Valuation Inputs
As might be expected, pricing a first-to-default basket (FTDB) is more complicated than pricing a single-name credit-default swap. Any theoretical model of pricing basket swaps would include the following key inputs:
* number of reference entities;
* probability of default of reference entities and protection seller;
* default correlations between reference entities;
* default correlations between reference entities and protection seller;
* maturity of swap;
* expected recovery value of the reference entities.
The basket premium depends not only on the probability of default of each credit in the basket but also on the default correlation between these credits.
As the seller of protection on a basket, an investor is essentially paid for a single default plus the increased likelihood of the occurrence of default. Given that reference credits are typically less than perfectly correlated, the credit risk of a basket is greater than a single-name default swap for any of the basket constituents. The seller should be compensated for this risk with a higher yield on the basket than any single-name swap. The weaker the correlation relationship, the greater the degree of additional compensation that is required.
The following boundary conditions should apply to the basket premium:
1. Basket premium should exceed the single-name default premium on the weakest credit in the basket. This compensates the seller for the increased likelihood of default relative to any single reference entity.
2. Basket premium should be less than the sum of the premiums available for single-name default swaps for each credit in the basket, assuming the reference entities are positively correlated. This condition should be satisfied because the buyer is not buying protection on all the names in the basket but only on the first one to default.
Unlike a single-name swap, a first-to-default basket cannot be replicated in the cash market making it difficult to price this instrument from arbitrage relationships between the cash and the derivative markets. The practical approach to pricing a basket is derived from the dynamic hedging behaviour of dealers who buy protection on FTDBs as described below.
Dynamic Hedging Of The Basket
The hedging behaviour of a dealer provides some intuition behind the actual basket premium. A dealer that buys protection on a basket from an investor would normally hedge this transaction by selling default protection on each individual name in the basket. Chart 1 illustrates the hedge.
The amount of protection sold by the dealer in each name is known as the delta or the hedge ratio of that name. Among other factors, hedge ratios depend on default correlations and relative premiums of the single-name default swaps of the underlying credits. If single-name default swaps trade at similar levels, all credits would have similar hedge ratios assuming similar recovery rates.
As the underlying default premiums shift, the deltas will change and the hedges will need to be rebalanced dynamically. The efficiency with which the hedge can be managed is a key factor that determines the basket premium. For small movements in the hedge ratio, the dealer may not be able to sell or buy protection and may instead buy or sell bonds to hedge, thus taking on basis risk.
Following a credit event, the dealer will be forced to unwind the hedges on the other credits (assuming non-zero deltas for these credits). The cost of unwinding the hedge would depend on the spread movement for each of the non-defaulted credits. This, in turn, would depend on the correlation between the defaulted and the non-defaulted credits.
The greater this correlation, the greater the expected spread widening for a non-defaulted single-name default swap. This would imply a greater cost of unwinding the hedge. The dealer would therefore maintain a lower delta i.e., sell a lower amount of protection, to minimise losses from the unwind. This would, in turn, provide a lower premium to pay for the basket protection.
On the other hand, a low correlation would imply a lower expected spread change in a non-defaulted credit in the event of default and consequently a lower cost of unwinding that hedge. The hedger could therefore maintain a higher delta to manage the hedge i.e., sell a higher amount of protection. This provides a higher premium to pay for the basket protection.
Negative Carry & Long Gamma Trade
Consider the following basket example:
* Three-credit basket, each five-year single-name default swap trades at 100 basis points.
* At 50% correlation, model-implied breakeven basket premium1 is 236bps.
* The hedge ratio for each name in the basket is 68.4%. The hedge carry is therefore 205bps (68.4% x 100 x 3). Hedge carry is defined as the sum of the premiums received from selling single-name default swaps of credits in the basket.
* The hedge carry is less than the breakeven basket coupon and thus the dealer has a negative carry of 31bps.
For typical baskets, the hedge is a negative carry trade for the dealer, i.e., the breakeven basket premium is greater than the hedge carry. This is due to a positive net expected gain2 following a credit default.
Basket swaps cannot be fully replicated using only single-name default swaps. In other words, single-name default swaps cannot be used to hedge simultaneously the stochastic process of the spread movement of individual credits and the stochastic process of the actual default of any one of the credits. Dealers typically hedge only the spread process and are thus less than fully hedged. As a result, they pay a negative carry in return for a net expected gain on default. The difference also reflects the fact that the dealer is long gamma as described below.
Gamma is defined as the rate of change of delta. As the spread of an underlying credit widens, the dealer needs to sell more protection on that credit to rebalance the hedge. Thus the hedge ratio or delta for this credit increases, i.e. gamma is positive. The dealer's hedge is a long gamma trade and dynamic hedging benefits the dealer in the following way:
* If a reference credit widens, the delta increases and the dealer sells more protection increasing the carry on the trade.
* If a reference credit tightens, the delta decreases and the dealer buys more protection thus booking gains and reducing risk.
For typical baskets, a static hedge would have a negative carry that can be recaptured in the process of dynamic hedging. From an arbitrage perspective it is intuitively satisfying to infer that if the dealer is hedging only the spread process of underlying reference credits, the hedge should have a negative carry.
Default Correlation
Default correlations are key determinants of hedge ratios which determine basket premiums that dealers are willing to pay. The boundary conditions for the basket premium can be restated in terms of the default correlation as follows:
1. If the default correlation among the credits is equal to zero, the basket premium should be equal to the sum of all the single-name default premiums.
2. If the default correlation among credits is equal to one, the basket premium should be equal to the widest single-name default premium (or the lowest quality credit).
Basket premiums should, therefore, decline with an increase in correlation. A basket of uncorrelated credits trading at similar spreads produces the largest relative increase in premium compared to the average single-name default swap premium.
Default correlations impact the likelihood of multiple defaults up to a given time horizon. In practice, there is a lack of historical data that could be used to extract default correlations. Instead, market players use the asset correlation to calculate default correlation.
Asset correlations can be extracted from the ability-to-pay process of a portfolio of firms. Such a process is modeled for an individual firm as its market value of assets minus liabilities. Market inputs are equity and debt data. The asset correlation derived in this manner is deterministically related to the default correlation, i.e. one can be transformed into the other.
Another approach is to apply jump models. In these models, a spread correlation is used to determine the expected spread widening (or mathematically, a jump in the annualized default rate) of the non-defaulted credits in case one credit in the basket defaults.
This week's Learning Curve was written by Atish Kakodkar, v.p. in the credit derivatives research department at Merrill Lynchin London.
1 The breakeven basket premium is one that makes the expected value of the trade zero on day one. Different market players use different mathematical models to derive this premium.
2 Net expected gain following a credit default = Expected gain on the defaulted credit less the expected loss on unwinding the surviving credits.
