages were first suggested in interest rate modeling by Malkiel (1966).
- 12.
It is irrelevant for this discussion whether or not zero is accessible; if zero is not accessible for r then clearly neither r nor x can become negative. By comparison to the CIR process, however, we can determine that 0 is accessible for r (though not x). Specifically compare the CIR process with θ < 2σ2 ∕ κ to the process in Equation (102.21). The diffusion terms are identical and the excepted change under the bivariate process is smaller than for the CIR process whenever both r and x are less than θ. Since 0 is accessible for the CIR process, it must be accessible for the dominated bivariate process. The fact that r and x can be larger than θ does not alter this conclusion as the accessibility of 0 depends only on the behavior of r and x near 0.
- 13.
Only r is locally stochastic, so the risk premium is proportional to P r ∕ P and independent of P x ∕ P. The risk-neutral and true processes are equivalent if and only if ψ1 < κ so that r remains positive under the risk-neutral process as well.
- 14.
In particular,
$$\begin{array}{ll} b(\tau +\Delta \tau ) \approx b(\tau ) \\ \qquad -\left [\frac{1} {2}{\sigma}^{2}{b}^{2}(\tau ) + (\kappa + {\psi}_{0})b(\tau ) -\delta c(\tau ) - 1\right ]\Delta \tau & \quad b(0) = 0 \\ c(\tau +\Delta \tau ) \approx c(\tau ) -\left [({\psi}_{1} - \kappa )b(\tau ) +\delta c(\tau )\right ]\Delta \tau & \quad c(0) =0. \end{array}$$ - 15.
If κ ≥ 0 in the PW model, then the interest rate is eventually trapped at zero with probability one. If κ < 0, then the expected interest rate and variance become infinite, \(\mathbb{E}[{r}_{t}\vert {r}_{0}] = {r}_{0}{e}^{-\kappa t} \rightarrow \infty,\mathrm{Var}[{r}_{t}\vert {r}_{0}] = {r}_{0}{\sigma}^{2}\left ({e}^{-\kappa t} - {e}^{-2\kappa t}\right )\left /\right. \kappa \rightarrow \infty \), and there is an atom of probability for r ∞ = 0 equal to exp(2κr 0 ∕ σ2).
- 16.
In addition, the risk-neutral process must also be equivalent to the true process so all state variables must remain nonnegative under the latter as well.
- 17.
The central tendency parameter, θ, and the adjustment parameter, κ, can be zero just as in the PW model. For the three-halves process, the origin remains inaccessible even in these cases, and yields are still bounded below by zero. There is, however, no finite-variance steady-state distribution.
- 18.
Since the drift term is zero at an interest rate of zero, r t = 0 is technically an absorbing state. However, zero is inaccessible for all parameter values so r is never trapped there. To verify this define z = 1 ∕ r. Then using Itô’s Lemma, the evolution of z is \(\mathit{dz} = (\kappa + {\sigma}^{2} - \kappa \theta z)\mathit{dt} - \sigma \sqrt{z}d\omega \). Since z = ∞ is inaccessible for the square root process with linear drift, zero is inaccessible for r = 1 ∕ z. Note also that 2(κ + σ2) > σ2 so zero is inaccessible for z guaranteeing that ∞ is inaccessible for r in Equation (102.31).
- 19.
- 20.
We require that \({\psi}_{2} \geq -(\kappa + \frac{1} {2}{\sigma}^{2})\) so that the true and risk-neutral processes are equivalent. If this condition is not satisfied then the risk-neutral process is explosive, and the interest rate can become infinite in finite time. As shown in footnote 18, the risk-neutral process for z ≡ 1 ∕ r has \(\hat{\mathbb{E}}[\mathit{dz}] = (\hat{\kappa} + {\sigma}^{2} -\hat{\kappa}\hat{\theta}z)\mathit{dt}\). So if ψ2 violates the condition given, \(2(\hat{\kappa} + {\sigma}^{2}) < {\sigma}^{2}\), and 0 is accessible for z implying that ∞ is accessible in finite time for r under the risk-neutral (though not true) process.
- 21.
See Abramowitz and Stegum (1964) for the properties of the gamma and confluent hypergeometric functions.
- 22.
Holding σ constant, \(\hat{\kappa}\delta\) increases from 0 to 1 when \(\hat{\beta}\) ranges from 1, its lowest value, to ∞.
- 23.
When ψ2 > κ, the asymptotic bond price is \(P(r,\infty ) = \Gamma (\nu -\delta )\left /\right. \Gamma (\nu ){[-2\hat{\kappa}/{\sigma}^{2}]}^{\delta}M(\delta,\nu, 2\hat{\kappa}/{\sigma}^{2}) > 0\).
- 24.
The lower bound for any yield is zero since for every finite τ, there exists an interest rate r τ such that Y (r, τ) < ε for all r < r τ. The bound is not a uniform one for all τ, however, in that r τ depends on τ. The bound cannot be uniform since the asymptotic long rate is a positive constant.
- 25.
Zero can also be an inaccessible natural barrier for the process if μ(0) = ∞ even if σ(0)≠0.
- 26.
- 27.
The derivative of the hyperbolic cotangent is the negative hyperbolic cosecant function
$$d\coth x/\mathit{dx} = -{\mathrm{csch}}^{2}x = -4{({e}^{x} - {e}^{-x})}^{-2}.$$The third equality in Equation (102A.6) uses the identity coth2 x − csch2 x ≡ 1. The fourth equality follows from the definition of γ.
- 28.
The “half-angle” identity is \(\coth \frac{1} {2}x = (\cosh x + 1)\left /\right. \sinh x\). The final equality in Equation (102A.13) follows from the identity cosh2 x − sinh2 x = 1.
- 29.
The derivative of the confluent hypergeometric function is ∂M(a, b, x) ∕ ∂x = (a ∕ b)M(a + 1, b + 1, x). The asymptotic behavior as x → ∞ is M(a, b, − x) = Γ(b) ∕ Γ(b − a)x − a[1 + O(1 ∕ x)]. See Abramowitz and Stegum (1964).
- 30.
Note that \(\delta (\nu -\delta - 1) = {\sigma}^{-4}\left \{{\left [{\left (\hat{\kappa} + \frac{1} {2}{\sigma}^{2}\right )}^{2} + 2{\sigma}^{2}\right ]}^{1/2} +\hat{\kappa} + \frac{1} {2}{\sigma}^{2}\right \}\left \{{\left [{\left (\hat{\kappa} + \frac{1} {2}{\sigma}^{2}\right )}^{2} + 2{\sigma}^{2}\right ]}^{1/2} -\hat{\kappa} -\frac{1} {2}{\sigma}^{2}\right \} = 2{\sigma}^{-2}\) and \(\nu -2\delta - 2 = 2\hat{\kappa}/{\sigma}^{2}\)
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Ingersoll, J. (2010). Positive Interest Rates and Yields: Additional Serious Considerations. In: Lee, CF., Lee, A.C., Lee, J. (eds) Handbook of Quantitative Finance and Risk Management. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-77117-5_102
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