Having generated in section 6.2.2, page 117, through Monte-carlo simulations, the distribution of interest rate swaps over the interval (t, T], we are interested to assess the impact of the modelling of credit contagion with the phase-type distribution on those claims. In order to do so, with replicate the scenarios of credit contagion that we used in the marginal distribution (Q(τi > t)) illustrations in section B.2, page 159, with the 6 following scenarios:
• counterparty C and investor I are low risk obligors i.e. aC =aI = 0.005
• counterparty 1 and investorI are normal risk obligors i.e. aC =aI = 0.015
• counterparty 1 and investorI are high risk obligors i.e. aC =aI = 0.03
• counterparty 1 is a low risk obligor and investorI is a high risk obligors i.e. aC = 0.005, aI = 0.03
• counterparty 1 is a high risk obligor and investorI is a low risk obligors i.e. aC = 0.03, aI = 0.005
• counterparty 1 is a very high risk obligor and investor I is a low risk obligors i.e. aC = 0.04, aI = 0.005
with aC, aI being the respectively the base intensity of counterparty C and investor I. The variation of the contagion level through the jump intensity b{i,j} will be as varied on a range of value to assess the impact of contagion between obligors.
We will consider a two payers 15-year and 7-year interest rate swap as in figure B.2, page 162. The valuation using proposition 6.2.1, page 115 is considered under the assumption of indepen- dence between the probability distributions Q(τi > t), i ∈ T = {I, C} and the FT-measurable claim X. In this case, the CVA and DVA value in equation 6.41 is adjusted with the separation of the probability and the valuation of the claim X in the future and the BVA are calculated with the discounted distribution of the defaultable claim on the interval (τ, T] weighted by the probability of default in the trigger set T over the same interval. We assume also time bucketing by postponing the default timeτ to the first ti following τ.
In terms of implementation, as specified in [Cesari09], it is necessary when aggregating claims with different underluyings into a credit risk systems that the discretisation steps are identical in numbers and in time reference thus to enable consistent calculations and potential netting of risk positions. In accordance with the discretisation in section 6.2.2 for the valuation of the non defaultable claims, we use an identical n-step time-bucketing with a weekly step for the generation of the probability distributions of Q(τ > t). Thus, we implement the following equations
BCV A = CV A−DV A = LGDC n X i=1 (αeTti−1G C1−αeTtiGC1)E(βtiX(ti) +) − LGDI n X i=1 (αeTti−1G I1−αeTtiGI1)E(βtiX(ti) − ) = n X i=0 (αeTti−1G
C,I1−αeTtiGC,I1)E(βti[ED(ti)]) (6.44)
Thus, the figures 6.1 and 6.2 contain those CVA/DVA absolute values att= 0 and immediately show the sensitivity of the metrics as a function of the intersection of several features such as:
• the maturity of the financial claim,
• the structural exposure (payer vs receiver).
More importantly, the graphs 6.1 and 6.2 highlight the non-linear deterioration of both the DVA and CVA values and, thus, the difficulty in quantifying the direction or the magnitude of the change. However, in the multivariate phase-type as presented in the illustration in section B.2, page 159, the contagion impact is feeding on maturity higher than 10 years thus justifying the higher exposure of the 15-yr swap.
Additionally, the previous figures show that the impact of contagion-correlation or potential wrong-way risk is far wider than the volatility adjustment of α = 1.4 required by Basel II.It also confirms the limitation of this fixed volatility adjustment parameter α. This results are consis- tent with previous work from [Brig07a] that investigates the CVA volatility behaviour through the correlation of the default intensity and the underlying drivers like interest rate. However, [Brig07a] encapsulates a wider contagion scenario describing the state of the world where empiri- cally high period of defaults are consistent with higher interest rates. Same results for exotics like bermudan swaptions and CMS spread options are obtained with [Brig07a] and similar aspects of impact of volatility and correlation for credit and commodities derivatives can be found in [Brig06], [Brig08b], [Brig10b] and [Brig08a].
We believe an overlooked problem in Counterparty Credit Risk is the case of “wrong-way risk”where the credit contagion among a pool of obligor affects at the same time the value of the claim, specifically defaultable ones, and the probability of defaults in the trigger set T in a feedback loop. Most of the analysis use a correlation as previously of underlying drivers, we hope that the current multivariate framework under permutations π ∈ Π will highlight this feedback effect as pointed in the articles [Braithwaite12] and [Carver11].
Comment: In the current section, the implementation is done using m = 2 obligors since the probabilities Q[P
π∈ΠC1t<{τ(π−)<τC<τ(π+)} | Ft] is not different than Q[1t<τC} | Ft] since, by
Figure 6.1: Contagion Illustration: CVA and DVA effect for a 15y-payer, semi-annual, interest- rate swap as a function of the credit contagion bI,C for different scenarios of credit riskiness of the counterparty C and the investor I.
Figure 6.2: Contagion Illustration: CVA and DVA effect for a 7y-payer, semi-annual, interest-rate swap as a function of the credit contagion bI,C for different scenarios of credit riskiness of the counterparty C and the investor I.
τC can be viewed as an arbitrary higher intensity aC. However, in the next section on defaultable claims, this simplification is no longer possible in a multivariate setting.