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:: Introduction to Credit Risk Models ::

In this article we will cover the basics of credit risk (also known as default risk), it's  importance in financial modelling and the 2 main market techniques for incorporating credit risk into pricing options. We start right at the beginning by defining credit risk, then introducing some of the jargon used by credit risk analysts, before moving onto the modelling aspects.

What is credit risk? In short, it is the risk arising from the possibility of firms defaulting and being forced to file for bankruptcy and liquidate. Any agreement involving two counterparties involves such risks; e.g. corporate bond holders must accept the chance that the issuing company may default, (OTC) option holders must acknowledge the possibility of the option writer defaulting on the payoff payment, etc. This is referred to as counterparty credit risk and a risk taker must be rewarded for taking on this additional risk; this reward almost always manifesting itself as a lower price for the risky instrument, compared to the risk free counterpart. This is a fundamental property for any rational market and the reason for this is simple: a cheaper instrument gives the holder a higher yield, or return on investment. In many cases, credit risk is not counterparty related. This is particularly true for exchange traded derivatives where the cash flow is guaranteed by the exchange. Here, the credit risk extends from the default of the underlying. Hence, although the option writer (the counterparty) may not default on the payout for a call option, the issuer of the actual stock underlying  the call option may default, whereby the stock price crashes and the option expires either worthless (if a call) or deep in the money (put). The exact nature of what happens (the option can even become void when the underlying stock defaults) is written into the agreement between the option writer and buyer. In this (and successive articles) we shall predominantely be concerned with credit risk of the former type.  

Why do we need to worry about credit risk? To arrive at a better price for our bond/option! Significant default risks result in lower prices and allowances for this must be made. The question we must ask our selves is how significant is the risk of default? This depends on the market we are in, the actual counterparty we are dealing with and the nature of the contract being considered. In emerging markets the risk of default is accepted to be higher than in developed markets. Hence, you may safely assume that the risk of default is much more significant for a, say, South American corporate than a FTSE-100 listed corporate. The nature of the contract is also an important factor to consider. OTC derivative contracts involve significantly more risks to the contract holder than an exchange traded contract. A classic example is the Forward versus Futures contract. The former are OTC, whereby the cash flows occur directly between the 2 parties involved at the maturity of the contract. The Futures contract in contrast are exchange traded and the cash flows are almost guaranteed by the exchange (this is achieved via margin payments). The credit issue is thus much less significant (in most cases it is dropped altogether). We note here that in the case of bonds, although they are exchange traded, they are not guaranteed by the exchange as they are not derivatives. In the case of bonds, the default risk thus always persists.

How is credit risk characterised? In the markets, this is done via the credit rating (creditworthiness/credit quality) of the company in question. The 2 main credit rating agencies are Moody's and Standard and Poor's (S&P), who 'rate' the debt issued by listed companies in accordance to the risk of these debts not being fully redeemed due to default. Hence, a corporate of lower credit quality than another must offer an investor a higher yield on its debt (debt=bond). In the developed financial markets this is often measured by comparing the yield on the bond to the yield from a risk free government bond: the credit spread of the risky debt over the risk free yield. A corporate bond with a high credit spread is deemed to be risky (how high is 'high' is another question which will not be answered here).

What happens when a company defaults? While an accountant can much better answer this question, here we will highlight the key steps. Once bankruptcy is declared then the company must liquidate and the debts repaid. All outstanding debt is ranked by 'seniority'. This is analogous to the difference between 'preference shares' and normal shares that companies sometimes issue (the former are paid dividends before the latter in the event of economic hardship/default and are given higher precedence). 'Debt' here, does not only necessarily mean bonds. It involves ALL capital that is owed, e.g. bank loans, derivative settlements, etc. In the case of options the payout after default will often be a percentage of the actual terminal value of the option. This is termed the recovery rate. Hence, the credit risk comes down to the probability of realising the full payout versus the recovery rate.

Modelling credit default

There are 2 approaches to modelling credit risk.Modelling credit risk means modelling the occurrence of default,when it will happen.The first model is the Firm value model proposed by Merton.The intuition simple: A company defaults when its share price (or the company's 'value') falls below a prescribed barrier.Two approaches can be taken. Default can be defined as the first time that the share price hits the barrier, in which case the actual dynamics of the share price path is important. Alternatively, only the final share price is considered. At this time, if the share price is below the prescribed barrier then the company is deemed to be in a state of default.

The advantage with the firm value approach is the ability to hedge default using the counterparty share price. The simple binomial framework above will be used as example. The price of the bond had there been no risk of default would have been 1/(1+0.05). Now,we introduce the possibility of the bond issuer defaulting; when the issuer's share price falls below (and C(d) < ) the issuer is deemed to be in a state of default.  At this point, the bond is redeemed at below par ( < 1).  This is now a standard binomial pricing problem:  it is identical to a derivative on the share price C(t), which pays out 1 if the upper node is reaches and otherwise.  If you wanted to model the possibility of default occurring before time T then a more granular tree is required, and we could decide whether default has occurred at each node while discounting backwards through the tree, much like the technique adoped for American options.  Notice also how closely the properties of this derivative resemble that of Barrier options.

The example above assumed constant interest rates for simplicity. In practice this is also random and the pricing is more complicated in that there are now 2 sources of randomness:the interest rate and the default time.

The second approach (Intensity approach) models default as a sudden, unpredictable event.This is achieved using the first arrival time (the first jump time) of a Poisson process (see earlier article). In this model default hedging is not always possible as there is not always a traded instrument that characterises the arrival of default. In short, the Binomial model reduces to:

In this model the default cannot be hedged, as there is no traded instrument that characterises its default. How then do we price the risky bond? The key is to determine the risk neutral probability p. Recall that this is a probability that can be used solely for pricing derivatives and needs not reflect the actual probability of default occurring. The mechanics of this will not be discussed here,we will only mention that once the probability is determined, then the risky bond can be prices as the discounted expectation of the final possible payouts from the bond. Notice also that this value will always be less than the risky free bond price, provided 0 < d < 1.

In Summary:

• Credit risk is important when dealing with financially weak counterparties. OTC derivatives introduce this risk as exchange traded derivatives are usually guaranteed by the exchange.
• For a derivative on a stock S, written by a party A,the credit risk is 2 fold. Either the issuer of the stock can default (stock price crash) or the counterparty A may default on the payout of the derivative. The main theme that will be addressed in future articles is the latter.
• Two types of credit risk modelling: Firm value (Structural) models and Intensity models. The former allow default hedging, but are unpopular due to difficulties in calibrating to market data. The latter are very popular but make hedging difficult (if not impossible).

   

Written by Samy Mohammed.

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