Risk Latte - A Credit Default Swap (CDS) is a proxy for a Put Option on the Assets of a Firm

A Credit Default Swap (CDS) is a proxy for a Put Option on the Assets of a Firm

Rahul Bhattacharya
October 5, 2008

Perhaps no other financial derivative instrument in history has generated so much buzz and debate as the credit default swaps (CDS) have in the last year or so. Traders, risk managers, regulators, U.S. Congressman, financial journalists, financial TV anchors..., almost everyone has been talking of CDS of late. It seems that today everyone believes - erroneously so, however - that all hell in the financial markets is breaking loose largely due to these credit default swaps and CDS linked instruments.

What is a credit default swap (CDS)? It is a form of insurance against a company defaulting; in other words, if you have bought the bonds of a company then you'd want to buy protection - an insurance policy - such that if the company defaults on those bonds (the principal amount) then you get compensated by a certain amount (of the notional principal of the bonds).

A typical CDS is valued as a swap with one side being the premium payment from the protection buyer to the protection seller and the other side of the swap being the contingent benefit (in the event of a default) payment from the seller to the buyer. The CDS value - in basis points (100 basis points equals one percentage point) - expresses the cost of default, or the cost of insurance for the underlying bonds of the company.

However, in the classical Mertonian world (if there ever was any such thing as a Black-Scholes world or a Mertonian world) - a world where Black-Scholes-Merton's option dynamics hold - then the equity of a company is a call option on the assets of the company with the face value of the debt being the strike price (here, of course, the simplistic assumption is that the firm has only one type of debt and that is a zero coupon bond; and also, the firm pays no dividend). Mirroring the call is the put option which is actually the cost of default of the company. Of course, as mentioned above this holds only in a Black-Scholes-Merton economy with risk neutral probabilities. Let's see how that is the case.

If AT is the asset of the firm and K is the face value of the debt (i.e. the principal amount that needs to be repaid back to the debt holders at maturity) then equity of the firm equals (the payoff of the call at maturity, T):

And the value of the bonds (payoff at maturity) will be the minimum of the assets and the strike price, which is the face value of the bonds. This is because, either the bond holders will be repaid their principal whereby the company remains a going concern or they get a claim on the assets of the firm in the event the company goes bankrupt. Therefore, the value of the bonds equal:

This can be further simplified as follows given the min() and max() identity:

This means that by going long on bonds the bondholders are long the face value of the bond and short a put option on the assets of the firm with the strike price being the face value (principal) of the bonds.

Therefore, if the bond holders buy the bonds of a company and simultaneously buy a put option with strike of the put being the face value of the bonds then they are, theoretically, and at time t = 0, fully hedged. A long bond plus a long put equals the face value of the debt (the principal amount that the company has repay to the bond holders at maturity). The put option acts as an insurance against a default by the company.

In practice, however, the price of a CDS contract is never equal to a vanilla put option on the assets of the company because almost all of the assumptions of Black-Scholes-Merton are violated. In fact, other than perhaps the credit rating specialists who use KMV type models, very few, if any, practitioners even bother to estimate the value of a put option on the assets of a company let alone try to match it to CDS values.

But fundamentally, and as far as the theoretical construct goes, the two concepts are identical. A CDS and a vanilla put option on the assets of the company are both measures to express the cost of default of the company. Therefore, if one is willing to make adjustments to the risk neutral assumptions of Black-Scholes-Merton and model the credit of a firm using implied probabilities of default, then he can use a CDS as a proxy for the put option on the assets of the firm.

Politicians and other agents, such as regulators and especially financial TV commentators are wondering aloud as to how the total issuance of CDS contracts can outstrip the total outstanding value of bonds in our economy. After all, a CDS is an insurance policy against a company's default and therefore, for every dollar of a bond issued there should be no more than a dollar of CDS outstanding.

What is overlooked by many these days is that a CDS no longer is - has been for the past couple of years - just a hedging instrument. It is an instrument of speculation by virtue of it being nothing but a vanilla (i.e. simple) put option on the assets of a company, as shown above. If a speculator - a hedge fund, for instance - believes that a company's chances of defaulting is high then he'll go long a put option on the company's assets. But since no such instrument exists in the market, he will use a CDS as a proxy for such a put option.


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