CIR Yield Curves and Parameter Sensitivities

From CIR Bond Prices to Yields and Forwards

In the CIR model, the short rate is nonnegative and mean-reverts toward a long-run level. Its parameters shape the entire zero-coupon yield curve, not just the rate at one instant. The affine bond formula makes that dependence explicit: one coefficient multiplies the current short rate, while another captures the accumulated effect of the long-run level. Differentiating those coefficients gives instantaneous forward rates and parameter sensitivities.

Work under a risk-neutral measure \(\mathbb{Q}\) on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{Q})\) carrying a Brownian motion \(B_t\). The CIR short rate satisfies \(dr_t=\kappa(\theta-r_t)\,dt+\sigma\sqrt{r_t}\,dB_t\), where \(\kappa>0\), \(\theta>0\), and \(\sigma>0\). The stochastic integral here is an Ito integral: on every finite horizon its integrand is progressively measurable and belongs to \(\mathcal{L}^2\), meaning \(\mathbb{E}^{\mathbb{Q}}\int_0^T \sigma^2r_s\,ds<\infty\). We use the nonnegative CIR solution, for which the first moment is finite on finite horizons. The drift is \(\kappa(\theta-r_t)\); the diffusion coefficient is \(\sigma\sqrt{r_t}\).

The zero-coupon bond price at time \(t\) for maturity \(T\) is the risk-neutral price \(P(t,T)=\mathbb{E}^{\mathbb{Q}}[e^{-\int_t^T r_s\,ds}\mid\mathcal{F}_t]\). Write \(\tau=T-t\) for time to maturity and \(r=r_t\) when viewing the price as a function of the current rate. The affine CIR bond formula, obtained by reducing the pricing equation to Riccati equations, is

Here \(\tau\ge0\), and \(A(0)=1\), \(B(0)=0\). Define the continuously compounded yield \(y(\tau,r)=-\log P(t,T)/\tau\) for \(\tau>0\). Define the instantaneous forward rate \(f(\tau,r)=-\partial_T\log P(t,T)\), with \(t\) and \(r_t\) held fixed. Since \(\partial_T=\partial_\tau\), the affine form gives \(f(\tau,r)=B'(\tau)r-A'(\tau)/A(\tau)\).

The bond coefficients satisfy the Riccati identities \(B'=1-\kappa B-\tfrac12\sigma^2B^2\) and \(A'/A=-\kappa\theta B\), with \(B(0)=0\), \(A(0)=1\). These identities follow by differentiating the affine formula; they are also the coefficient equations in the affine pricing reduction. In particular, they yield convenient expressions for both curve quantities:

The integral in the yield expression is an ordinary deterministic integral. The equality follows by integrating \(-A'/A=\kappa\theta B\) from \(0\) to \(\tau\), using \(A(0)=1\). These formulas separate the effect of the current rate from the effect accumulated through the model parameters.

A Worked CIR Curve

Parameter Sensitivities

For fixed \(\tau>0\), the yield is differentiable in the model parameters because the displayed affine coefficients are smooth for \(\kappa,\theta,\sigma>0\). The long-run level has a particularly direct effect: \(B\) does not depend on \(\theta\), so

The coefficient \(B(s)\) is positive for \(s>0\), as is seen from its explicit formula. Therefore the yield increases strictly with \(\theta\) for every positive maturity. The same formula shows that the yield increases with the current rate at rate \(\partial y/\partial r=B(\tau)/\tau>0\).

The volatility effect can also be signed for nonnegative current rates. Let \(U(\tau)=\partial B(\tau)/\partial\sigma\). Differentiate the Riccati equation with respect to \(\sigma\), holding \(\kappa\) fixed:

Proposition: If \(r\ge0\), then \(\partial y(\tau,r)/\partial\sigma<0\) for every \(\tau>0\).

Proof. Define \(a(s)=\kappa+\sigma^2B(s)\), which is positive. Multiplying the linear equation for \(U\) by the integrating factor \(\exp(\int_0^\tau a(v)\,dv)\) and integrating from \(0\) to \(\tau\) gives

For every \(\tau>0\), \(B(s)>0\) when \(s>0\); all other factors in the integrand are positive. Thus \(U(\tau)<0\). Differentiating the yield formula gives \(\partial y/\partial\sigma=[rU(\tau)+\kappa\theta\int_0^\tau U(s)\,ds]/\tau\). The first term is nonpositive because \(r\ge0\), and the integral term is strictly negative because \(U(s)<0\) for \(s>0\). This proves the claim. \(\square\)

Mean-reversion sensitivity is less straightforward. If \(V=\partial B/\partial\kappa\), then differentiating the same Riccati equation gives \(V'=-(\kappa+\sigma^2B)V-B\), with \(V(0)=0\); hence \(V(\tau)<0\) for \(\tau>0\). But the yield also contains the factor \(\kappa\) multiplying the integral, so

The positive middle term competes with the other terms. Consequently, the sign cannot be inferred merely from \(V<0\). The maturity limits below give a concrete reason that mean-reversion sensitivity need not have one sign across maturities.

Short- and Long-Maturity Limits

At short maturities, \(B(0)=0\) and the Riccati equation gives \(B'(0)=1\). Hence \(B(\tau)=\tau+O(\tau^2)\). Integrating gives \(\int_0^\tau B(s)\,ds=\tfrac12\tau^2+O(\tau^3)\). Therefore \(y(\tau,r)\to r\) and \(f(\tau,r)\to r\) as \(\tau\downarrow0\). More precisely, \(y(\tau,r)=r+\tfrac12\kappa(\theta-r)\tau+O(\tau^2)\). If \(r>\theta\), this short-maturity yield initially decreases as \(\kappa\) increases; its first-order sensitivity is \(\tfrac12(\theta-r)\tau<0\).

For long maturities, the exponential terms in the affine formula give \(B(\tau)\to B_\infty=2/(\kappa+\gamma)\). Also, the denominator in \(A\) is asymptotic to \((\kappa+\gamma)e^{\gamma\tau}\), so

The last equality follows from \(\gamma^2-\kappa^2=2\sigma^2\). Since \(B(\tau)\) stays bounded, \(rB(\tau)/\tau\to0\); thus the long-maturity yield tends to \(q_\infty=2\kappa\theta/(\kappa+\gamma)\). The forward rate has the same limit: its formula and the Riccati equation give \(f(\tau,r)\to\kappa\theta B_\infty=q_\infty\).

This limiting yield increases with \(\kappa\), unlike the short-maturity sensitivity when \(r>\theta\). Indeed, differentiating \(q_\infty=2\theta\kappa/(\kappa+\gamma)\), using \(\partial\gamma/\partial\kappa=\kappa/\gamma\), gives \(\partial q_\infty/\partial\kappa=4\theta\sigma^2/[\gamma(\kappa+\gamma)^2]>0\). So a statement such as “more mean reversion always raises yields” is not valid at every maturity.

Common Pitfall: Confusing Yield with Forward Rate

A yield is an average over the bond’s full remaining life; an instantaneous forward rate is a maturity derivative. They coincide in the short-maturity limit and in the long-maturity limit in this time-homogeneous CIR model, but need not coincide at an intermediate maturity. A common incorrect calculation is to set \(y=rB(\tau)/\tau\), dropping the \(A\)-coefficient. In the finite-maturity example this gives \((7/17)/\log 2\), whereas the correct yield is \([\tfrac14\log(17/12)+7/17]/\log 2\). The omitted term is nonzero because the accumulated mean-reversion level affects the bond price through \(A\). The forward rate has a different calculation again: it uses the derivative \(B'(\tau)\), not \(B(\tau)/\tau\).