Investigating the Performance of Non-Gaussian Stochastic Intensity Models in the Calibration of Credit Default Swap Spreads

Abstract

Most important financial models assume randomness is explained through a normal random variable because, in general, use of alternative models is obstructed by the difficulty of calibrating and simulating them. Here we empirically study credit default swap pricing models under a reduced-form framework assuming different dynamics for the default intensity process. We explore pricing performance and parameter stability during the highly volatile period from June 30, 2008 to December 31, 2010 for different classes of processes driven by Brownian motion, three non-Gaussian Lévy processes, and a Sato process. The models are analyzed from both a static and dynamic perspective.

Appendix

Stability Analysis of the Regularized Optimization Problem

The non-linear least square optimization problem defined in Eq. (16), that is

$$\begin{aligned} \hat{\varTheta } = \min _{\varTheta } \big (\textit{RMSE}(\varTheta )\big )^2 \end{aligned}$$

(21)

has neither a closed-form solution nor a global minimum. A numerical optimization routine is needed to find a relative minimum also because the gradient vector and the Hessian matrix related to the problem are difficult to express in closed-form: even if they can be computed, they have a messy expression. In this paper we follow the practical approach described in Fang et al. (2010); indeed we define the regularized problem

$$\begin{aligned} \hat{\varTheta } = \min _{\varTheta } \big (\textit{RMSE}(\varTheta )^2 + \rho \Vert \omega \cdot (\varTheta - \varTheta _0)\Vert ^2\big ). \end{aligned}$$

(22)

where $\rho$ is a constant term and $\omega$ is a vector defined to provide a comparable parameter sensitivity as the parameters may differ significantly in magnitude. The vector $\omega$ is given by $(1/\varTheta _0^1, \dots , 1/\varTheta _0^N)$, where $N$ is the length of $\varTheta$ and with “$\cdot$” we indicate the inner products of vectors. The choice is aimed at achieving a satisfactory calibration error and parameter stability over time.

In this appendix we study how, by increasing the value of the parameter $\rho$, the parameters, calibration errors and computational time vary. The selection of a proper $\rho$ is itself an optimization problem which has to be solved to find a solution to the original least squares problem. As already observed in Sect. 5.1, $\rho$ depends on the data at hand and on the level of error present in it. In Table 2 we report the results of the empirical study conducted over time and across all the 117 companies analyzed. More precisely, we show the lag-5 autocorrelation computed by considering the parameter time series of each company. Then we compute median and mean values across all companies. As expected, the parameter stability increases by increasing $\rho$, even if some parameters are more volatile than others. Additionally, we report median and mean values of the RMSE, of the average relative percentage error (ARPE) and of the number of function evaluations into the optimization routine. The number of function evaluations is a proxy for the computational time. These values are computed both over time and across all the 117 companies analyzed. By increasing the value of $\rho$, we find that the calibration error increases in the CIR and in the Gamma-OU cases, it remains quite stable in the Sato Gamma case, and in the IG-OU and in the VG-OU cases it reaches the minimum value when $\rho {=}10$. The computational time decreases in the CIR case and, conversely, increases in the Sato-Gamma case. The value $\rho {=}100$ shows a good balance between the calibration error and the parameter stability and for this reason we selected this value in the main text of the paper. In the empirical study we solve a large number of problems of the form of (22): for each model and across the 117 companies we consider 655 daily observations for a total of more than 380,000 daily calibration exercises. For this reason, even if the selected value of $\rho$ may be not the optimal value, it is sufficient for our purposes as it provides us with an acceptable calibration error and parameter stability. As shown in Table 2, when $\rho$ is equal to 100, the median values for the ARPE are just over 2 per cent (less than 1.5 per cent if we do not consider the Sato based model) and the median lag-5 autocorrelations are all above 0.9. Finally, we note that Table 2 confirms that the VG-OU model outperforms its competitor models while having a comparable degree of parameter stability over time and of computational complexity, not only when $\rho$ is equal to $100$, but also for all other selected values of $\rho$.

Table 2 Lag-5 autocorrelation of the parameters, calibration error and computing time