4.3 Structural Model of CVA for CDS
4.3.1 Numerical Results
Hull and White calculate the initial spread S for the value of a CDS con- tract with unilateral CVA. That is S, the insurance premium, that makes the contract between protection buyer and protection seller equal to zero. Hull and White then see how S changes with the correlation between the reference credit and protection seller. This is equivalent to the approach of Brigo. In Brigo the credit spread is assumed to have been contractually fixed at some earlier time and therefore in his formulation he is not so much valuing the CDS as re-valuing the contract after intermediary credit activ- ity. For any t > 0, S as a constant can be made the subject of the CDS valuation formula by setting its value equal to zero. The value can then be compared to the original S paid and the change in value to credit risk and time evaluated. Thus is doesn’t matter whether one uses the payment streams between counterparties or S to value the CDS. In practise CDS values are quoted inS.
Notice that it is the lack of optionality in Hull and White’s framework that allows them through simple algebra to makeS the subject of the for- mula. This is not possible in Brigo’s model due to the fact that CDS prices are embedded in options. In Brigo’s model they may have to be deduced numerically.
The valuation att= 0 ofS is thus just a particular case of the valuation methodology of Brigo. Hull and White restrict themselves in their numerical work to the valuation of CVA att= 0 and they calculate it in S-space that is in terms of the difference between the credit spread of CDS without credit risky protection seller and that with.
They find that, as with Brigo, as the correlation is increased the differ- ence increases, i.e. the CVA term grows. Their model thus captures the wrong-way-risk effect. The dip effect found in Brigo for high correlation and deterministic credit spreads is not replicated by Hull and White. The volatility parameters for their credit indices are set to one and are thus two high to realise the case where one always defaults before the other.
Chapter 5
Conclusion
In this investigation, an attempt has been made to give an account of val- uation adjustment under credit risk, CVA. The basic models in credit risk modelling and the model independent calculation of CVA were developed and two particular models and their experimental results investigated. The market standard model of Brigo based upon his general CVA pricing frame- work was found to be flexible and accurate. Its experimental results proved that it took account of all the risk dynamics associated with CDS pricing under counterparty risk. The model of Hull and White was also found to be simple and easy to calibrate and to effectively capture the wrong-way-risk effect as their experimental results verify.
Chapter 6
Appendix
6.0.2 A Proof. Recall, Π(t, T) = π(t, T) = 1{T≤min{τ0,τ2}}π(t, T) + 1{τ0≤min{τ2,T}}π(t, T) + 1{τ2≤min{τ0,T}}π(t, T)Then by the linearity of E-operator, Et[ΠD(t, T)] = Et[Π(t, T)] + Et[LGD01{τ2≤min{τ0,T}}D(t, τ0)[−N P V(τ0)] +] − Et[LGD21{τ0≤min{τ2,T}}D(t, τ0)[−N P V(τ2)] +] becomes, Et[ΠD(t, T)] = Et[Π(t, T) + LGD01{τ2≤min{τ0,T}}D(t, τ0)[−N P V(τ0)] + − LGD21{τ0≤min{τ2,T}}D(t, τ0)[−N P V(τ2)] +]
Then using (103) becomes,
Et[ΠD(t, T)]
= Et[1{τ0≤min{τ2,T}}π(t, T) + 1{τ2≤min{τ0,T}}π(t, T) + 1{T≤min{τ0,τ2}}π(t, T)
+ LGD01{τ2≤min{τ0,T}}D(t, τ0)[−N P V(τ0)]
+
− LGD21{τ0≤min{τ2,T}}D(t, τ0)[−N P V(τ2)]
= E[1{τ0≤min{τ2,T}}π(t, T) + 1{τ2≤min{τ0,T}}π(t, T) + 1{T≤min{τ0,τ2}}π(t, T) + LGD01{τ2≤min{τ0,T}}D(t, τ0)[−N P V(τ0)] + − LGD21{τ0≤min{τ2,T}}D(t, τ0)[−N P V(τ2)] +] = E[1{τ0≤min{τ2,T}}π(t, T)−LGD21{τ0≤min{τ2,T}}D(t, τ0)[−N P V(τ2)] + 1{τ2≤min{τ0,T}}π(t, T) + LGD01{τ2≤min{τ0,T}}D(t, τ0)[−N P V(τ0)] + + 1{T≤min{τ0,τ2}}π(t, T)]
using the definition of LGDi = 1−RECi,
= E[1{τ0≤min{τ2,T}}π(t, T)
