FACTORES QUE DETERMINAN LA CALIDAD DE LA CARNE

In the case of early default, the default procedure may take several days to be completed. The time elapsed between the default event and the completion of the close-out procedure is called the margin period of risk. During this time period, the mark-to-market of the derivative may change considerably, resulting in a large mismatch between the posted collateral and the exposure. Moreover, the surviving party may default during this period, which should also be considered when we compute the on-default cash-flow.

In this section, we continue our discussion for the CCP cleared or bilateral CSA trades following the analysis carried out in Brigo and Pallavicini [30] and discuss the cash flows occurring upon an early default event taking into consideration the margin period of risk.

6.3.1

On-default cash-flow

The case of trades where there was a bilateral CSA was discussed in Section 2.1.2. Here, we extend the study and consider also the initial margins that are posted

to cover the additional risks. In the case of CCP cleared trades, if the client “C” defaults first, the clearing member “I” will take responsibility for the position as in a bilateral trade and evaluate the close-out amount.

Assume that the default procedure takes time δ to be completed. The default procedure can be considered as though the surviving party at the default time τ

enters a deal with a cash-flow θ and maturityτ +δ. This cash-flow will depend on the close-out amount ετ+δ (with consideration of the margin period risk) and the

value of (the pre-default) variation and initial margin accounts denoted respectively as Mτ−,NC

τ− and N_τI−.

Applying the results in Corollary 6.2.1, the adjusted price of such a deal denoted as ϑ at timeτ can be expressed as (we take the conditional expectations under the filtration Gτ, using the same technique as in Section 2.3.2),

ϑτ :=E˜hτ h θτ+δ ετ+δ, Mτ−, N_τC−, N_τI− D(τ, τ +δ; ˜fS) i , (6.25)

where the expectation is taken under the probability measure _Q˜h_{, and ˜f}S _{is the}

effective funding rate of the surviving party that is funding such a deal.

The above pricing equation depends on the funding rates of both parties. How- ever, practically, one cannot know the other party’s liquidity policy. [30] approxi- mates the discount factors by assuming that the payment takes place at the default time τ without modelling the funding rates of both parties and points out that the effects of this approximation are second order compared to the uncertainties of the recovery rates and close-out values. Therefore, instead of (6.25) we write

ϑτ :=E ˜ h τ θτ+δ ετ+δ, Mτ−, NC τ−, N_τI− . (6.26)

Bare in mind that in the case of a CCP cleared trade we haveNI τ− = 0.

6.3.2

Close-out netting rule

In order to determine the cash-flow θτ+δ, we need to investigate the close-out net-

that may happen upon the first default event. Starting with the case where the counterparty/client default first τ =τC < τI, we analyse the following scenarios.

When ετ+δ ≥ 0 and Mτ− ≥ 0, the investor has a positive exposure and the

counterparty has posted variation margin. We then have the following cases: 1. The exposure is netted with the variation and initial margins posted by the

counterparty, but the collateral is not enough to cover the exposure, and the investor can get back his initial margin:

1{ετ+δ≥Mτ−+NτC−}(RC(ετ+δ−Mτ

−−NC

τ−)−N_τI−).

2. The exposure is covered by the variation and initial margins. The investor does not face a loss and gets back his initial margin:

1_{ε τ+δ<Mτ−+N C τ−} (ετ+δ−Mτ−−NC τ−−N_τI−).

3. In the case where the exposure is completely covered by the variation margin, we need to consider two more scenarios:

• The investor does not default or defaults after the margin period. He faces no loss and gets back his initial margin:

1{ετ+δ<Mτ−}1{τI>τC+δ}(ετ+δ−Mτ−−N

C

τ−−N_τI−).

• The investor defaults during the margin period. In this case, the in- vestor’s initial margin can be used to reduce losses:

1{ετ+δ<Mτ−}1{τI≤τC+δ} (ετ+δ−Mτ−−N I τ−)+ +R0_I(ετ+δ−Mτ−−NI τ−)−−N_τC− .

