**4. Arbitrage-Free Nelson–Siegel**

The Arbitrage-Free Nelson–Siegel (AFNS) static model for daily yield curve fitting was derived by Christensen et al. (2011) from the standard continuous-time affine Arbitrage-Free formulation of Duffie and Kan (1996). The AFNS model almost matches the NS model except by the yield-adjustment term −*<sup>C</sup>*(*<sup>τ</sup>*,*τM*) *τM*−*τ* . In fact, the definition of the AFNS static model in Christensen et al. (2011) is given by

$$AFNS(\tau) = \beta\_1 L\_1(\tau) + \beta\_2 L\_2(\tau) + \beta\_3 L\_3(\tau) - \frac{C(\tau, \tau\_M)}{\tau\_M - \tau}. \tag{5}$$

The AFNS model built by Christensen et al. (2011) considers the mean levels of the state variable under the Q-measure at zero, i.e., Θ*Q* = 0. Thus, −*<sup>C</sup>*(*<sup>τ</sup>*,*τM*) *τM*−*τ* have the form

$$\begin{aligned} -\frac{\mathbb{C}(\boldsymbol{\tau}, \boldsymbol{\tau}\_{\mathcal{M}})}{\tau\_{\mathcal{M}} - \boldsymbol{\tau}} &= -\frac{1}{2} \frac{1}{\tau\_{\mathcal{M}} - \boldsymbol{\tau}} \\ \sum\_{j=1}^{3} \int\_{\boldsymbol{\tau}}^{\tau\_{\mathcal{M}}} \left( \boldsymbol{\Sigma}^{\prime} \boldsymbol{B} (\boldsymbol{s}, \boldsymbol{\tau}\_{\mathcal{M}}) \boldsymbol{B} (\boldsymbol{s}, \boldsymbol{\tau}\_{\mathcal{M}})^{\prime} \boldsymbol{\Sigma} \right)\_{j,j} ds. \end{aligned}$$

Considering a general volatility matrix (not related to the dynamic model for forecasting the yield curve)

$$
\Sigma = \begin{pmatrix}
\sigma\_{11} & \sigma\_{12} & \sigma\_{13} \\
\sigma\_{21} & \sigma\_{22} & \sigma\_{23} \\
\sigma\_{31} & \sigma\_{32} & \sigma\_{33}
\end{pmatrix},
$$

Christensen et al. (2011) show that an analytical form of the yield-adjustment term can be derived as

$$\frac{\mathbb{C}(\tau,\tau\_M)}{\tau\_M-\tau} = \frac{1}{2} \frac{1}{\tau\_M-\tau} \int\_{\tau}^{\tau\_M} \sum\_{j=1}^3 \left( \Sigma' B(s,\tau\_M) B(s,\tau\_M)' \Sigma \right)\_{j,j} ds.$$

They also estimated the general volatility matrix for maturities measured in years as

$$
\Sigma = \begin{pmatrix}
0.0051 & 0 & 0 \\
0 & 0.0110 & 0 \\
0 & 0 & 0.0264
\end{pmatrix}
$$

and *λ* ˆ = 0.5975 for independent factors AFNS model, solving the yield-adjustment equation for arbitrage-free conditions.

Note that the adjustment-term *<sup>C</sup>*(*<sup>τ</sup>*, *<sup>τ</sup>M*) is only time-independent. In other words, it is a deterministic function that depends only on the maturity of the bond. Thus, let the auxiliary function <sup>Γ</sup>(*τ*) be

$$
\Gamma(\tau) = -\frac{\mathcal{C}(0,\tau)}{\tau}\_-
$$

As in the Dynamic Nelson–Siegel model, the Dynamic AFNS model describes the AFNS static model evolving over time. The Dynamic AFNS model is defined by

$$\begin{aligned} s\_t(\boldsymbol{\pi}) &= \beta\_{1,t} L\_1(\boldsymbol{\pi}) \\ &+ \beta\_{2,t} L\_2(\boldsymbol{\pi}) \\ &+ \beta\_{3,t} L\_3(\boldsymbol{\pi}) + \Gamma(\boldsymbol{\pi}) + \boldsymbol{\epsilon}\_t \\ t &= 1, \dots, T \end{aligned} \tag{6}$$

where the loadings *<sup>L</sup>*1(*τ*), *<sup>L</sup>*2(*τ*) and *<sup>L</sup>*3(*τ*) are the usual functions of (3) and the coefficients *β<sup>i</sup>*,*<sup>t</sup>* are autoregressive processes described by

$$
\beta\_{i,t} = c\_i + \phi\_i \beta\_{i,t} + \eta\_{i,t} \qquad i = 1,2,3 \tag{7}
$$

where the parameters *ci* and *φi* and the coefficients *β<sup>i</sup>*,*<sup>t</sup>* are estimated and predicted as described in Section 3.

If *β<sup>i</sup>*,*<sup>t</sup>* is a linear function of *βj*,*<sup>t</sup>* where *i* = *j*, or there is a cointegration, the component *β<sup>i</sup>*,*<sup>t</sup>* can be predicted as a linear function of *βj*,*t*, and the model can be simplified.
