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at time i. Notice again that we are discounting the future payments of the
swap using the LIBOR curve and that we are omitting the repayment of
principal at the end of the contract.
We are now in a position to derive the market value of the IRS embedded
in the asset swap. From the point of view of the asset swap buyer, who
receives the ļ¬‚oating and pays ļ¬xed, we have:

V IRS (0, N ) = V LL (0, N ) ā’ V XL (0, N ) (5.5)

and thus we now have all necessary ingredients for valuing the entire asset
swap.

6
D(0, i) corresponds to discount factors that reļ¬‚ect the credit quality of the reference
entity.
5.4 Valuation Considerations 61

Remember again that the market practice is to choose a value for A,
the asset swap spread, such that the asset swap has zero market value at
its inception. To see how we can ļ¬nd this value of A, called the par asset
spread, let us rewrite (5.1) using the results derived thus far:

V AP (0, N ) = [V B (0, N ) ā’ P ] + [V LL (0, N ) ā’ V XL (0, N )] (5.6)

Ė
Finding the par asset swap spread, A, amounts to solving the above
equation for A while setting V AP (0, N ) to zero. To do this note that we
can rewrite (5.4) as

N N
ā— ā—
(Ī“i Dā— (0, i)) P (5.7)
(D (0, i)Ī“i F (0, i ā’ 1, i)) + A
LL
V (0, N ) =
i=1 i=1

Adding and subtracting Dā— (0, N )P to equation (5.7), and remembering
that

N
(Dā— (0, i)Ī“i F ā— (0, i ā’ 1, i)) + Dā— (0, N ) P
i=1

is nothing more than the market value of a par ļ¬‚oater with a zero spread
over LIBORā”see equation (4.3) and recall the deļ¬nitions of Dā— (0, i) and
F ā— (0, i ā’ 1, i)ā”we can write:

N
(Ī“i Dā— (0, i)) ā’ Dā— (0, N ) P
LL
V (0, N ) = 1 + A (5.8)
i=1

After substituting the above expression into (5.6) and rearranging some
terms, one obtains

N
Ī“i Dā— (0, i)P ā’ V XL (0, N ) ā’ Dā— (0, N )P
B
0 = V (0, N ) + A (5.9)
i=1

Note now that the last two terms of the above expression have a natural
ļ¬nancial interpretation. They represent the present discounted value of the
cash ļ¬‚ows of the bond underlying the asset swap, where the discount factors
were derived from the LIBOR curve, instead of from the bond issuerā™s yield
curve. Let us denote this quantity V Bā— (0, N ), which is consistent with the
62 5. Asset Swaps

ā—
notation of using the superscript to denote variables derived from the
LIBOR curve. Then we can write
N
Ī“i Dā— (0, i)P
0 = V (0, N ) ā’ V
B Bā—
(0, N ) + A (5.10)
i=1

Ė
and it is easy to see that the par asset swap spread A is

Ė V (0, N ) ā’ V (0, N )
Bā— B
A= (5.11)
N ā—
i = 1 D (0, i)Ī“i P

which has the intuitive interpretation that the par asset swap spread will
be positive if V Bā— > V B , which in turn means that the discount factors
associated with the reference entity, D(0, i), are lower than those derived
from the LIBOR curve, Dā— (0, i). Recall that lower discount factors mean
heavier discounting of future cash ļ¬‚ows. The fact that the cash ļ¬‚ows of
the bond issued by the reference entity are discounted more heavily than
the cash ļ¬‚ows of the ļ¬xed leg of the swap leads to V Bā— (0, N ) > V B (0, N ),
which would imply that the issuer has a lower credit quality than that
embedded in LIBOR and thus its asset swap spread should be positive.

5.4.2 Comparison to Par Floaters
Let us look at the cash ļ¬‚ows associated with the par asset swap
just described, focusing on the investorā™s (asset swap buyerā™s) position.
As Figure 5.1 showed, the investor paid par at the time of inception of the
asset swap, and in return was promised a net cash ļ¬‚ow of L + A for as
long as the reference entity does not default. Recall that, by the nature of
the contract, the cash ļ¬‚ow received from the bond is exactly oļ¬set by the
ļ¬xed-rate payments made by the investor to the dealer. Should the refer-
ence entity default while the asset swap is still in place, the investor loses
the bondā”assume for simplicity that the bond has no recovery valueā”and
is left with a position in an interest rate swap that may have either positive
or negative value.
Notice that the net cash ļ¬‚ow of the asset swap looks very much similar
to the cash ļ¬‚ows of a par ļ¬‚oater. As discussed in Chapter 4, the par ļ¬‚oater
buyer pays par for the note and, assuming the note issuer does not default
on its obligations, receives variable interest rate payments based on L + S
each period until the the ļ¬‚oaterā™s maturity date. Should the par ļ¬‚oater
issuer default during the life of the ļ¬‚oater, and continuing to assume zero
recovery, the investor is left with nothing.
Thus, setting aside the topic of counterparty credit risk in the interest
rate swap for the moment, were it not for the possibly non-zero value of
5.4 Valuation Considerations 63

TABLE 5.1
Cash Flows of a Par Floater and a Par Asset Swapa
(Assuming that the swap is terminated upon default by the reference entity,
without regard to its market value.)

Year Floater Asset Swap Diļ¬erence

(1) (2) (3) (2) minus (3)

A. Assuming no default by the reference entity

ā’1 ā’1
0 0
Sā’A
1 L+S L+A
Sā’A
2 L+S L+A
Sā’A
3 L+S L+A
Sā’A
4 1+L+S 1+L+A

B. Assuming default by the reference entity at t = 2.b

ā’1 ā’1
0 0
Sā’A
1 L+S L+A
2 0 0 0
a From the perspective of the buyer. Notional amount = \$1.
b Assuming default occurs at the very beginning of year 2.

the interest rate swap at the time of the reference entityā™s default, the cash
ļ¬‚ow of the asset swap would be identical to that of this par ļ¬‚oater. To see
this, let us take a look at Table 5.1 where we consider an asset swap where,
unlike the asset swap described in Figure 5.1, the embedded interest rate
swap is terminated without regard to its market value upon default by the
reference entity.
Table 5.1 shows the cash ļ¬‚ows of two hypothetical par products, a ļ¬‚oater
paying a spread of S over LIBOR and an asset swap with a spread of A, also
over LIBOR. We assume that the asset swap is written on a bond issued by
the same reference entity that issued the ļ¬‚oater and that they have the same
maturity, payment dates, and principal amounts. For simplicity, we assume
that payments are annual. If there is no default by the reference entity, the
ļ¬‚oater and the asset swap will pay L + S and L + A, respectively each
period. In addition, the last period includes the par values of the ļ¬‚oater
and asset swap. This is shown in the upper panel of the table.
Suppose now the reference entity defaults on its obligations at year 2
before making its annual interest rate payments and assume that these
obligations have no recovery value. The cash ļ¬‚ows of the two products are
shown in the lower panel of Table 5.1.
64 5. Asset Swaps

What is the relationship between the par ļ¬‚oater and par swap spreads
shown in Table 5.1? Notice that the two investment alternatives have the
same initial cost, same payment dates, and same risk exposure. In either
case, the investor is exposed to the credit risk of the reference entity. Thus,
it must be the case that S ā’ A = 0, i.e., they must generate the same
cash ļ¬‚ow. To see this, consider what would happen if, say, S > A. In
this case one could buy the ļ¬‚oater and sell an asset swap. Notice that this
investment strategy costs nothing at year 0; one can ļ¬nance the purchase of
the ļ¬‚oater with the sale of the asset swap. One would then enjoy a positive
net income of S ā’ A per period during the life of the swap for as long as the
reference entity remains solvent. If the reference entity does default during
the life of the swap, both the ļ¬‚oater and the asset swap become worthless.
The point here is that this is an arbitrage opportunity that cannot persist in
an eļ¬cient market. Other investors would see this opportunity and attempt
to exploit it, putting downward pressure on the ļ¬‚oater spreadā”faced with
strong demand for its ļ¬‚oaters, the ļ¬‚oater issuer would be able to place
them in the market with a lower spreadā”and upward pressure in the asset
if people are rushing to sell them. Thus, the asset swap and par ļ¬‚oater
spreads illustrated in Table 5.1 should be the same.7
Unlike the hypothetical example in Table 5.1, however, the typical asset
swap does not specify the automatic termination of its embedded IRS with-
out regard to its market value upon default by the reference entity. As a
result, the par asset swap spread will generally not be equal to the par
ļ¬‚oater spread of the reference entity. In particular, the asset swap spread
will take into account the fact that the value of the embedded IRS will
commonly be nonzero and thus the asset swap buyer has some exposure to
the value of the IRS should the reference entity default. In practice, how-
ever, such exposure tends to be small, especially when the reference bond
is trading at close to par at the time of inception of the asset swap and
so par asset swap and par ļ¬‚oater spreads tend not to diverge signiļ¬cantly
from one another.8
Thus far, we have abstracted from counterparty credit risk in the embed-
ded IRS, or the risk that one of the IRS counterparties will default on its
IRS obligations. If that risk were signiļ¬cant, it could drive a substantial

