In the previous section we spoke at length of default, credit risk, and in Equation 3.2 we have introduced, albeit without much explanation, St, the survival probability at time t of a certain borrowing entity. Any model of credit must somehow try to estimate the value of this probability, either explicitly or in the form of a forecast of the default of the borrowing entity.

Credit models can be assigned to three broad families. We shall briefly mention all of them, but we will concentrate on the last one since it will be a useful introduction to the pricing of bonds shown in later chapters.

Structural models of credit can be traced to Merton's work (see Merton [62] or Hull et al. [51] for an interesting analysis) in the mid-1970s. In these types of models, as the name suggests, we look at the balance sheet structure of a company and we observe the debt amount owed at a time T and the assets of the company. We consider the company in default if the value of the assets falls below the promised debt amount. The equity of the company is seen as a (European) call option on the asset of the company with a strike given by the debt amount.

As soon as we mention a financial option we know that there must be an implied volatility associated with it, in this case linked to the risk- neutral probability of default. While elegant in its simplicity, this model is more suited to an estimate of default made by an economist rather than an assessment made in a trading environment (the situation in which we position ourselves). The inputs needed for the model are the value of the company assets and their volatility, two values that are far from clear. A way of implementing it is to assume that the market capitalization represents the company's assets, and its equity's instantaneous volatility represents its volatility. All the debt is mapped onto a single payment at time T. While simple, easy, and often able to reproduce information found in the market, this family of models is not as suited to our needs as other trading-based models.

Most of us are familiar with credit rating agencies. These agencies, of which there are many–over 90% of the market is dominated by three, Standard & Poor's, Moody's, and Fitch–observe the health of a company and issue a rating of its debt. An investor holding the company's debt should use the rating as an indication of the company's creditworthiness.

The ratings are made up of a combination of letters and numbers starting with AAA (for all three agencies) for the highest-rated bonds/institutions. The big distinction is between investment grade debt, that is, a debt that is worth considering as an investment and junk status debt where the name is descriptive enough. Note that junk status does not mean that the debt (or the company to be more precise) has defaulted;[1] it simply means that it is very risky. In the 1980s the trading of junk bonds showed that a lot of profit could be made out of them. The change in credit ratings is usually fairly slow (compared to the change in other financial data) because it should be the outcome of a thorough analysis on the part of the credit agencies.

Ratings are very important because they affect many things in a secondary way. We have seen in Section 2.4.4 that AAA-rated institutions have a special collateral regime. Since we have seen that collateral can take the form of cash or liquid instruments, institutions that do pay collateral can offer AAA-rated securities. Should the debt they have offered be downgraded, it does not qualify anymore and something else needs to be posted. (This is something that was feared when Standard & Poor's downgraded U.S. debt, but did not materialize.) Another example is investment on the part of institutions or portfolio/fund managers. Investment managers are given guidelines on the percentage of debt they can hold in each rating category.

Despite their acceptance by market participants, the role of rating agencies is not without critique, as was the case after the U.S. downgrade of 2011 (a downgraded entity is usually the first to complain) or after the financial crisis of 2007 to 2008 where it was revealed that many securities that proved to be worthless were rated AAA.

It seems natural to wonder whether credit ratings can be used to estimate the probability of default of a company. They can and this is undertaken through credit rating models. In these models (see Altman and Kao [3] and Lando and Skodeberg [59] for an interesting example of empirical data and implementation) the key concept is the one of transition and the probability associated with it, that is, the probability of the transition of the rating of a company from one value to another. This is visualized through a transition matrix with ratings on both dimensions: the highest values will be the diagonal ones, that is, the highest probability that a rating will not change; the next highest will be the near diagonal values, that is, the probability that a rating will go up or down by one notch; and finally low values will be given to the probability that a rating will jump across many rating grades.

Although these models can be integrated (see Schonbucher [73] and [74]) within a risk-neutral framework, they are not particularly suited to pricing derivatives, which is the focus of our attention since, let us stress again, we are concerned with the activities of a trading institution. The amount of information available, in terms of transition data, is enormous, but it is all historical and, while forward looking in nature (after all ratings are given to forecast future events), not suited to obtain implied quantities. To price an option we do not estimate the volatility of a stock from its time series but we use a volatility surface implied from traded option prices:[2] in a similar way rating transitions are historical data that cannot easily be used to price derivatives.

The third and final family is the one made of spread-based models. We will be concentrating on this one as it is the most suited to the pricing of derivative instruments and also because it is the one most suited to the understanding of someone familiar with the fixed-income world.

While structural models and credit rating models can reproduce observable results, this is not quite the same as the rigorous calibration someone familiar with risk-neutral pricing is used to. As shown in great detail by Brigo and Mercurio [19], in the spread-based framework an analogy between interest rate models and credit models can be taken very far.

The same way we have a variety of short rate models in fixed income, we have a set of intensity models in credit ranging from simpler Hull and White-like models to more complex models involving jump diffusion (see, for example, Brigo and Alfonsi [17] or Duffle et al. [33]). The same way we have forward rate models in fixed income, such as HJM or LMM, in credit we have forward credit spread models claiming to have a more immediate relation to the market (see, for example, Brigo and Morini [20] or Das and Sundaram [29]).

Between these two subfamilies we shall concentrate on the former as our needs are simpler than the majority of market participants. While it is important to obtain an understanding of wider credit issues, our immediate goals are the understanding of how credit affects a bond price (laying the foundation for the subsequent chapters dealing with bond pricing) and the discounting of the risky cash flows of a loan. More advanced topics, such as the pricing of exotic credit derivatives of, say, the range accrual type, which would benefit from a forward rate model, are beyond our scope.

  • [1] Debt past default is not rated and is often referred to, somewhat euphemistically, as nonperforming.
  • [2] Sometimes if implied volatilities are not available, historical ones are used, but it is done knowing that it violates the principles of risk neutrality.
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