No accurate method for pricing credit derivatives has yet been developed, and current models often rely on data that may be neither available nor relevant. Jessica James describes the challenge facing banks today

Seminars and courses with titles such as "Practical pricing of credit derivatives" are all the rage. They attract bankers keen to return to their desks and build a spreadsheet to price credit derivatives, using algorithms provided in the course and feeds from a wire service such as Reuters.

There may, these delegates imagine, be some quantities – analogous to implied volatility in the options market – that are needed, but they presume these will be available from feeds or traders. Instead, the courses may provide a lot of mathematics, or discussions on the costs of hedging. The one thing that delegates will almost certainly not learn is a practical way to price credit derivatives. And a phrase they will learn to dread will be "assuming that the probability of default is known".

Such courses are run because there is great demand for information on pricing credit derivatives. Why? Because, as yet, there is no robust way of finding the fair value of a credit derivative. There are academics, who know, in theory, how to price them if the default probability is known or there is a market variable which is purely indicative of the default probability. And there are market experts, who know how to hedge a credit derivative exactly, so that pricing it becomes irrelevant.

This is not to say no-one in the market has a pricing model. A bank may sell a credit derivative that has been priced using a model – but this price is probably different from the price that a different bank with a different model would quote. Both these prices are probably very different from the price at which either bank would agree to repurchase the credit derivative. And if the primary market in credit derivatives is illiquid, the secondary market is almost non-existent.

The first question when trying to price anything is: "Can I use arbitrage-based pricing?" This method uses other liquid instruments in the market to construct a value for the one being priced. If there were sufficient liquid credit instruments available to make this kind of pricing methodology feasible, it would be used. But the credit derivatives market is currently so sparsely populated that arbitrage based pricing is found rarely, if at all.

If the primary market in credit derivatives is illiquid, the secondary market is almost non-existent

In other markets, instruments are priced by taking the expected value and including the price of risk, which is derived from interest rates. If the probability of default of a company were known, it would be possible to arrive at the expected value of the credit derivative. However, to price the credit derivative, not only is it necessary to be able to arrive at an expected value, it is also essential to have a value for the price of risk in the credit market. Financial economic theory states that there is only a risk premium in equilibrium if default is correlated with a variable economic factor, such as consumption. Given that default likelihood is affected by economic conditions and cycles, it must be assumed that to price credit derivatives accurately, there is a price of risk in the credit market that must be known. If there is no price of risk, the risk preferences of market participants become irrelevant. If there is, they are important.

To illustrate risk preference, consider house insurance against fire. We pay more than the expected value for house insurance, because not only must the insurance company pay all its claims from the accumulated premiums, it must also pay its employees. But we are willing to pay the extra because we are not risk-neutral to the possibility of losing our homes – we are highly risk-averse.

The question, then, when considering the pricing of credit derivatives, is: are the buyers of credit derivatives risk-neutral or risk-averse? After all, the main reason for most early credit derivatives was that portfolio managers were averse to the credit risk held in their portfolios. Indeed, a significant bar to the development of the market has been that people are far keener to buy protection than sell it. However, as the market grows, it becomes apparent that investors want exposure to high yielding assets. It is also possible that short-term market makers will be risk-neutral. A reasonable conclusion is that a price of risk exists and must be included in pricing calculations, just as in markets such as interest rates or equities.

Calculating the price of risk for credit derivatives would be a major challenge. It is possible that the term structure of credit spreads might yield information about it, but not until some time in the future. Thus, it seems all efforts to find the correct value for a credit derivative, using both the expected value and the price of risk, will be in vain in the short term. The next step is to decide whether an approximation can be made using only the expected value and not the price of risk. It is important to realise that such a price can only be a ballpark estimate, and in some cases it may well be very wide of the mark. However, for many institutions, such an estimate of only the expected value of the credit derivative would be very useful. This initially does not seem so difficult. But problems do arise.

To derive an estimate of only the expected value of a credit derivative, consider a company that will receive a payment of $100, in a year’s time, from a credit-risky institution. The company wants to buy protection from a bank. The bank estimates there is a 5% chance of the company defaulting in the next year, and therefore a 5% chance that it will have to make a payment to the company. This payment will be $100, less any recovered amount, which it estimates, for the type of agreement in place, will be about $40 or 40% of the amount. Thus there is a 5% chance the bank will have to pay $60 to the company, and therefore the fair value of the default swap is $3. This seems easy, but several assumptions have been made which are not necessarily true. These are:

The default probability is accurately known;

The payment is a fixed amount ($100) independent of market rates; and

The recovery rate is accurately known.

In fact there are serious problems with all of the above, which will be addressed in turn.

The default probability
There are several methods to obtain the probability of default of an institution on its obligations.