7
Note that we have just encountered the ļ¬rst example of pricing a credit derivative
via the static replication approach discussed in Chapter 1. We will do a lot more of this
in the remainder of the book.
8
With the bond trading at close to par at the time of inception of the asset swap,
the market value of the embedded IRS will be close to zero. That would suggest about
even odds that future values of the IRS will be positive to the asset swap buyerā”if the
IRS were more/less likely to have positive than negative value to the asset swap buyer
in the future, the market value of the IRS to the asset swap buyer would have been
positive/negative.
5.4 Valuation Considerations 65

wedge between par asset swap and par ļ¬‚oater spreads as investing on a par
ļ¬‚oater involves only the credit risk of the ļ¬‚oater issuer, whereas entering
into the asset swap would encompass both this risk and the risk associated
with default in the embedded IRS. In reality, however, counterparty credit
risk tends to be mitigated in the asset swap market via the use of credit
enhancement mechanisms such as netting and collateralization.
To sum up, similar to the pricing of a par ļ¬‚oater, the main determinant
in the pricing of a par asset swap is the credit quality of the reference
entity. Nonetheless, other factors potentially enter into the pricing of asset
swaps, such as counterparty credit risk and the default-contingent exposure
of the asset swap buyer to the market value of the embedded interest rate
swap. Still, the contribution of these factors to the determination of the
asset swap spread tends to be small relative to that of the credit risk of the
reference bond. That is why asset swaps are viewed essentially as synthetic
ļ¬‚oating-rate notes. Having said that, however, as mentioned in Chapter 1,
other ātechnicalā factors do aļ¬ect asset swap spreads in practice, such as
diļ¬erences in liquidity between the corporate bond and asset swap markets,
market segmentation, and other supply and demand inļ¬‚uences.
6
Credit Default Swaps

Credit default swaps (CDS) are the most common type of credit derivative.
According to diļ¬erent surveys of market participants, which were summa-
rized in Chapter 2, CDS are by far the main credit derivatives product in
terms of notional amount outstanding. Their dominance of the marketplace
is even more striking in terms of their share of the total activity in the credit
derivatives market.1 As actively quoted and negotiated single-name instru-
ments, CDS are important in their own right, but their signiļ¬cance also
stems from the fact that they serve as building blocks for many complex
multi-name products.2
The rising liquidity of the credit default swap market is evidenced by the
fact that major dealers now regularly disseminate quotes for such contracts.
Furthermore, along with risk spreads in the corporate bond market, CDS
quotes are now commonly relied upon as indicators of investorsā™ perceptions
of credit risk regarding individual ļ¬rms and their willingness to bear this
risk. In addition, quotes from the CDS market are reportedly increasingly
used as inputs in the pricing of other traditional credit products such as
bank loans and corporate bonds, helping promote greater integration of
the various segments of the credit market.

1
CDS transactions are much more common than multi-name credit derivatives such as
synthetic collateralized debt obligations, but the latter have substantially larger notional
amounts.
2
We shall examine some multi-name instruments in Chapters 9 through 11 and in
Part IV.
68 6. Credit Default Swaps

6.1 How Does It Work?
A credit default swap is a bilateral agreement between two parties, a buyer
and a seller of credit protection. In its simplest āvanillaā form, the protec-
tion buyer agrees to make periodic payments over a predetermined number
of years (the maturity of the CDS) to the protection seller. In exchange,
the protection seller commits to making a payment to the buyer in the
event of default by a third party (the reference entity). As such, a credit
default swap shares many similarities with traditional insurance products.
For instance, car owners generally go to an insurance company to buy pro-
tection from certain car-related ļ¬nancial losses. The car insurance company
collects a stream of insurance premiums from its customers over the life of
the contract and, in return, promises to stand ready to make payments to
customers if covered events (accidents, theft, etc.) occur.
Figure 6.1 illustrates the basic characteristics of a credit default swap.
In a typical credit default swap, a protection buyer purchases ādefault
insuranceā from a protection seller on a notional amount of debt issued
by a third party (the reference entity). The notional quantity, in eļ¬ect,
represents the amount of insurance coverage. In the credit default swap
market, the annualized insurance premium is called the ācredit default swap
spread,ā or ācredit default swap premium,ā which is quoted as a fraction
of the notional amount speciļ¬ed in the contract and generally set so that
the contract has zero market value at its inception. As an illustration, the
credit default swap represented in the diagram has a notional amount of
\$100 and an associated premium of 40 basis points. Thus, the protection
buyer pays 40 cents per year for each \$100 of notional amount in exchange
for protection against a default by the reference entity. Typically, CDS
premiums are paid quarterly so that, in this example, the protection seller
agrees to pay 10 cents per quarter for each \$100 of desired credit protection.

FIGURE 6.1. Example of a Credit Default Swap
6.1 How Does It Work? 69

In the event of default by the reference entity, a CDS can be settled
physically or in cash, with the settlement choice determined upfront when
entering the contract. In a physically settled swap, the protection buyer
has the right to sell (deliver) a range of defaulted assets to the protection
seller, receiving as payment the full face value of the assets. The types of
deliverable assets are also prespeciļ¬ed in the contract. For instance, the
typical CDS determines essentially that any form of senior unsecured debt
issued by the reference entity is a deliverable asset, and thus any bank loan
or bond that meets this criterion is a deliverable asset.
In a cash settled swap, the counterparties may agree to poll market
participants to determine the recovery value of the defaulted assets, and the
protection seller is liable for the diļ¬erence between face and recovery values.
The asset or types of assets that will be used in the poll are prespeciļ¬ed
in the contract. Cash settlement is more common in Europe than in the
United States, where, by far, the majority of CDS are physically settled.
As mentioned in Chapters 1 and 3, the adoption of standardized doc-
umentation for credit default swap agreements has played an important
role in the development and greater liquidity of the CDS market. The use
of master agreements sponsored by the International Swaps and Deriva-
tives Association (ISDA) is now a common market practice, signiļ¬cantly
reducing setup and negotiation costs. The standard contract speciļ¬es all
the obligations and rights of the parties as well as key deļ¬nitions, such
as which situations constitute a ācredit eventāā”a default by the refer-
ence entityā”and how a default can be veriļ¬ed. Regarding the former, for
instance, CDS contracts generally allow for the following types of default
events:

ā¢ bankruptcy
ā¢ failure to pay

ā¢ debt moratorium
ā¢ debt repudiation

ā¢ restructuring of debt
ā¢ acceleration or default

Some of these events are more common in contracts involving certain types
of reference names. For instance, moratorium and repudiation are typically
covered in contracts referencing sovereign borrowers. In addition, especially
in the United States, CDS contracts are negotiated both with and without
restructuring included in the list of credit events. In Chapter 24 we discuss
the standardized ISDA contract for credit default swaps in some detail,
including the speciļ¬c deļ¬nitions of each of these credit events.
70 6. Credit Default Swaps

The maturity of a credit default swap does not have to match that of
any particular debt instrument issued by the reference entity. The most
common maturities of credit default maturities are 3, 5, and 10 years, with
the ļ¬ve-year maturity being especially active.
It is possible, and increasingly easier, to terminate or unwind a credit
default swap before its maturity (and this is commonly done) in order to
extract or monetize the market value of the position. Typically, unwinding
a CDS position involves agreement by both parties in the contract regard-
ing the market value of the position.3 The party for whom the position
has negative market value then compensates the other accordingly. Alter-
natively, a party may be able to close out its position by assigning it to
a third party, but this generally requires mutual approval of both new
counterparties.

