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Appendix A
The Recovery of Market Value Condition (RMV)
The RMV condition is inspired by the recovery rules of OTC derivatives (see Schönbucher (2003)). The ISDA master agreement for swap contracts specifies that at default of one counterparty, the other counterparty’s claim is the market value of a non-defaulted, but otherwise equivalent, security (if the value of this security is positive). He is paid a fraction 1 RR− of this claim.
Now in order to price a credit default swap, we need to follow some procedure.
First, we assume that defaultable zero-coupon bonds are the underlying for default swaps. Define the price of a defaultable zero-coupon bond at time t as ( )Z t . Thus in mathematical terms, the pricing recursion under the RMV condition is:
{
( ) ( ) (1 ( )) ( ) 1} (
( ) ( ))
( )
( ) 1
e rh q t Z t h q t Z t h t t RR
Z t Z T
π π
− + + + − − + − +
=
=
Here, ( )q t is the probability of an up-move conditioned on no default occurring,
( )
Z+ t+h and Z−(t+h) are the two states that the bond price evolves to in the next period, ( )π t is the probability of default, and RR is the recovery rate. Notice that the recovery value is a fraction of the value of the bond, if and when default occurs. Next, we compute the default leg of the default swap contract with the following recursive equation:
{ } [ ]
( ) ( ) ( ) (1 ( )) ( ) 1 ( ) ( )(1 ) ( )
( ) 0
rh
L L L
L
PV t e q t PV t h q t PV t h t Z t RR t
PV T
π π
− + + + − − + − + −
=
=
The first part above is the present value of future possible losses on the default swap, given that default has not occurred at time t. The second part is the expected value of the loss-given-default. As for the fixed spread leg of the contract, the expected present value of a $1 payment at each node is calculated, defined below:
{ } [ ]
1 1 1
1
( ) ( ) ( ) (1 ( )) ( ) 1 1 ( )
( ) 0
PV t e rh q t PV t h q t PV t h t
PV T
π
− + + + − − + + −
=
=
Finally, the spread is calculated with Eq. (4) in Chapter II.
Appendix B
One-Factor Gaussian Copula
Copulas are functions that express dependence among random variables. The one-factor Gaussian copula is constructed by the standard multivariate normal distribution with correlationρ. The method is summarized below:
1. Let , , X1 X2 ..., X be N N independent random variables, each distributed as N(0,1).
2. Define random variables , , Y Y1 2 ..., Y as N
= 1 2
i i i i
Y a M + −a X , i=1, 2, ...,N
where M ~ (0,1)N , independent of all X . In Gaussian copula models, it is i often assumed that the variables have the same pair-wise correlation ρ, such that ai =aj = ρ .
3. Given the default probability p at a specific time t , firm i defaults if ( )Φ Yi ≤ . p
The systematic risk factor M can be viewed as an indicator of the state of the business cycle. The idiosyncratic factor X is a firm-specific factor, which is used to i describe the quality of the management, or the financial situation of the firm, etc. The relative sizes of the systematic and idiosyncratic components are controlled by the correlation coefficient ρ. If ρ= , then the market condition has no direct influence 0 on the firms. While if ρ= , then M is the only driver of defaults, and the 1 individual firm has no control.
Note here that we are assuming ρ is positive so that the coefficient of M for the Y ’s is a real number. Effectively, we assume that all firms in the same economy i are positively related to the macroeconomic factor M , since there are nearly few industries that do not comply with the business cycle. When the market is good, there should be lower chance of default, and vice versa. The correlation between two firms is actually attributed to the systematic factor, and has nothing to do with their idiosyncratic counterparts in this model. Still, there exist realistic cases where the default behavior of two firms may be negatively related. An example is the existence of the “competition effect” (see Jorion and Zhang (2007)), which refers to the situation where a firm benefits from the demise of its rival. However, one-factor Gaussian copulas cannot directly capture this effect. The idiosyncratic components for