CDS-GARCH

Motivation

A Credit default swap (CDS) is a financial instrument to hedge and trade credit risk. The buyer of the CDS contract makes a series of payments (the CDS spread or premium) to the seller up until the maturity date of the contract. In return, the seller agrees to pay off the debt of a third party (the reference entity) if this party defaults on the debt. A CDS can be viewed as insurance against corporate default. Measuring and forecasting the volatility of CDS helps us answer important questions like:

• How much short run risk is an investor incurring by buying or selling a CDS?
• How volatile is the asset value of a firm?
• How volatile is the probability of default?

The short run risk of the CDS owner is the change in the "up-front" payment when he trades the CDS. The correct volatility concept for measuring the short run risk is therefore the volatility of the percent change in "up-front" payment which is approximately proportional to the change in spread, not the change in log spread. For other purposes the volatility of the change in log spread may be of most interest. Fortunately we have $${\mathrm{Vol}}_{t-1}\left(\Delta {\mathrm{spread}}_t\right)\approx {\mathrm{Vol}}_{t-1}\left(\Delta \text{log}\left({\mathrm{spread}}_t\right)\right)\times {\mathrm{spread}}_{t-1}$$

Definition

$\Delta \text{log}\left(\mathrm{spread}\right)$ and $\Delta \mathrm{spread}$ both have very high kurtosis and some serial correlation. Contrary to the case for many financial assets, there is little asymmetry in the volatility of CDS. Motivated by these statistical properties, we estimate a GARCH model for $\Delta \text{log}\left(\mathrm{spread}\right)$ with moving-average errors and fat-tailed innovations. Formally, we define $r_t^{\mathrm{CDS}}=\Delta \text{log}\left({\mathrm{spread}}_t\right)$ and adopt the standard GARCH specification:$$r_t^{\mathrm{CDS}}=\mu +{\epsilon }_t$$$${\epsilon }_t={\sigma }_t{z}_t$$$${\sigma }_t^2=\omega +\alpha {\epsilon }_{t-1}^2+\beta {\sigma }_{t-1}^2$$with the exception that the error term $z_t$ now follows a moving-average (MA) process$$zt={\zeta }_t+{\theta }_1{\zeta }_{t-1}$$and that the white noise innovation series $\left\{{\zeta }_t\right\}$ follows a Student's $t$ distribution with 4 degrees of freedom.

Estimation

V-Lab estimates all the parameters $\left(\mu \text{,}\omega \text{,}\alpha \text{,}\beta \text{,}{\theta }_1\right)$ simultaneously, by maximizing the log likelihood. The CDS-GARCH model captures volatility clustering in the CDS return data. The volatility is more likely to be high at time $t$ if it was also high at time $t-1$. Another way of seeing this is noting that a shock at time $t-1$ also impacts the variance at time $t$. Moreover, the MA structure and the fat-tailed distribution of the innovations capture the large excess kurtosis and the serial correlation observed in the CDS return data. The Student's $t$ distribution with 4 degrees of freedom has finite fourth moments which is required for consistency of standard errors. Tails fatter than this may be difficult to estimate.

Prediction

Let $r_t^{\mathrm{CDS}}$ be the last observation in the sample, and let $\hat{\omega },\hat{\alpha },\hat{\beta }$ be the Maximum Likelihood (ML) estimates of parameters $\omega ,\alpha ,\beta$ respectively. The CDS-GARCH model implies that the forecast of the conditional variance at time $T+h$ is:$${\hat{\sigma }}_{T+h}^2=\hat{\omega }+\hat{\alpha }{\hat{\sigma }}_{T+h-1}^2{\hat{z}}_{T+h-1}^2+\hat{\beta }{\hat{\sigma }}_{T+h-1}^2$$with the ${\hat{z}}_{T+j}^2$ specified by $${\hat{z}}_T^2={\left({\hat{\zeta }}_T+{\hat{\theta }}_1{\hat{\zeta }}_{T-1}\right)}^2$$$${\hat{z}}_{T+1}^2={\hat{\theta }}_1^2{\hat{\zeta }}_T^2+2$$ $${\hat{z}}_{T+j}^2=2\cdot \left(1+{\hat{\theta }}_1^2\right)\forall j\ge 2$$Where ${\hat{\zeta }}_T$, the last two residuals of the fitted MA(1) model, are readily available from the estimation. Thus we can forecast the conditional variance for any horizon $h$ by applying the above formula iteratively. Then, the forecast of the compound volatility at time $T+h$ is$${\hat{\sigma }}_{T+1\text{:}T+h}=\sqrt{\sum _{i=1}^h{\hat{\sigma }}_{T+i}^2}$$

CDS-GARCH-DYN

The specific model just described can be generalized to account for more MA terms and ARCH terms. The CDS-GARCH-DYN model considers that:$${\sigma }_t^2=\omega +\sum _{i=1}^p{\alpha }_i{\epsilon }_{t-i}^2+\beta {\sigma }_{t-1}^2$$$$z_t={\zeta }_t+\sum _{i=1}^{p^{\text{'}}}{\theta }_i{\zeta }_{t-i}$$and select $p$ and $p^{\text{'}}$ by Bayesian Information Criterion (BIC), also known as Schwarz Information Criterion (SIC).