6.2 Common Uses
From the analogy with traditional insurance products, it becomes obvious
what the most direct uses of credit default swaps are. At the most basic
level, protection sellers use credit default swaps to buy default insurance,
and protection sellers use them as an additional source of income. In prac-
tice, however, market participantsā™ uses of credit default swaps go well
beyond this simple insurance analogy. Indeed, because credit default swaps
are the main type of credit derivative, we have indirectly discussed many of
their most common applications in the general discussion of uses of credit
derivatives in Chapter 3. We shall only brieļ¬‚y review them here, devoting
more space in this chapter to those applications that are more germane to
the credit swaps or that were not speciļ¬cally covered in Chapter 3.

As we saw in Chapter 3, credit derivatives in general, and credit default
swaps in particular, allow banks and other holders of credit instruments
to hedge anonymously their exposure to the credit risk associated with
particular debtors. Thus, while the credit instruments may remain in the
holderā™s balance sheetā”which may be important particularly to banks for
relationship reasonsā”the associated credit risk is transferred to the protec-
tion seller under the CDS contract. From the perspective of the protection
buying end of the market, here is where the car insurance analogy works
best; you can keep your car while shifting some of the associated ļ¬nancial
risk to an automobile insurance company.

3
The valuation of CDS positions (marking to market) is discussed in Part III.
6.2 Common Uses 71

As discussed in Section 3.4, however, some market participants may want
to buy protection through credit default swaps even if they have no expo-
sure to the reference entity in question. In particular, buying protection
is akin to shorting the reference entityā™s debt as the market value of the
protection buyerā™s position in the swap would increase in the event of a
subsequent deterioration in the credit quality of the reference entity.

6.2.2 Protection Sellers
On the other side of the market, as we also discussed in Chapter 3, sellers
of default protection see the credit default swap market as an opportu-
nity to enhance the yields on their portfolios and diversify their credit risk
exposure. Here again there is a straightforward analogy to selling tradi-
tional insurance policies. For as long as the events covered in the contract
do not occur, protection sellers receive a steady stream of payments that
Of course, prospective protection sellers could, in principle, simply buy
debt instruments issued by the desired reference entities directly in the
marketplace in order to obtain potentially yield enhancement and port-
folio diversiļ¬cation beneļ¬ts similar to those provided by credit default
swaps. Furthermore, buying credit risk through outright long positions
in, say, corporate bonds and loans, has the advantage of not exposing one
to counterparty credit risk in the CDS contract.
One may thus ask the following question: In addition to the fact that
there are so-called natural buyers of default protection, what motivates
someone to sell protection in the CDS market? As noted in Section 3.3, the
unfunded nature of many credit derivatives, including typical credit default
swaps, distinguishes them importantly from cash market instruments such
as bonds and bank loans.
For instance, credit default swaps allows an investor to obtain, say, expo-
sure to \$10 million worth of debt issued by XYZ Corp. with essentially no
upfront cost other than a possible initial posting of collateral. In contrast,
that same exposure would have required a sizable initial cash outlay by the
investor if the exposure were obtained in the form of a direct purchase of
bonds or loans issued by XYZ Corp. In other words, the investor would
have to use its scarce capital to fund its purchase of credit risk if that risk
were obtained via an outright purchase of bonds or loans, whereas credit
risk obtained through credit default swaps involves essentially little or no
funding requirement.4

4
If the investor could ļ¬nance the cost of the outright purchase in the repo market
for bonds or loans issued by XYZ Corp., the transaction could also be characterized
as one requiring little or no funding. The investor would buy the bond or loan for its
market price of, say, Y and immediately repo it out, essentially using the bond or loan
72 6. Credit Default Swaps

In Section 3.3 we also mentioned the use of credit derivatives to create
synthetic long positions in corporate debtā”instead of holding the credit
risk assets outright, one sells protection in the credit derivatives market.
This use of credit derivatives highlights the fact that, in addition to their
unfunded nature, credit default swaps might be particularly attractive to
investors in situations where outright positions in the cash market regard-
ing an individual reference entity are diļ¬cult to establish. Consider, for
instance, a ļ¬rm whose debt is closely held by a small number of investors.
For an investor who wants to obtain some credit risk exposure to that ļ¬rm,
but who cannot buy its debt instruments directly in the cash market, selling
protection via a CDS contract becomes a potentially appealing alternative.
Our ļ¬nancial intuition should tell usā”and this will be conļ¬rmed belowā”
that the income that the investor will receive under the CDS contract will
reference entityā™s debt directly.

We shall consider two simple, speciļ¬c applications of credit default swaps
from the perspective of both sellers and buyers of protection. First, we
examine the case of a highly rated investor who wants to generate some
additional income while minimizing the exposure to credit risk. Second,
we discuss the situation of a below-investment grade investor who wants
exposure to high-quality credits, but who faces high funding costs in the
ļ¬nancial markets. These examples highlight the importance of investorsā™
funding costs, an issue we will turn back to in the discussion of valuation
issues regarding credit default swaps.

Synthesizing a (relatively) riskless asset. Consider an investor who estab-
lishes a hedged position in a certain credit: The investor buys bonds issued
by a given reference entity and simultaneously buys protection in a credit
default swap that references that same entity. The investor funds the pur-
chase of the bonds by borrowing in the ļ¬nancial market. The investorā™s
net income on the bond is then given by the spread between the yield paid
by the bond and its funding costs. If that spread is wide enough, i.e., if
the investorā™s creditworthiness is strong enough that it can fund itself at a
relatively low cost, the net income derived from the purchase of the bond

as collateral for a loan in the amount of Y . The net result of these two simultaneous
transactions would be that an initial cash outlay of zero for the investorā”the purchase of
the bond was ļ¬nanced by the proceeds of the repo transaction involving the bondā”and
a net income of Cy ā’ Cr , where Cy is the coupon payment made by the bond and Cr
is the payment made by the investor to its repo counterparty. While this works well in
theory, in practice the repo market for corporate debt is still at a very early stage of
development even in the United States, where the market for corporate debt instruments
is generally more developed than in other parts of the world.
6.3 Valuation Considerations 73

may well exceed the payments made by the investor under the terms of the
credit default swap. The net result of this hedged credit position is thus a
synthetic asset that is relatively free from credit risk. We say relatively free
because the investor is still exposed to the credit risk of its default swap
counterparty, although the extent of that exposure can be mitigated via
netting and collateralization.

Adding highly rated assets to oneā™s portfolio. The previous example works
well for investors with relatively low funding costs. For a less creditworthy
investor, the net income derived from holding the bond on the balance
sheet may well be lower than the cost of buying protection in a credit default
swap, resulting in a net negative income stream from the hedged portfolio
position. Worse still, if the yield on the corporate bond is lower than the
investorā™s cost of funds, it might make little sense for the investor to buy the
bond in the ļ¬rst place. What if the investor wanted to add the credit risk of
this particular issuer to its portfolio? A potentially more cost eļ¬ective way
to do it would be to sell protection in a CDS that references that issuer.
Because CDS are unfunded instruments, the investor can eļ¬ectively bypass
the funding market, where its costs are high, and more proļ¬tably āaddā
higher-quality credits to its portfolio.

6.3 Valuation Considerations
Suppose you are asked to estimate the cost of buying protection against
default by a given reference entity. Consistent with the car insurance anal-
ogy discussed at the beginning of this chapter, you know that the higher
the credit risk associated with the entity, the higher the price of protection.
(A car that is more prone to accidents will command a higher insurance
premium.) But you need to come up with a speciļ¬c number. How high
should the credit default swap premium be for this particular reference
entity?
As we did in Chapter 5, when we priced a simple variant of the asset
swap, we will make use of the static replication approach to valuing ļ¬nancial
assets. That approach tells us that if we can devise a portfolio made up of
simple securities that replicates the cash ļ¬‚ows and risk characteristics of
the contract we want to price, the price of that contract is, in the absence
of arbitrage opportunities, simply the price of setting up the replicating
portfolio.5

5
There are other technical conditions that the replicating portfolio must satisfy, such
as the requirement that it must constitute a self-ļ¬nancing investment strategy, but we
will just assume that all these conditions are satisļ¬ed here. Baxter and Rennie (2001)[6]
provide an intuitive discussion of this topic. For a more rigorous, but still accessible
exposition of replicating strategies, see, e.g., Bjork (1998)[7].
74 6. Credit Default Swaps

We shall consider two highly stylized, though increasingly realistic, exam-
ples. Together, they provide us with some basic insights regarding the
valuation of credit default swaps, in particular, and the static replication
approach in general.6

Example 6.1 Consider an investor who is oļ¬ered the choice of either of
two portfolios
ā¢ a long position in a risky ļ¬‚oater yielding Rf +S combined with a short
position in a riskless ļ¬‚oater yielding Rf ;
ā¢ a short position (protection seller) in a CDS written on the risky
ļ¬‚oater.