When ετ+δ ≥ 0 and Mτ− < 0, the investor has a positive exposure and the

4. If the initial margin posted by the counterparty is not enough to cover the investor’s exposure, the investor faces a loss and gets back the initial margin and the variation margin if it is not rehypothecated:

1_{ετ+δ≥NC τ−}

(RC(ετ+δ−NτC−)−R0_CM_τ−−NI

τ−).

5. If the initial margin is enough to cover the investor’s exposure, the investor gets back his initial margin and does not suffer a loss unless the variation margin is rehypothecated:

1_{ε

τ+δ<N_τC−}((ετ+δ−Mτ− −N

C

τ−)−−R_C0 (ε_τ+δ−M_τ−−N_τC−)+−N_τI−).

When ετ+δ < 0 and Mτ− ≥ 0, the investor has a negative exposure and the

counterparty has posted variation margin. We then have the following cases: 6. The counterparty expects to get back the variation and initial margins.

• If the investor does not default or defaults after the margin period, the counterparty gets back the collateral in full:

1{τI>τC+δ}(ετ+δ−Mτ−−N

C

τ−−N_τI−).

• If the investor defaults before the margin period, the counterparty may face a loss depending on where the collateral is rehypothecated:

1{τI<τC+δ} RI(ετ+δ−NτI−)−+R_I0 (ε_τ+δ−N_τI−)+−M_τ− − + (ετ+δ−NτI−)+−M_τ− + −N_τC− .

When ετ+δ <0 and Mτ− <0, the investor has a negative exposure and the in-

vestor has posted variation margin. The exposure is netted with the posted collateral unless the variation margin is rehypothecated, in which case the initial margin is used to reduce the losses. We then have the following cases:

7. If the initial margin is not enough to cover the losses due to the rehypotheca- tion, the investor suffers a loss and gets back his initial margin:

1{ετ+δ−Mτ−≥NτC−}(R

0

C(ετ+δ−Mτ−−N_τC−)−N_τI−).

8. If the initial margin is enough to cover the losses due to the rehypothecation, the investor gets back his collateral in full:

1_{ε

τ+δ−Mτ−<NτC−}

(ετ+δ−Mτ−−N_τC−−N_τI−).

9. If the investor has to pay a greater exposure, we consider the following two cases:

• If the investor does not default or defaults after the margin period, the investor gets back his initial margin:

1{ετ+δ<Mτ−1{_{τI >τC}+δ}}(ετ+δ−Mτ−−N

C

τ−−N_τI−).

• If the investor defaults before the margin period, the investor’s initial margin can be used to reduce the losses:

1{ετ+δ<Mτ−}1{τI<τC+δ} (ετ+δ−Mτ−−N I τ−)+ +RI(ετ+δ−Mτ−−N_τI−)−−N_τC− .

Similarly, we can list the cash flows when the investor defaults first. If we sum up all the cash flows for all the possible scenarios, we can reach an expression for the on-default cash-flow θτ+δ (see [30] for more details). Substitutingθτ+δ in (6.26)

we see that the price of the on-default cash-flow can be expressed as follows,

ϑτ =E

˜

h

τ[ετ+δ]−CVA(τ, T;M, NC, NI) + DVA(τ, T;M, NC, NI), (6.27)

collateralized CVA and DVA terms with CVA(τ, T;M, NC, NI) :=_E ˜ h τ 1{τC<τI+δ}ΠCVAcoll(τ) , DVA(τ, T;M, NC, NI) :=_E ˜ h τ 1{τI<τC+δ}ΠCVAcoll(τ) , (6.28) and Π_CVAcoll(s) = LGD_C (ε_τ+δ−N_τC−)+−M+ τ− + + LGD0_C (ε_τ+δ−N_τC−)−−M− τ− + , Π_DVAcoll(s) = −LGD_I (ε_τ+δ−N_τI−)−−M_τ−− − + LGD0_I (ε_τ+δ−N_τI−)+−M_τ+− − . (6.29) Observe that if rehypothecation of the collateral is not allowed, the terms multiplied by LGD0_C and LGD0_I drop out of the CVA and DVA calculations.

In the case where the trade is cleared by a CCP, only the client posts initial margin, so we can set NI = 0 in equation (6.27). Moreover, if upon the default of a clearing member the transaction will be transferred to a backup clearing member, we can then assume that the loss given default for the clearing member is close to zero.