# Discretizing the conditional variance in the Arbitrage Free Dynamic Nelson Siegel model

for my thesis I am trying to fit the correlated factor arbitrage free dynamic Nelson Siegel model to yield data. I use the Kalman filter to model this but since the model is in continuous time, I need to discretize the conditional mean and conditional variance. The conditional mean was not difficult but I can't succeed in discretizing the variance. The expression for the conditional variance is: $$V[X_t|Y_{t}] = \int_0^{\Delta t} \exp(-K^P s)\Sigma \Sigma' \exp(-[K^P]'s) ds$$ where $\Delta t = 1 / 252$ and

$$K^P = \begin{bmatrix} k_{11} & k_{12} & k_{13} \\ k_{21} & k_{22} & k_{23} \\ k_{31} & k_{32} & k_{33} \end{bmatrix}$$ and $$\Sigma = \begin{bmatrix} \sigma_{11} & 0 & 0 \\ \sigma_{21} & \sigma_{22} & 0 \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix}$$

I hope this is the place to ask this question and that you guys can help me, thanks in advance!

• What about performing a numerical integration? I guess your $\Delta t$ is not that big, right? – JejeBelfort Jun 2 '17 at 10:40
• That sounds like a good option. My $\Delta t$ is indeed quite small. Since I use daily data my $\Delta t$ is only 1/252. Thanks for your suggestion. – Pim Jun 2 '17 at 10:46
• Can you point to some references on this 'Arbitrage Free Dynamic Nelson Siegel model'? I am very interested. – user25064 Jun 2 '17 at 11:28
• This is the article I am studying: Christensen, Jens HE, Francis X. Diebold, and Glenn D. Rudebusch. "The affine arbitrage-free class of Nelson–Siegel term structure models." Journal of Econometrics 164.1 (2011): 4-20. – Pim Jun 2 '17 at 12:16

Regarding the comment we had, and since $\Delta t$ is rather small, numerical integration could be suited for purpose.

According to this Wikipedia article, three options are available.

Denoting by $f(s)$ the integrand, that is:

$$f(s) = \exp \left( -K^P s\right) \Sigma \Sigma' \exp \left( - [K^P]' s\right),$$ the quantity $V \left[ X_t | Y_t \right]$ can be approximated by the following rules (ranked in ascending order in terms of complexity and accuracy):

• Rectangle rule:

$$\int_0^{\Delta t} f(s)ds \approx \Delta t f\left( \frac{\Delta t}{2}\right)$$

• Trapezoidal rule:

$$\int_0^{\Delta t} f(s)ds \approx \Delta t \left( \frac{f(0) + f(\Delta t)}{2}\right)$$

• Decomposition rule for $n > 1$:

$$\int_0^{\Delta t} f(s)ds \approx \frac{\Delta t}{n} \left( \frac{f(0)}{2} + \sum_{k=1}^n f\left(k \frac{\Delta t}{n}\right) + \frac{f(\Delta t)}{2}\right)$$