We assume that both ļ¬‚oaters have the same maturity, coupon dates, and
face values (\$1), and that they sell at par immediately after their coupon
payment dates. To keep things even simpler, let us postulate further that
the recovery rate on the risky ļ¬‚oater is zero and that default can only occur
immediately after the coupon payment dates.7
What are the cash ļ¬‚ows associated with each portfolio? For as long as
the issuer of the risky ļ¬‚oater does not default, the ļ¬rst portfolio yields S
every period. As for the second portfolio, the CDS has a cash ļ¬‚ow of Scds
every period, where Scds is the CDS premium.
In the event of default, the holder of the portfolio of ļ¬‚oaters ends up
with a short position in the riskfree ļ¬‚oater, which translates into a liability
of \$1, given that the ļ¬‚oater is valued at par on its coupon payment dates.
The protection seller in the CDS is liable for the CDS payoļ¬, which is also
worth \$1. Thus, when there is a default, both portfolios have the same
payoļ¬.
At this point, we should pause to make two key points:
ā¢ With time-varying interest rates, the static replication argument out-
lined above would generally fail if, instead of using ļ¬‚oating-rate notes
to replicate the CDS cash ļ¬‚ows, we had used ļ¬xed-rate notes. This
occurs because a ļ¬xed-rate note is not generally valued at par after it
is issued and thus the liability of the short seller in the event of default
could well be diļ¬erent from \$1.
ā¢ Neither portfolio required a cash outlay when they were set up: The
proceeds of the short sale of the riskless ļ¬‚oater were used to ļ¬nance
the purchase of the risky ļ¬‚oater, and it costs nothing to enter into a
vanilla CDS.

6
These examples are extracted from Bomļ¬m (2002)[11].
7
These assumptions can be relaxed and the basic results would still hold.
6.3 Valuation Considerations 75

Given the same initial cost, the same payoļ¬s in the event of default,
and the same risk exposure of the CDS transaction and the portfolio of
ļ¬‚oaters, it must be the case that the CDS and the ļ¬‚oater portfolio must
have the same cash ļ¬‚ow in the absence of a default by the reference entity.
This requires that Scds = S. Thus, under the conditions set out above, the
premium that should be speciļ¬ed in a CDS written on a given corporation
is the same as the risk spread associated with a par ļ¬‚oater issued by that
corporation.
The above example made an important point, highlighting the tight cor-
respondence between the CDS spread for a given reference entity and the
borrowing costs facing that entity. However, we are still missing a few
important aspects of reality. For instance, we have thus far ignored the
fact that the ļ¬rst portfolio ultimately has to be funded on the balance
sheet, whereas the CDS does not. We turn now to a slightly more realistic

Example 6.2 Using the same notation and assumptions of example 1,
consider the following two scenarios:
ā¢ The investor ļ¬nances the purchase of the risky ļ¬‚oater, by repoing it
out, paying the repo rate Rf + F . (Alternatively, we can think of
Rf + F as the rate at which the investor can obtain ļ¬nancing for the
portfolio.) Assuming no default by the issuer of the risky ļ¬‚oater, the
investor receives Rf + S every period and pays out Rf + F to its
repo counterparty. In the event of default, the risky ļ¬‚oater becomes
worthless, and the investor ends up owing \$1 to its repo counterparty.
To sum up, the investorā™s cash ļ¬‚ows are: S ā’ F (no default) and ā’\$1
(default).
ā¢ The investor sells \$1 worth of protection in a CDS written on the issuer
of the same risky ļ¬‚oater considered in the previous scenario. The cash
ļ¬‚ows associated with such a CDS position are: Scds (no default) and
ā’\$1 (default).
Again, notice that neither strategy required an initial cash outlay and
both have the same payoļ¬ in the event of default. Thus, in the absence of
arbitrage opportunities and market frictions, it must be the case that they
have the same payoļ¬ in the absence of default, i.e., the CDS premium Scds
must be the equal to the diļ¬erence between the risky ļ¬‚oater spread S and
the investorā™s funding cost F :

Scds = S ā’ F (6.1)

where the above diļ¬ers from the result obtained from example 1 because
we are now explicitly taking into account the fact that the ļ¬rst portfolio
76 6. Credit Default Swaps

has to be funded on the balance sheet of a leveraged investor whereas the
CDS is an unfunded instrument.
To bring the discussion of the above examples even closer to the real
world, we should note the following: Although the above approach for
pricing a CDS relied on rates on par ļ¬‚oaters issued by the reference entity,
most corporate debts issued in the United States are ļ¬xed-rate liabilities.
In practice, however, one can circumvent this problem by resorting to the
asset swap marketā”see Chapter 5.8 In particular, the above examples can
be made more realistic as illustrations of how to obtain an (approximate)
value for the CDS premium if we (i) substitute the par ļ¬‚oater spread,
S, with the par asset swap spread associated with the reference entity
in question and (ii) redeļ¬ne Rf as a short-term LIBOR rate in order to
conform with the quoting convention for asset swaps.

6.3.1 CDS vs. Cash Spreads in Practice
We can use observed market quotes to verify how well the data support
the simple valuation relationships uncovered by the above examples. As an
illustration, the four panels in Figure 6.2 show quotes on credit default
swaps and asset swaps for four investment-grade reference entities that
underlie some of the most frequently negotiated credit default swaps of the
late 1990s: Bank of America, General Motors Acceptance Corp. (GMAC),
Tyco International, and Walmart, which, at the end of the period shown
in the ļ¬gure, were rated A+, A, Aā’, and AA by Standard and Poorā™s,
respectively.
The CDS and par asset swap spreads shown for Bank of America and
GMAC do line up closely and are thus broadly consistent with the results of
the static replication approach outlined in the previous section. In contrast,
charts for Tyco and Walmart show CDS spreads that are substantially
above what would be suggested by the asset swap market, displaying what
market participants call āpositive biasā or a positive āCDS-cash basis.ā
The divergence between CDS and asset swap spreads for the reference
entities shown in Figure 6.2 highlights the role that market segmentation
and idiosyncratic supply and demand factors still play in the CDS market.
For instance, the substantial positive bias associated with Tyco during the
period shown in the ļ¬gure was attributable in part to strong demand by
issued substantial amounts of convertible debt during the period featured in
the chart, but the investors who bought such bonds were focusing primarily
on the cheapness of embedded call options on Tycoā™s stock. In particular,

8
When neither ļ¬‚oaters nor ļ¬xed-rate instruments are actively traded, valuation
approaches based on credit risk models become particularly relevant, as discussed in
Section 6.3.3.
6.3 Valuation Considerations 77

FIGURE 6.2. An Informal Test of the Static Replication Approach
Source: Bomļ¬m (2002)[11]

they used the CDS market to shed the credit risk associated with Tyco
and liquidity and market segmentation factors led to a widening of the
CDS-cash basis.
premiums in practice, and even CDS for reference entities that borrow
at LIBOR ļ¬‚at or below, such as Walmart in the late 1990s, tend to be
slightly positive. Another factor that contributes to positive bias is the fact
that participation in the CDS market is limited either by some investorsā™
lack of familiarity with credit derivatives or by regulatory restrictions and
78 6. Credit Default Swaps

internal investment policies of certain institutional investors. In addition,
for some reference entities, a liquidity premium on CDS, reļ¬‚ecting the
poorer liquidity of the CDS market relative to the cash (corporate bond)
market for those entities, may also be a factor leading to positive bias.

