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Credit spread derivatives refer to the credit spread relative to the risk-free benchmark or to the differential of credit spreads of two risky assets. Credit spread derivatives isolate the effect of spreads as opposed to contracts on prices subject to the entire change yield, combining both interest rate variations and spread variations. Credit spread options provide the right to buy a future credit spread, as do other forward and option contracts.

The seller of a credit spread swap pays to the buyer a predetermined fixed spread while the buyer pays to the seller the spread of a risky asset over the risk-free rate or another security. The buyer is insensitive to spread variations. The seller gains if the spread widens and suffers a loss if the risky spread narrows.

Credit spread options are options whose payoff depends on the spread between a risky bond and a risk-free bond. Cash transactions do not disentangle the price variations due to interest rate variations from spread variations. A put on the risky debt does not work because of the effect of interest rate risk in addition to that of the spread risk. This is a protection against a fall in the price of the asset if the yield, inclusive of credit spread, moves in the opposite direction.

Only credit spread options strip the risk and isolate spread risk. Credit spread options have the usual value drivers: notional, maturity, underlying, strike spread, and premium. In addition, duration is important because it measures the sensitivity to a credit spread change.

Basket swaps refer to credit events of assets within a portfolio (basket) rather than single asset credit events. A "first to default" derivative is triggered by the first default event of any asset in the basket.

First-to-default products are correlation products. Diversification reduces the risk of the basket but it does not reduce the risk of first-to-default derivative. In fact, it is the reverse. The probability of triggering a basket derivative is higher than for a single asset. Consider a basket of two assets only. The rationale is based on joint survival probability[1]. The probability of zero default within the basket is the joint probability that all survive. The joint survival probability is higher when correlation is positive. The complement to one of the joint survival probability is the probability that any one of the assets defaults. Equivalently it is the probability that asset one, or asset two, or that the two assets default. If correlation increases, the joint survival probability increases, and the basket risk decreases. The probability that at least one asset defaults, triggering the derivative, also decreases. The credit risk of the basket varies inversely with correlation.

If we have two independent assets of default probabilities, 1% and 2%, the joint default probability is the product of the two default probabilities. The survival probability of both is the joint survival probability, or 99% x 98% — 97.02%. The probability that either one of the two assets defaults is simply 1- 97.02% — 2.98% if the events are independent. This is higher than any one of the two single default probabilities, which shows that the risk of first-to-default baskets is greater than for single assets. If the credit correlation is positive the joint survival probability increases above 97.02% and the probability that any one defaults is lower than 2.98%. The probability of triggering a first-to-default product is inversely related to correlation. Sometimes, the seller of such products is considered as selling default correlation, and is "short" correlation. Conversely, the buyer is "long" correlation.

This analysis explains the rationale for such derivatives. From the protection buyer standpoint, there might be very large exposures of high-grade obligors to hedge. Putting them together makes sense. From the protection seller standpoint, there is some comfort in finding high-grade exposures in the basket. In fact, the portfolio of high-grade exposures has a lower credit quality in terms of first-to-default events. Nevertheless, it remains an eligible investment in low-risk assets with an enhanced return compared to that of individual high-grade assets.

Other N-to-default products provide payments when at least N assets default. There are triggered when several assets default, such as 2, 3 or more assets of a portfolio. When N defaults are reached, the seller pays the loss to the buyer.

• [1] The discussion of joint probabilities and conditional probabilities is detailed in Chapter 31.

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