Default probabilities from rating agencies, such as Moody’s Investors Service or Standard & Poor’s: these tend to be accurate with hindsight, but may lag behind current events, often changing after they should have. They are useful as estimates of default likelihood, but hardly market variables. Additionally, they specifically refer to the probability of default on debt which may differ significantly from the probability of default on other obligations. On top of this, not all companies are rated.

Credit spreads: this method looks at the spread between the interest paid for company debt and that paid on government debt of the same maturity. It assumes the sole reason for the higher rate of interest payable on company debt is that the company has a certain probability of default, while the government has none. Thus the company has, in effect, an option to default and the credit spread contains the value of this option. In theory, it then becomes possible to extract the probability of default of the company. But this is not easy using available models, which do not obtain realistic results. In addition, the credit spread is affected by many other issues as well as the perceived risk of default. Like using rating agency probabilities, this method is only strictly applicable to company debt and not to other agreements.

KMV Expected default frequency (EDF) model: this is a very different idea, and rests upon the assumption that a company goes bankrupt and defaults when its liabilities exceed its value (as calculated by its shares). Using the share price as an indicator of the probability of default is very appealing, not least because the default indicator is a market variable, affected by all the same factors that might be expected to influence the probability of default. But it does assume all movements in share price are caused by changes in default probability, which is not true. Share prices are affected by a huge number of variables and the default probability is difficult to extract cleanly using this method.

Default instruments in the market: this method currently holds great promise, and may well be the choice for future calculations as the market becomes more liquid. In a process analogous to the extraction of implied volatilities from the vanilla options market, the price of simple default instruments may provide a market estimate of the perceived likelihood of default. Despite question marks remaining over the best estimate of the recovery amount, there are signs that the market is moving in this direction. Indeed, it may be the only way to achieve a market value for a credit derivative, although the market is not yet liquid enough to make this generally possible.

The payment amount
In our example, we assumed that the payment amount being insured was fixed. In fact, many default swaps guarantee to replace some kind of floating cashflow, and so the market rates at the time become important.

The recovery rate
This was assumed to be known or easy to estimate. In fact, data on recovery rates comes from company debt rather than any other type of obligation, and is highly dependent on the nature of the debt, ranging from 80% for senior debt to 20% for junior subordinated debt. It is difficult to relate these probabilities to other transactions.

Hedge hunters
Today, a few brave institutions are using theories such as those outlined above to price and risk manage credit derivatives. However, the majority of institutions dealing in credit derivatives operate on the principle that "we do it where we can hedge it".

This robust, if somewhat limiting, hedging method can be illustrated with a total return swap: a bank’s counterparty wishes to gain exposure to a particular asset, without ever actually buying it. The counterparty undertakes a total return swap, whereby the bank pays it the total return (interest, dividends and any increases in value) and the counterparty pays the bank any decrease in value.

The bank could find a perfect hedge for this by actually holding the reference asset. If this is feasible, then it is ideal. A variation would be to purchase an option to buy the asset, and include this in the price of the credit derivative. Any bank sticking to this kind of policy with credit derivatives will always be fully hedged, but will be limited in the number and type of derivatives it can trade.

Perhaps the closest analogy with the credit derivatives market today is the options market some 25 years ago, shortly before the Black-Scholes pricing model was developed. At that time, the price of an option was simply whatever the trader said it was. At-the-money options would probably have some kind of agreed market value, because traders communicated well between themselves, but in- or out-of-the money options were illiquid and different banks would give them different price tags. Then along came the Black-Scholes model, with the idea that, in fact, what traders were doing to price options was to make instinctive guesses about how likely rates were to move to the strike in the future – in other words, to estimate the future volatility of the underlying rates.

Perhaps the closest analogy with the credit derivatives market today is the options market some 25 years ago

Almost overnight, the new formalism was adopted, with the new "trader-defined" market variable being the implied volatility rather than the price. This not only rationalised the pricing of vanilla options but, at a stroke, enabled a wide range of more complex products to be priced easily using the same principles. Today, the price and volatility of an option are almost synonymous in the market.

The credit derivatives market seems to be waiting for a similar revolution. There is a feeling that just around the corner lies some "Black-Scholes model for credit derivatives" and, after it is found, pricing will become simple.

However, the situation is not the same. There are plenty of models out there and there has been some excellent work on how to price credit derivatives as long as there is a clean way of extracting default probability information from the market. So far, however, there is no such data, and therefore market liquidity remains constrained. Credit derivatives are not yet at the same stage that the already liquid, vanilla options market was when Black-Scholes arrived.

To achieve true liquidity, the market needs liquidity. Only when an implied default probability can be easily extracted from a market instrument will credit derivatives have a true market price, allowing a liquid secondary market to develop. At present, only a few market screens show any credit derivative information at all, and it may be some time before pricing and risk managing credit derivatives becomes any easier.

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