6.3.2 A Closer Look at the CDS-Cash Basis
We used Figure 6.2 to highlight the fact that the theoretical result that
suggests the equality of CDS premiums and par asset swap spreads for the
same reference entity does not always hold in practice. In other words, the
so-called CDS-cash basis, deļ¬ned as

is often nonzero.
Should an arbitrageur who sees, for instance, a negative CDS-cash basis
for a given reference entity (par asset swap spread above CDS premium)
hopes that the gap between the two will close? Not necessarily. In many
realistic situations, a nonzero CDS-cash basis can be perfectly justiļ¬ed by
fundamental factors that were not included in the stylized examples dis-
cussed in the beginning of this section. The arbitrageurā™s challenge is then
to identify those movements in the basis that are driven by fundamentals
from those that are the result of temporary supply and demand dislocations
that can be proļ¬tably exploited.
Fundamental factors behind a nonzero CDS-cash basis include
ā¢ cheapest-to-deliver feature of CDS contracts,
ā¢ default-contingent exposure in asset swaps,

ā¢ accrued premiums in CDS contracts,
ā¢ funding risk in asset swaps,
ā¢ counterparty credit risk,

ā¢ liquidity risk diļ¬erentials.

As noted, most CDS contracts are physically settled and allow for a wide
range of deliverables (typically all senior unsecured debt). In principle, in
the event of default, all obligations of the reference entity that meet the
deliverability criterion should have the same recovery value. This would
which assets are actually delivered to settle the CDS contract. As is often
the case, things are not so simple in the real world.
6.3 Valuation Considerations 79

In many realistic circumstances, the values of deliverable obligations can
diļ¬er at the time of settlement of the CDS contract. The most obvious case
is that of a CDS triggered by a restructuring of the reference entityā™s debt
because restructuring tends to aļ¬ect the market values of bonds and loans
diļ¬erently as they are most often applied to loans. In eļ¬ect, this means
that the protection buyer is long a cheapest-to-deliver (CTD) option, i.e.,
the buyer can look at the full range of deliverables and hand over the
cheapest ones to the protection seller. Now consider the buyer of an asset
swap. If the reference entity defaults for whatever reason, the asset swap
buyer will receive the post-default value of the speciļ¬c ļ¬xed-rate bond or
loan underlying the asset swap. There is no cheapest-to-deliver option!
(The same argument would apply to someone who bought a par ļ¬‚oater
directly instead of synthesizing one in the asset swap market.)
As the old saying goes, āthere is no free lunch,ā and thus protection
sellers in the CDS market will āchargeā for the embedded CTD option in
their product by demanding a higher CDS premium than the spread paid
in either the par asset swap or par ļ¬‚oater markets. Thus, other things
being equal, the embedded CTD option in a credit default swap results
in a positive CDS-cash basis that is perfectly in line with economic and
ļ¬nancial fundamentals. In such cases, the positive CDS-cash basis is not
indicative of an arbitrage opportunity. We should point out, however, that
the value of the embedded CTD option has likely diminished in recent years
in light of changes in the way restructurings are treated in CDS contracts
(see Chapter 24).
But other things are not equal. In particular, as we saw in Chapter 5,
the asset swap buyer has a default-contingent risk exposure to the marked-
to-market value of the interest rate swap embedded in the asset swap.
To recap brieļ¬‚y, unlike the credit default swap, the asset swap does not
completely terminate with a default by the reference entity. In particular,
the interest rate swap embedded in the former continues even after the
reference entity defaults. Thus, it could well be the case that, in addition
to losing the diļ¬erence between the par and recovery values of the bond
underlying the asset swap, the asset swap buyer may ļ¬nd itself with a
position in an interest rate swap that has negative market value. When
this is likely, the CDS-cash basis has a tendency to be negative, going in
the opposite direction of the CTD eļ¬ect.
is the fact that, in the event of default by the reference entity, the protection
seller in a CDS still receives that portion of the CDS premium that accrued
between the last payment date of the CDS and the time of default. The
asset swap buyer does not enjoy that beneļ¬t and thus must be compensated
in the form of a higher asset swap spread than would otherwise be the case.
Also contributing to a negative CDS-cash basis is the fact that the asset
swap buyer is generally subject to funding risk. This stems from the fact
80 6. Credit Default Swaps

that the asset swap buyer may have to fund the purchase of the underlying
bond through a short-term loanā”for instance, terms in the repo market
rarely go beyond a few monthsā”and roll over the loan for the duration of
the asset swap at uncertain future costs. In contrast, participants in a CDS
contract face no such uncertainties.
Lastly, we mentioned liquidity and counterparty credit risk as factors
that may potentially aļ¬ect the CDS-cash basis. For certain reference
entities, such as some US corporations with large amounts of bonds out-
standing, the CDS market may be less liquid than the cash market. That
would tend to push CDS premiums above corresponding cash market
spreads as protection sellers would have to be compensated for the greater
illiquidity they face.9 Regarding counterparty credit risk, one should be
aware that, while it is a potential factor in the pricing of both credit
default and asset swaps, that is certainly not the case for conventional
need to be considered accordingly.
To sum up, a number of factors drive a wedge between CDS premiums
and spreads in the par ļ¬‚oater and asset swap markets, some contributing
to a positive CDS-cash basis, some to a negative one. As a result, if you are
are deļ¬nitely a good place to start, but they are almost certainly not going
to give you the whole answer.

6.3.3 When Cash Spreads are Unavailable...
Thus far, our main inputs for determining the fair value of a CDS premium
have been spreads in the cash market, such as par asset swap spreads
and par ļ¬‚oater spreads. Certain reference entities, however, may not have
marketable debt outstanding, or the market for their debt may be very
illiquid and available quotes may be uninformative.
An alternative approach to valuing credit default swaps that is especially
useful when reliable spreads in the cash market are not available is the one
based on credit risk models.10 For standard credit default swaps, an impor-
tant starting point is the basic insight that, because the contract has zero
market value at its inception, the CDS premium is set such that the value
of the āprotection legāā”deļ¬ned as the present value of the expected pay-
ment made by the protection seller in the event of default by the reference
entityā”is equal to the value of the āpremium legāā”deļ¬ned as the present

9
The reverse has reportedly been true for some sovereign reference names, where
liquidity in the cash markets at times has fallen short of liquidity in the CDS market.
10
In Part III of this book, we discuss some modeling approaches.
6.3 Valuation Considerations 81

As a preview of what is to come, suppose we have a model that gives us
the default probabilities associated with a given reference entity. Consider
now the (extremely) simple case of a one-year CDS with a \$1 notional
amount and a single premium payment, Scds , due at the end of the contract.
Let us make the artiļ¬cial assumption that a default by the reference entity,
if any, will only occur at the maturity date of the contract. (To keep things
even simpler, assume no counterparty credit risk and no market frictions
such as illiquidity or market segmentation.)
The current value of the protection leg is simply the present value of the

where P V [.] denotes the present value of the variable in brackets.
How about the present value of the protection leg? Let Ļ denote the
probability that the reference entity will default in one yearā™s time. The pro-
tection seller will have to pay 1 ā’ X with probability Ļ and 0 otherwise,
where X is the recovery rate associated with the defaulted instrument.11
Thus we can write the present value of the protection leg as

PV[protection] = PV[Ļ Ć— (1 ā’ X) + (1 ā’ Ļ) Ć— 0] (6.4)

If the CDS is to have zero market value at its inception, the present values
in equations (6.3) and (6.4) must be equal, and that will happen when

Scds = Ļ Ć— (1 ā’ X) (6.5)

and we get the result that the cost of protection Scds is increasing in the
probability of default and decreasing in the recovery rate associated with
the reference entity. In particular, in the limiting case of no recovery, the
CDS premium is equal to the probability of default. Thus, if we have a
theoretical model that gives us the default probabilities associated with
the reference entity, we can price a CDS written on that entity accordingly.
As we shall see later in this book, these results can be generalized, with
a few modiļ¬cations, for more realistic cases, such as multi-period credit
default swaps.