+ (REC2−1)1{τ0≤min{τ2,T}}D(t, τ0)[−N P V(τ2)]]
+ Et[1{τ2≤min{τ0,T}}π(t, T) + (1−REC0)1{τ2≤min{τ0,T}}D(t, τ0)[−N P V(τ0)]
+]
+ Et[1{T≤min{τ0,τ2}}π(t, T)]
Now develop the three expectations,
1{τ2≤min{τ0,T}}π(t, T) + (1−REC0)1{τ2≤min{τ0,T}}D(t, τ0)[−N P V(τ0)]
+ = 1{τ2≤min{τ0,T}}π(t, T) + 1{τ2≤min{τ0,T}}D(t, τ0)[−N P V(τ0)] + − 1{τ2≤min{τ0,T}}D(t, τ0)REC0[−N P V(τ0)] +
taking the expectation conditional onτ0, and noticing thatπ(t, T) =π(t, τ0)+
D(t, τ0)π(τ0, T) asπ(s, u) are discounted to s. And also using N P V(τ0) =
Eτ0[π(τ0, T)], gives Eτ0[1{τ2≤min{τ0,T}}π(t, T) + 1{τ2≤min{τ0,T}}D(t, τ0)[−N P V(τ0)] + − 1{τ2≤min{τ0,T}}D(t, τ0)REC0[−N P V(τ0)] +] = 1{τ2≤min{τ0,T}}Eτ0[π(t, τ0) +D(t, τ0)π(τ0, T) + D(t, τ0)[−Eτ0[π(τ0, T)]] + − D(t, τ0)REC0[−Eτ0[π(τ0, T)]] +] = 1{τ2≤min{τ0,T}}[π(t, τ0) + D(t, τ0)Eτ0[π(τ0, T)] +D(t, τ0)[−Eτ0[π(τ0, T)]] + − D(t, τ0)REC0[−Eτ0[π(τ0, T)])] +]
Then using (−.)+ =−(.)−, with (.)−= inf(0, .) the ”negative−part− of”-operator, = 1{τ2≤min{τ0,T}}[π(t, τ0) + D(t, τ0)Eτ0[π(τ0, T)]−D(t, τ0)[Eτ0[π(τ0, T)]] − − REC0[−Eτ0[π(τ0, T)]] +] = 1{τ2≤min{τ0,T}}[π(t, τ0) + D(t, τ0)[NPV(τ0)]+−REC0[−N P V(τ0)]+]
Now conditioning on tand using Et[Eτ0[.]] =Et[.] as t < τ0 gives,
Et[1{τ2≤min{τ0,T}}π(t, τ0) +D(t, τ0)[NPV(τ0)]+−REC0[−N P V(τ0)]+] similarly we obtain, Et[1{τ0≤min{τ2,T}}[π(t, τ2)] +D(t, τ2)(REC2(NPV(τ2))+−(−NPV(τ2))+)] as required
Bibliography
[1] Algorithmics, 2009. Credit Value Adjustment and the chang-
ing environment for pricing and managing counterparty risk
http://www.algorithmics.com/EN/media/pdfs/Algo-WP1209- CVA- Survey.pdf
[2] D Brigo F Mercurio, 2006.Interest Rate Models - Theory and Practise
Springer Finance
[3] F Black J C Cox, 1976. Valuing corporate securities: Some effects of bond indenture provisions J. Finance. 31 (1976) 351-367.
[4] C Blanchet-Scalliet, 2008.Counterparty risk valuation for CDSpreprint [5] D Brigo A Capponi, 2009. Bilateral Counterparty risk valuation with stochastic dynamical models and application to Credit Default Swaps
www.damianobrigo.it
[6] D Brigo K Chourdakis, 2008. Counterparty Risk for Credit De-
fault Swaps: impact of spread volatility and default correlation
www.damianobrigo.it
[7] D Brigo M Morini M Tarenghi, 2009.Credit Calibration with Structural Models: The Lehman case and Equity Swaps under Counterparty Risk
www.damianobrigo.it
[8] D Brigo, A Pallavicini, V Papatheodorou, 2010.Bilateral counterparty risk valuation for interest rate products: impact of volatility and corre- lation www.damianobrigo.it
[9] D Brigo M Tarenghi, 2005.Credit Default Swap Calibration and Coun- terparty Risk Valuation with a Scenario based First Passage Model
www.damianobrigo.it
[10] D Brigo M Tarenghi,2004. Credit Default Swap Calibration and Eq- uity Swap Valuation under Counterparty Risk with a Tractable Struc-
tural Model Proceedings of the FEA Conference at MIT, Cambridge,
Massachussets, November 8-10. Available at ssrn.com, arXiv.org and defaultrisk.com.
[11] M Crouhy D Galai R Mark, 2000. A comparative analysis of current credit risk models Journal of Banking and Finance 24, 59-117
[12] J Gregory, 2009.Being Two-Faced Over Counterparty Credit Risk Risk [13] J Hull, A White, 2001. Valuing Credit Default Swaps II: Modelling
Default Correlations
[14] Merton, 1974.On the pricing of corporate debt: the risk structure of interest rates Journal of Finance, 29, 449-470
[15] M Morini, A Prampolini, 2010.Risky funding: a unified framework for counterparty and liquidity charges
[16] M Pykhtin S Zhu, 2007.A guide to modelling counterparty credit risk
Global Association of Risk Professionals 37 (July/August)
[17] P J Schonbucher, 2003. Credit Derivatives Pricing, Models and Imple-
mentation John Wiley and Sons
[18] H J Stein K P Lee, 2011.Counterparty Valuation Adjustments inCredit Risk Frontiers: subprime crisis, pricing and hedging, CVA, MBS, rat- ings and liquidity Bloomberg: Financial Series