Consistent Re-Calibration in Yield Curve Modeling: An Example

Abstract

Popular yield curve models include affine term structure models. These models are usually based on a fixed set of parameters which is calibrated to the actual financial market conditions. Under changing market conditions also parametrization changes. We discuss how parameters need to be updated with changing market conditions so that the re-calibration meets the premise of being free of arbitrage. We demonstrate this (consistent) re-calibration on the example of the Hull–White extended discrete-time Vasiček model, but this concept applies to a wide range of related term structure models.

Appendix: proofs

Proof

(Theorem 1 ) The theorem is proved by induction.

(i) Initialization $t=k+1$. We initialize by calculating the first term $b_{t}^{(k)}=b_{k+1}^{(k)}$ of $\mathbf {b}^{(k)}$. We have $A^{(k)}(k+1,k+2)=0$. This implies, see (5),

$$\begin{aligned} A^{(k)}(k,k+2)=-b_{k+1}^{(k)} B(k+1,k+2) +\frac{\sigma ^2}{2} B(k+1,k+2)^2. \end{aligned}$$

From (8) we have

$$\begin{aligned} A^{(k)}(k,k+2)=r_kB(k,k+2)- 2\varDelta y^{\mathrm{mkt}}(k,k+2). \end{aligned}$$

Merging the last two identities and using $r_k=y^{\mathrm{mkt}}(k, k+1)$ provides

$$\begin{aligned}& b_{k+1}^{(k)} B(k+1,k+2) \\&\quad \,\,\,=\frac{\sigma ^2}{2} B(k+1,k+2)^2- y^{\mathrm{mkt}}(k, k+1)B(k,k+2)+ 2\varDelta y^{\mathrm{mkt}}(k,k+2)\\&\quad \,\,\,=z_1(\beta ,\sigma ,\mathbf {y}^{\mathrm{mkt}}_{k}). \end{aligned}$$

This is exactly the first component of the identity

$$\begin{aligned} \mathsf{C}(\beta ) \mathbf {b}^{(k)}=\mathbf {z}(\beta , \sigma , \mathbf {y}_k^\mathrm{mkt}). \end{aligned}$$

(38)

(ii) Induction step $t \rightarrow t+1 < M$. Assume we have calibrated $b^{(k)}_{k+1}, \ldots , b^{(k)}_t$ and these correspond to the first $t-k$ components of (38). The aim is to determine $b^{(k)}_{t+1}$. We have $A^{(k)}(t+1,t+2)=0$ and iteration implies

$$\begin{aligned} A^{(k)}(k,t+2)=-\sum _{j=k+1}^{t+1} b_{j}^{(k)} B(j,t+2)+\sum _{j=k+1}^{t+1}\frac{\sigma ^2}{2} B(j,t+2)^2. \end{aligned}$$

From (8) we obtain

$$\begin{aligned} A^{(k)}(k,t+2)=r_kB(k,t+2)- (t+2-k)\varDelta y^{\mathrm{mkt}}(k,t+2). \end{aligned}$$

Merging the last two identities and using $r_k=y^{\mathrm{mkt}}(k, k+1)$ provides

$$\begin{aligned}&\sum _{j=k+1}^{t+1} b_{j}^{(k)} B(j,t+2)\\&\qquad = \sum _{j=k+1}^{t+1}\frac{\sigma ^2}{2} B(j,t+2)^2 -y^{\mathrm{mkt}}(k, k+1)B(k,t+2)\\&\qquad \quad + \,(t+2-k)\varDelta y^{\mathrm{mkt}}(k,t+2)\\&\qquad =z_{t+1-k}(\beta ,\sigma ,\mathbf {y}^{\mathrm{mkt}}_{k}). \end{aligned}$$

Observe that this exactly corresponds to the $(t+1-k)$th component of (38). This proves the claim. $\square$

Proof

(Theorem 2 ) Using (12) and (10) for $t=k+1$ we have

$$\begin{aligned}& \left( m-(k+1)\right) \varDelta ~ Y(k+1,m)\\&\quad \,\,= -A^{(k)}(k+1,m)+ \left( b^{(k)}_{k+1} + \beta _k r_{k} + \sigma _k \varepsilon _{k+1}^*\right) B^{(k)}(k+1,m).\nonumber \end{aligned}$$

We add and subtract $-A^{(k)}(k,m)+r_k B^{(k)}(k,m)$,

$$\begin{aligned} (m-(k+1))\varDelta ~ Y(k+1,m)= & {} -~A^{(k)}(k,m)+r_k B^{(k)}(k,m)\\&+~A^{(k)}(k,m) -A^{(k)}(k+1,m)-r_k B^{(k)}(k,m)\\&+ ~\left( b^{(k)}_{k+1} + \beta _k r_{k} + \sigma _k \varepsilon _{k+1}^*\right) B^{(k)}(k+1,m). \end{aligned}$$

We have the following two identities, the second simply follows from the definition of $A^{(k)}(k,m)$,

$$\begin{aligned} -A^{(k)}(k,m)+r_k B^{(k)}(k,m)= & {} (m-k)\varDelta ~ Y(k, m),\\ A^{(k)}(k,m) -A^{(k)}(k+1,m)= & {} -b_{k+1}^{(k)} B^{(k)}(k+1,m)+\frac{\sigma ^2_k}{2} B^{(k)}(k+1,m)^2. \end{aligned}$$

Therefore, the right-hand side of the previous equality can be rewritten and provides

$$\begin{aligned} (m-(k+1))\varDelta ~ Y(k+1,m)= & {} (m-k)\varDelta ~Y(k,m) +\frac{\sigma ^2_k}{2} B^{(k)}(k+1,m)^2\\&+~ \sigma _k B^{(k)}(k+1,m)\varepsilon _{k+1}^*\\&-~r_k \left( B^{(k)}(k,m)- \beta _kB^{(k)}(k+1,m)\right) . \end{aligned}$$

Observe that the bracket on the third line is equal to $\varDelta$ and that $r_k=Y(k,k+1)$. This proves the claim. $\square$