11
We are being intentionally vague here regarding the nature of Ļ and the discount
factors implicit in P V [.]. We will address the issues of discounting and risk-neutral vs.
objective probabilities in Part III. For now, let us simply assume that market participants
ĀÆ
are risk-neutral, i.e., they are indiļ¬erent between, say, receiving Y for sure and receiving
ĀÆ.
an uncertain amount Y , where the expected value of Y is Y
82 6. Credit Default Swaps

6.4 Variations on the Basic Structure
There are several variations on the āvanillaā CDS discussed thus far in
this chapter, but none of these variants are nearly as liquid and widely
negotiated as the standard form of the contract. We shall very brieļ¬‚y
discuss three structures that are closely related to the basic CDS contract.
Binary or ļ¬xed-recovery credit default swaps, also called digital credit
default swaps, are similar to vanilla CDS contracts except that the payoļ¬
in the event of default by the reference entity is known ahead of time and
written into the contract. (Recall that in the vanilla CDS, the payoļ¬ is
equal to the notional amount of the contract minus the post-default value
of the underlying assets, but this value is only known following the default.)
While the binary CDS eliminates the uncertainty about recovery rates, it
is generally a less eļ¬ective hedging vehicle than its vanilla cousins. One use
of binary credit default swaps is to enhance the yield on oneā™s portfolio:
Selling protection in a binary CDS with an implied ļ¬xed recovery rate that
is lower than the market consensus should result in a higher premium than
in a vanilla CDS.
Certain credit default swaps, especially those written on reference entities
that are viewed as potentially headed for trouble, require upfront payment
of at least some of the protection premiums. (Recall that no money changes
hands in the inception of a vanilla CDS.) In the case of highly distressed
reference entities, the upfront payments help attract protection sellers to a
market that could otherwise be severely one-sided.
An alternative to buying protection through a vanilla CDS is to buy an
option on credit default swap, commonly referred to as a credit default
swaption. As the name suggests, CDS options are contracts that give their
buyers the option, but not the obligation, to enter into a CDS at a future
date if the CDS premium on the reference entity goes higher than some
āstrike level.ā12

12
We discuss the closely related topic of spread options in Chapter 8. The valuation
of credit default swaptions is addressed in Chapter 18.
7
Total Return Swaps

In a total return swap (TRS), an investor (the total return receiver) enters
into a derivatives contract whereby it will receive all the cash ļ¬‚ows asso-
ciated with a given reference asset or ļ¬nancial index without actually ever
buying or owning the asset or the index. The payments are made by the
other party in the TRS contract, the total return payer. Unlike an asset
swap, which essentially strips out the credit risk of ļ¬xed-rate asset, a total
return swap exposes investors to all risks associated with the reference
assetā”credit, interest rate risk, etc.1 As such, total return swaps are more
than just a credit derivative. Nonetheless, derivatives dealers have custom-
arily considered their TRS activity as part of their overall credit derivatives

7.1 How Does It Work?
Total return swaps come in diļ¬erent variations. We shall describe the most
basic form ļ¬rst. Like other over-the-counter derivatives, a TRS is a bilat-
eral agreement that speciļ¬es certain rights and obligations for the parties
involved. In the particular case of the TRS agreement, those rights and
obligations are centered around the performance of a reference asset.

1
The fact that an asset swap involves the actual purchase of the asset is another
diļ¬erence between the asset swap and the total return swap.
84 7. Total Return Swaps

For instance, suppose an investor wants to receive the cash ļ¬‚ows associ-
ated with a ļ¬xed-rate bond issued by XYZ Corp., but is either unwilling
or unable to purchase the bond outright. The investor approaches a deriva-
tives dealer and enters into a total return swap that references the desired
XYZ bond. The dealer promises to replicate the cash ļ¬‚ows of the bond and
pay them out to the investor throughout the maturity of the swap, pro-
vided, of course, the issuer of the reference bond does not default. What
does the dealer get in return? The investor promises to make periodic pay-
ments to the dealer, where the payments are tied to short-term LIBOR
plus a ļ¬xed spread applied to the same notional amount underlying the
coupon payments made by the dealer. This basic arrangement is shown in
exactly match the timing and size of the reference bondā™s coupons (C) and,
in return, pays LIBOR (L) plus the TRS spread T to the dealer (the total
return payer).
Figure 7.1 looks very much like the lower panel of Figure 5.1, where we
illustrated the interest rate swap embedded in a par asset swap. But there
are some important diļ¬erences. First, the investor is now making a ļ¬‚oating-
rate payment to the dealer, as opposed to making ļ¬xed-rate payments in the
asset swap. Second, the reference asset in a total return swap need not be a
ļ¬xed rate asset; it could actually be a ļ¬‚oating-rate asset. Thus, in principle,
the exchange of payments between dealer and investor in Figure 7.1 could
well be an exchange of two ļ¬‚oating-rate payments. Lastly, unlike the asset
swap buyer, the total return receiver has not bought the reference asset.
After all, not having to purchase the asset outright is typically a major
reason for the TRS contract.

FIGURE 7.1. Total Return Swap
7.2 Common Uses 85

What happens at maturity of the TRS? Assuming no default by the
reference entity, the total return receiver is paid the last coupon of the
bond, along with the diļ¬erence between the market value of the bond at
the maturity of the TRS and the market value of the bond at the inception
of the TRS. If that diļ¬erence is negative, the total return receiver pays
that amount to the total return payer. As a result, the TRS replicates not
just the coupon stream of the bond but also the capital gain or loss that
would be experienced by an investor who had actually bought the bond at
the inception of the TRS and sold it at the maturity date of the TRS.
What if the issuer of the reference bond defaults? The fact that the TRS
is designed to replicate the cash ļ¬‚ows of the bond means that the total
return receiver will bear the default-related loss. Once again, the total
return receiver pays the diļ¬erence between the price of the bond at the
inception of the TRS and the recovery value of the bond at the time
of default. Typically, the TRS is terminated upon the reference entityā™s
default.
We should also make one additional remark about the workings of a TRS.
The maturity of the contract need not coincide with that of the reference
bond. Indeed, as we shall see, a TRS can be used to synthesize assets that
suit the maturity preferences of individual investors.

7.2 Common Uses
Investors with relatively high funding costs can use TRS contracts to
synthetically own the asset while potentially reducing their funding disad-
vantage. For instance, consider an investor who can fund itself at LIBOR
+ 120 basis points and who wants to add a given bond to its portfolio.
The investor can either buy the bond outright and fund it on its bal-
ance sheet or it can buy the bond synthetically by becoming a total return
receiver in a TRS, where, say, it would have to pay LIBOR+50 basis points
to the total return payer in exchange for receiving the cash ļ¬‚ows associated
with the same bond. In this example, it is clear that the investor would be
better oļ¬ by tapping the TRS market. The example also illustrates that
the total return payer is essentially providing ļ¬nancing to the total return
receiver so it can synthetically ābuyā the bond, and that the TRS market
makes it easier for investors to leverage their credit risk exposure.
From the perspective of highly rated counterparties, the TRS market
also oļ¬ers some potentially attractive opportunities. Suppose that the total
return payer in the above example is an AA-rated entity that funds itself at
LIBOR ļ¬‚at. To carry on with the funding analogy, the total return payer is
extending a synthetic loan to the total return receiver where it is earning a
50 basis point spread over its cost of funding. Thus, the TRS market allows
highly rated entities to beneļ¬t from their funding advantage.
86 7. Total Return Swaps

One might question the plausibility of the numerical example just dis-
cussed. Why would the total return payer provide ļ¬nancing to the TRS
counterparty essentially at a below-market spread over LIBOR? After all,
the cash market requires a 120 basis point spread over LIBOR as com-
pensation for the credit risk associated with the total return receiver, but
the total return payer in the example seems perfectly willing to extend a
synthetic loan at the much lower spread of 50 basis points. In part, this
apparent inconsistency owes to at least two factors. First, counterparty
credit risk in the TRS contract can be mitigated via collateral and net-
ting arrangements that are common in other over-the-counter derivatives
contracts. Second, the TRS market oļ¬ers opportunities to total return
payers that, in practice, may not be easily available in the cash market.
The existence of such opportunities in one market but not the other may
create a wedge between otherwise comparable spreads. Indeed, as we shall
see below, market participantsā™ uses of total return swaps go well beyond
issues relating to their relative funding costs.
One opportunity aļ¬orded by a total return swap that may not be avail-
able in the cash market is the ability to short certain debt instruments.
Suppose, for instance, that a market participant has a negative view regard-
ing the future prospects of a given corporation. That participant may want
to express and (hopefully) proļ¬t from its view by taking a short position
in bonds issued by that corporation. For most corporate bonds, however,
that may not be a practical alternative given the inexistence of a fully
functional corporate repo market. A more feasible approach would be to
become a total return payer in a TRS contract that references a bond issued
by that corporation. For instance, assuming the participant is not actually
holding the reference bond, should the corporation default during the life
of the TRS, the total return payer would realize a proļ¬t equal to the dif-
ference between the market value of the bond at the time of inception of
the TRS and its recovery value.
Paying positions in total return swaps can be viewed as hedging vehicles
for investors who are actually long the reference asset. The investor holds
the reference entity, but essentially transfers all risks associated with the
asset to the total return receiver. Similar to a credit default swap, this
risk transfer can be done anonymously without requiring notiļ¬cation of
the reference entity, a feature that may be particularly attractive to a bank
that would like to diminish its exposure to particular customers without
risking bank relationships. Likewise, total return swaps can be used to
obtain exposure to debt instruments that may not be easily bought in the
cash market. For instance, one might want to be long the bonds issued by
a given corporation, but those bonds are in the hands of ābuy-and-holdā
investors who are not interested in selling. Through a receiverā™s position
in a total return swap, one may synthetically ābuyā those bonds in the
desired amounts. Lastly, as we mentioned above, because the maturity of
7.3 Valuation Considerations 87

the TRS contract need not coincide with that of the underlying bonds, TRS
contracts can be used to create assets that match the needs of individual
investors.

