# CDS Spreads and Equity Volatility

Why does CDS spreads track the implied volatility on equities? What is the fundamental relationship that would keep the two inline from deviating too far from each other?

My speculation: Could it be because CDS spreads are not necessarily “pure” compensation of default risk but also include spread volatility risk and that this spread volatility is a larger driver of the CDS spreads than default risk compensation and because it is highly correlated with equity volatility, that is the reason why bond insurance and stock insurance move in tandem. If it is the case, then CDS spreads are not “pure” default risk compensation, has anyone then built a way to filter out the spread volatility risk to arrive at the “Pure CDS Spread Default Compensation Risk ” for an issuer? Knowing the tight relationship between has anyone built an implied probability of default based on equity volatility for an obligor?

Would appreciate all your opinion on the topic.

For an individual firm, a theoretical model of the capital structure was developed by Robert Merton in 1974.

The simplest form of this model assumes the firm has zero-coupon debt maturing at some future time $T$. Default is defined as the condition where the value of the firm's assets fall below the outstanding debt. The firm equity is viewed as a call option on the assets expiring at $T$ with strike price equal to the face value of the debt. The model can be used to infer the probability of default and the appropriate credit spread for pricing the risky debt. The inputs to the model are the current value of the firm's assets, the debt structure and, of great relevance to your question, the volatility of the assets. In practice, the asset volatility must be inferred from the observable realized or implied volatility of the stock price.

The Merton model has been applied widely in systems designed to estimate default probability. KMV (now part of Moody's) was one of the first companies to develop a commercially available system of this type. While the basic Merton model underlies such systems, there are many nuances in its implementation.

So, in short, there are indeed practioners who try to infer implied default probability from equity volatility.

Such a model can also be the foundation for a trading strategy - so-called capital structure arbitrage. It does not always work out well as the theretical equity-credit relationship can diverge for extended periods or at the worst possible time.

A good example of how capital structure arbitrage can go astray are the events surrounding the downgrade of General Motors debt to junk in 2005. After the downgrade, models signaled a long bond / short equity position. These positions were squeezed badly as Kirk Kerkorian began making an offer on the stock in an attempt to take control of GM.

The relationship between volatility and CDS is very interesting. Volatility in finance is synonym of risk. There are many aspects of volatility. There are 2 primary ways to find CDS premium, one is using structural model and the other is reduced form or intensity based model. Structural models use equity valuation, outstanding debt and equity volatility to determine the probability of default.

The strong correlation is what prompts the use of structural models. However, the information needed for correct information is only known to insiders and is disclosed at certain intervals through SEC disclosures. In the mean time, investors gauge this from equity performance in the market; surely insider or informed players buy puts pending doom or bad news.

VIX may be a good proxy for volatility. It is, generally, a fear gauge. However, it moves in the wrong direction, eg. Between Oct 12-19, 2011 it moved up along with the market. Such brief periods of disconnections have been seen many times. Volatility increase does not always indicate doom. It can also go up if the market is expecting good news, like take over, good earnings news. Exploding VIX can also be interpreted at the bottom signalling an end downside and explosion to the upside sometime, depending upon when the upside comes.

In a low interest environment the number of defaults go down. This causes the CDS premium to go down. This, besides volatility, is also reflected in the structural models.There are many factor models that take into account various accounting ratio and volatility to predict defaults more accurately. There are in-house in-cycle and out-of-cycle credit rating models companies use for rating counter parties. But volatility is a large integral part.

All in all the VIX should correspond to CDS.

RRL's answer is entirely correct in terms of the theoretical reason underpinning the relationship between equity IV and CDS spreads.

"CDS spreads are not “pure” default risk compensation" - no they are not since the ISDA Quoted Spreads assume a homogeneous Poisson process (implying that instantaneous default risk is a constant over the life of a contract) as per Documentation of the ISDA CDS standard model. We know credit risk changes, so the likelihood and impact of these changes are subsumed into the ISDA homogeneous intensity (& therefore ISDA spread).

Real Hazard Rates (instantaneous conditional default probabilities) under the Q-measure contain similar risks to those "lived-out" under the P measure: a slow-ish change in credit quality (diffusion) and sudden events (jumps) implying the need for (at least) an affine stochastic model (a la Duffie) under the Q-measure which compensates the investor for the likelihood and impact of these credit risk changes (as well as the underestimation of even the current credit risk) by the "credit risk premium" as per papers: