The covariance matrix

After having plugged in historical data, solving equations (49) and (50) for the values $N_{i,m}$ gives us a time series $(N_{i,m})$ ( $i=1,\ldots,d+2$; $m=1-M,\ldots,0$) of hypothetical historical realizations of the normal random variables (48). Now, the values

$$\widehat{\Sigma}_{i,j} = \frac{1}{M}\sum_{m=1-M}^{0} N_{i,m}N_{j,m}$$ (54)

can be used as estimates for the entries of the covariance matrix $\Sigma$. However, there is still something missing since we can not get the historical realizations $x_{i,m}$ of the stochastic factors of the interest rate model (49) directly from the market. Instead, we use the affine term structure (38) to derive them from the interest data distributed by the German Federal Reserve (cf. Subsection 6.1). One has

$$\underbrace{ \begin{pmatrix} R(t,\tau_1) \\ R(t,\tau_2) \end{pmatrix}} = \underbrace{ \begin{pmatrix} -\frac{\log A_1(\tau_1)}{\tau_1} -\f... ...nd{pmatrix}} \cdot \underbrace{ \begin{pmatrix} x_1(t)\\ x_2(t) \end{pmatrix}}$$ (55)

$$R_t \hspace{6mm} = \hspace{17mm} M_A \hspace{16mm} + \hspace{12mm} M_B \hspace{10mm} \cdot \hspace{3mm} x(t) .$$ (56)

Hence, we obtain by

$$x(t) = M_B^{-1}(R_t-M_A)$$ (57)

a time series $x_{i,m}$ ($i=1,2$; $m = -M,\ldots, 0$) by inserting the time series of the respective spot rates into (57). Slightly different from Fischer, May and Walther (2003), our suggestion is

$$\tau_1 = 0.5 ,\quad \tau_2 = 10.0 .$$ (58)

Equation (57) also returns the starting values $x(0) = (x_1(0), x_2(0))$ for the simulation of the factors $x_1$ and $x_2$. The computation of the values $x(0)$ implies a mathematically continuous continuation of the history of the spot rates $R(.,\tau_1)$ and $R(.,\tau_2)$ by the CIR-2 model. For other maturities than $\tau_1$ and $\tau_2$ there might be jumps in the dynamics of the respective sport rate (cf. Fischer, May and Walther, 2003). A simulation study of the same authors showed that for realistic time horizons the starting values have significant influence on the means of the simulated interest rates. Hence, a proper calculation of starting values is important.

Having executed the explained procedure, one can compute the empirical covariance matrix $\widehat{\Sigma}$ by (54). At this point, a further problem arises. The CIR-2 model works with uncorrelated brownian motions (cf. subsection 4.1). Nonetheless, the upper left ${2\times 2}$-submatrix of $\widehat{\Sigma}$, which theoretically should be the two-dimensional identity, may differ from the theoretical values. To stay in the proposed model, one can adjust the estimate $\widehat{\Sigma}$ by setting the upper left $2\times 2$-submatrix to the identity matrix. Doing this, it is important to check whether the new matrix is still positively definite as we afterwards have to carry out the Cholesky decomposition. In cases where positive definiteness gets lost, one should choose a symmetric positively definite matrix close to the proposed matrix with the identity in the upper left corner.

The proposed technique for the computation of the covariance matrix and the starting values should be suitable for any stochastic interest rate model with an affine term structure as in (38) (e.g. Vasicek-2).