7.3 Valuation Considerations
Going back to Figure 7.1, most determinants of the cash ļ¬‚ows of a total
return swap come from outside the TRS market: The coupon (C) and the
initial and ļ¬nal prices of the reference bond come from the corporate bond
market, and the values of short-term LIBOR (L) throughout the life of the
contract are determined in the interbank loan market. The one exception
is the TRS spread T . Indeed, pricing a TRS is essentially synonymous to
ļ¬nding the value of T that would prevail in a competitive market.
As we have done in previous chapters, we shall rely on the static repli-
cation approach to price the TRS contract, but here we will take this
opportunity to look at the replicating portfolio from a slightly diļ¬erent
angle. In particular, if we can come up with a replicating portfolio for the
TRS contract, one can essentially combine long and short (or short and
long) positions in the TRS and the replicating portfolio, respectively, to
create a hedged portfolio, i.e., a portfolio that is completely riskfree. To see
this, simply recall that, by deļ¬nition, the replicating portfolio exactly
mimics the cash ļ¬‚ows of the TRS contract. Thus, any gains and losses
associated with, say, a long position in the contract, will be perfectly oļ¬set
by losses and gains in a short position in the replicating portfolio. In other
words, the replicating portfolio can be recast as that portfolio that pro-
vides āthe perfect hedgeā for the derivatives position. Thus, just as we
noted before that one can value a derivatives contract by looking at the
costs associated with establishing its replicating portfolio, we can now say
that determining the price of a derivatives contract essentially involves
determining the costs of setting up āthe perfect hedge,ā i.e., the costs of
hedging the derivatives position.
For the case of the total return swap, it is very straightforward to ļ¬nd
the replicating portfolio. Consider the position of the total return payer in
a market that is free of frictions and where counterparty credit risk is not
an issue. The TR payer pays a cash ļ¬‚ow to the total return receiver that is
supposed to mimic exactly the cash ļ¬‚ow of a given reference bond. Thus,
a short position in the reference bond is the replicating portfolio! Conse-
quently, if the TR payer is long the reference bond, he has a fully hedged
position in the TRS contract. To see this, consider a TRS contract between
two parties who can fund themselves at LIBOR + X. For additional sim-
plicity, assume that the reference bond is selling for its par value at time
0 and has the same maturity of the contract. Column (2) in Table 7.1
88 7. Total Return Swaps

TABLE 7.1
Determining the Total Return Swap Spreada

Cash Flows
TRS Ref. Bond
Year (TRS payer) (long position) TRS+Bond

(1) (2) (3) (2) plus (3)

A. Assuming no default by the reference entity

0 0 0 0
ā’C + L + T C ā’Lā’X ā’X
1 T
ā’C + L + T C ā’Lā’X ā’X
2 T
ā’C + L + T C ā’Lā’X ā’X
3 T
ā’C + L + T C ā’Lā’X ā’X
4 T

B. Assuming default by the reference entity at t = 2b

0 0 0 0
ā’C + L + T C ā’Lā’X T ā’X
1
ā’C + L + T + 1 ā’ R C ā’Lā’X +1ā’R T ā’X
2
a Negative cash ļ¬‚ows represent cash outlays. Notional amount = \$1.
b Assuming default occurs immediately after the coupon payments are made.

shows the cash ļ¬‚ows of the total return payer. Let us look at the upper
panel ļ¬rst, which shows cash ļ¬‚ows corresponding to a scenario involving
no default by the reference entity. At the inception of the contractā”time
0ā”the TR payer receives no cash ļ¬‚ow, but thereafter on each payment
date the TR payer will pay the reference bondā™s coupon C and receive
L + T from its counterparty in the TRS contract. Column (3) shows the
cash ļ¬‚ows associated with the outright purchase of the bond, funded on
the balance sheet of the TR payer. Again, at time 0 there is no cash out-
lay as the bond purchase is ļ¬nanced through a loan with an interest rate
of L + X. Thereafter, the long position in the bond entails receiving the
bond coupon C and paying the cost of funding that position L + X. The
last column in the table shows the cash ļ¬‚ow of a portfolio composed by
the payerā™s position in the TRS and a long position in the reference bond.
This portfolio involves no initial outlay at time 0 and a net cash ļ¬‚ow of
T ā’ X in all subsequent periods in the absence of a default by the reference
entity.
What happens in the event of default? The lower panel shows a scenario
where the bond issuer makes its coupon payment in period 2 and immedi-
ately proceeds to declare bankruptcy. In addition to the regular exchange
7.4 Variations on the Basic Structure 89

of payments in the TRS, the TR payer receives the diļ¬erence between
the value of the bond at time 0 (\$1) and its post-default value (R) and
the contract is terminated. Likewise, the long position entails the usual
coupon and funding cost ļ¬‚ows plus repayment of the \$1 loan obtained at
time 0 and receipt of the recovery value of the bond. Again we ļ¬nd that
the cash ļ¬‚ow of the portfolio in the last column is zero at the inception of
the contract and T ā’ X for as long as the contract remains in force.
What is the value of T ā’ X that is consistent with a market that is free
from arbitrage opportunities? Note that one spends nothing a time 0 and
is assured to receive T ā’ X during the life of the contract no matter what.2
Suppose T ā’ X is a positive quantity. One would be getting something for
sure out of nothing. That would clearly be an arbitrage opportunity. Indeed,
the opportunity to get something for nothing would make paying positions
in this TRS so attractive that prospective payers would be willing to accept
a lower TRS spread (T ) for as long as T ā’ X were positive. Consequently,
in equilibrium T would be such that T ā’ X = 0. The case where T ā’ X
is negative is entirely analogous: for as long as T ā’ X < 0, prospective
TRS payers would demand a higher T in order to enter into the TRS
contract.
The simple example in Table 7.1 yielded the basic insight that an impor-
tant factor in the pricing of a TRS contract is the funding cost of the total
return payer. While this is a key determinant of the TRS spread, other
factors also come into play under more realistic situations. In particular,
the total return payer may be concerned about the credit risk associated
with its TRS counterparty. As with other over-the-counter derivatives, this
risk can be mitigated via collateralization and netting arrangements.3

7.4 Variations on the Basic Structure
This chapter described the basic structure of a total return swap. We con-
clude by noting that there are several variations around this basic structure.
For instance, instead of terminating automatically upon default by the ref-
erence entity, the contract may continue until its maturity date. In addition,
instead of having a single bond or loan as the reference asset, the contract
might specify a given portfolio or a bond index as the reference āasset.ā

2
The reader can verify that, under the assumptions made in Table 7.1, the portfolio
consisting of the TR paying position and the bond would pay T ā’ X for as long as the
reference entity remains solvent regardless of when default occurs.
3
The total return receiver may also be concerned about counterparty credit risk, but
that too can be mitigated. For instance, the total return payer may post the underlying
bond as collateral.
90 7. Total Return Swaps

So-called index swaps can be a more eļ¬cient way of getting exposure to a
market aggregate, compared to buying all the individual securities in the
index. Other variants of the basic structure include the forward-starting
TRS, which allows investors to enter into a TRS today that will start only
at some future date at a predetermined spread, and contracts with an
embedded cap or ļ¬‚oor on the reference asset.
8

The credit derivatives instruments we have examined thus far have in com-
mon the fact that their ļ¬nal payoļ¬s are essentially tied to ādefault eventsā
involving the reference entity. Spread and bond options deviate from this
norm. Spread option payoļ¬s are generally speciļ¬ed in terms of the perfor-
mance of a reference asset relative to that of another asset. Bond options
are options to buy or sell bonds at a future date at a predetermined price.
Both types of options can be exercised regardless of whether or not the
issuers of the underlying assets have defaulted. They are credit derivatives
because they involve the yield spread of a credit risky asset over that of
some benchmark assetā”the spread optionā”or the market price of a risky
bondā”the bond option. As we shall see below, the basic structure of spread
and bond options is similar to that of standard call and put options.

8.1 How Does It Work?
To understand the workings of a credit option it is best to start with a sim-
ple example. Suppose you want to have the option, but not the obligation,
to buy a particular ļ¬ve-year asset swap one year from today.1 You want
the asset swap to reference a ļ¬xed-rate bond issued by XYZ Corp. and to
ĀÆ
have a prespeciļ¬ed par spread of A. You approach an options dealer and

1
Recall, from Chapter 5, that buying an asset swap means buying the underlying
bond and receiving ļ¬‚oating-rate payments in an interest rate swap.
92 8. Spread and Bond Options

agree to pay her an amount X today so that she will be ready to sell you
that asset swap in one year, should you decide to buy it then. We have just
described the basic structure of a simple spread option.
In the above example, the ļ¬ve-year XYZ asset swap is the āunderlying
ĀÆ
instrumentā in the spread option, the predetermined spread A is called the
āstrike spreadā of the option, and the upfront payment made to the dealer
is the āoption premiumā or the price of the option. The āexpiration dateā
of this option is one year from today. The āexercise dateā is also one year
from now, given that the contract only allows you to exercise the option in
one yearā™s time.2
Under which conditions would you decide to exercise your option to buy
ĀÆ
the asset swap? At the exercise date, you will compare the strike spread A
to the then prevailing asset swap spread on the underlying bond. You will
buy the underlying asset swap if the strike spread is above the prevailing
asset swap spread for the relevant bond, otherwise you let the option expire
unexercised. (If the prevailing spread is above the strike spread and you
still want to enter into a ļ¬ve-year XYZ asset swap, you would be better oļ¬
buying the asset swap in the open market and receiving the higher spread.)
Referring back to standard call and put options, one can think of the above
example as a put option on the spread. The holder of the option will beneļ¬t
if the spread prevailing in the marketplace at the expiry date of the option
is lower than the strike spread.
In some dimensions, a put option on the spread is analogous to a call
option on the underlying bond, where the latter option is a bilateral con-
tract in which one party pays for the option to buy a given bond at a future
date at a predetermined price. Consider a call option written on the same
bond that was referenced in the asset swap underlying the spread option
just discussed. For a given position of the LIBOR curve, a narrowing in
the asset swap spread, which we know would beneļ¬t the holder of the put
option on the spread, would be associated with an increase in the market
price of the bond, which would beneļ¬t the holder of a call option on the
bond. By the same token, a call option on the spreadā”for instance, an
option to sell an asset swap at a future date with a predetermined asset
swap spreadā”is akin to a put option on the reference bondā”or an option
to sell the bond at a future date at a predetermined price.
We have compared spread and bond options while holding the LIBOR
curve constant. But the analogy between spread and bond options breaks
down somewhat when we consider the eļ¬ects of shifts in the LIBOR curve.
For instance, suppose the LIBOR curve shifts up but the asset swap spread
for the underlying bond remains the same. In this case the values of the

2
The single ļ¬xed exercise date makes this a āEuropeanā option. We brieļ¬‚y discuss
other exercise date structuresā”āAmericanā and āBermudanā optionsā”towards the end
of this chapter.
8.2 Common Uses 93

bond and spread options are aļ¬ected diļ¬erently. Intuitively, one can think
of the eļ¬ect on the value of the spread option as being limited to the
discounting of its expected future payoļ¬, but the eļ¬ect on the value of the
bond option also stems from the fact that higher rates will drive the bond
price lower, aļ¬ecting the payoļ¬ directly. This highlights the fact that bond
options involve a joint bet on both the general level of interest rates and
the credit risk of the bond issuer, whereas spread options pertain mainly
to the latter.3
We mentioned above that the payoļ¬s of spread and bond options are
speciļ¬ed in terms of the performance of the reference asset relative to that
of a benchmark security. If this was not clear at the beginning of this
chapter, it should be so by now. Take, for instance, the spread option
examined above. Given that asset swap spreads are generally deļ¬ned in
terms of a spread over short-term LIBOR, the payoļ¬ of the spread option
can be thought of as being a function of how the synthetic ļ¬‚oating-rate note
embedded in the asset swap will perform during the life of the option vis-
`-vis a ļ¬‚oater that pays LIBOR ļ¬‚at. Likewise, the payoļ¬ of a bond option
a
may well depend on how its price moves relative to the prices of other
bonds; for instance, a decrease in the credit quality of the bond issuer will
lower the underlying bondā™s price and make it underperform relative to
other bonds. Nonetheless, as we noted above, the payoļ¬ of a bond option
also depends on the general level of interest rates.

8.2 Common Uses
Similar to other credit derivatives, spread and bond options can be used to
express a view on the future credit quality of a given issuer. Take the put
spread option discussed above and assume that the strike spread is set at the
current asset swap spread associated with the underlying bond. When the
option buyer is not holding the underlying bond, she is essentially placing
a bet that the credit quality of the issuer will improve during the coming
year. If she turns out to be right, she will proļ¬t from her view by being
able to buy an asset swap in one yearā™s time that pays a higher spread than
the one then prevailing in the market place. (Likewise, an investor with a
bearish view on the issuer might want to buy a call spread option struck
at or close to the current asset swap spread of the bond.)
As we noted above, pure credit views are more eļ¬ectively expressed
with spread options than with bond options because the payoļ¬ of
the latter also depends importantly on the general level of interest

3
This is analogous to the distinction between ļ¬xed- and ļ¬‚oating-rate notes, which
we discussed in Chapter 4.
94 8. Spread and Bond Options

ratesā”the āLIBOR curve.ā Nonetheless, if one wants to express a com-
bined credit and interest rate view, one might want to consider a bond
option. For instance, if one expects market interest rates to go higher and
the credit quality of a given reference entity to deteriorate, one might con-
sider buying a put option on a bond issued by that entity. Indeed, bond
options are commonly used by some hedge funds to place such joint bets,
which implicitly involve a view on the correlation between interest rate
levels and credit quality.
Spread and bond options can be used to express a view on volatility that
is independent of the direction taken by either the underlying spread or
bond price. Consider, for instance, someone who expects greater volatility
regarding both market interest rates and the prospects of a given ļ¬rm, but
who has no particular view on the direction of the resulting movements
in either market rates or the bond spreads associated with the ļ¬rm. That
investor can buy an option written on a bond issued by the ļ¬rm and hedge
it by taking short positions in the underlying bond. If the investor is appro-
priately hedged, should the bond price move, the investorā™s portfolio will
be little aļ¬ected as the change in the market value of the long position in
the option will be oļ¬set by the change in the value of the short position in
the underlying. Should uncertainty (volatility) surrounding the price of the
underlying bond increase, however, the value of the long option position
will increase, beneļ¬ting the investor.4 An analogous argument applies to
Investors who hold a particular bond, or who may be buyers in an asset
swap, can use spread and bond options to potentially increase the yield on
their portfolios. For instance, a bond investor may want to sell call options
on that bond with a strike price that is well above the current price of
the bond. After collecting the option premium, the investor waits for the
option buyerā™s decision on whether or not to buy the bond at the optionā™s
exercise date. Should the option buyer decide not to exercise its right to
buy the bond, the investor keeps both the bond and the premium. If the
buyer does exercise the option, the investor sells the bond to the option
buyer for the strike price, which could still be higher than the price the
investor originally paid for the bond.
Other than allowing investors to take ļ¬nancial positions that reļ¬‚ect their
views on prospective credit and interest rate developments, spread and
bond options are used as hedging vehicles by banks and other institutions
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