Bilateral Credit Valuation Adjustment for Large Credit Derivatives Portfolios Agostino Capponi Department of Applied Mathematics & Statistics Johns Hopkins University Finance and Stochastic Seminars, Imperial College London, London, January 15, 2013 Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University Counterparty Risk Pricing Dublin 2013 1 / 43
43
Embed
Bilateral Credit Valuation Adjustment for Large Credit ... · Credit Swisse Bank of America Bank of America Fixed coupon Variable coupon Interest Rates Swap Fixed coupon Variable
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Bilateral Credit Valuation Adjustment for Large CreditDerivatives Portfolios
Agostino Capponi
Department of Applied Mathematics & StatisticsJohns Hopkins University
Finance and Stochastic Seminars,Imperial College London,London, January 15, 2013
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 1 / 43
Talk Outline
Talk Outline
1 Introduction
2 Framework for Bilateral Counterparty Risk
3 Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
4 Numerical Analysis
5 Conclusions
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 2 / 43
Introduction
Talk Outline
1 Introduction
2 Framework for Bilateral Counterparty Risk
3 Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
4 Numerical Analysis
5 Conclusions
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 3 / 43
Introduction
What is Counterparty Risk?
Risk taken by an investor entering into a financial transaction withone (or more) counterparties having a relevant default probabilityAn economic loss would occur to investor if the transaction has apositive economic value for him at the time of defaultPervades all derivative transactions in over-the-counter markets
Credit Swisse
Bank of America
Fixed coupon Variable coupon
Interest Rates Swap
Bank of America
Variable coupon Fixed coupon
Credit Swisse
No Default Default
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 4 / 43
Introduction
The Credit Crisis
The recent credit crisis has demonstrated that systemically importantentities can pose serious threats to the economyDefault cascades triggered by individual firms can propagate widerinto the financial system and be responsible for huge lossesBasel Committee on Banking supervision identifies counterparty riskas the main driver of systemic failures
“During the financial crisis roughly two-thirds of losses attributed tocredit risk were due to counterparty valuation losses and only aboutone-third were due to actual defaults” [The Basel Committees December 2009
Proposals]
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 5 / 43
Introduction
Bilateral Counterparty Risk I
Unlike firm’s exposure to credit risk through a loan, where theexposure to credit risk is unilateral, counterparty credit risk creates abilateral risk of lossIn any bilateral transaction both parties exchange cash flows, henceboth counterparty and investor can defaultIt is standard market practise to take credit quality of investorsignificantly higher than counterparty =⇒ unilateral riskCredit crisis has shown that it is no longer realistic to take creditquality of investor for granted
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 6 / 43
Introduction
Bilateral Counterparty Risk II
The risk neutral price of bilateral counterparty risk, called BilateralCredit Valuation Adjustment, becomes:
Price of risk free Portfolio − BCVA = Price of risky portfolio
BCVA = cost of potential loss︸ ︷︷ ︸CVA (credit)
− benefit from investor ′s default︸ ︷︷ ︸DVA (debit)
Price of risky portfolio can be larger or smaller than price of risk-freeportfolio depending on whether credit quality of investor is lower orhigher than the one of counterpartyBilateral counterparty risk introduces symmetry in valuation: price ofthe portfolio to the investor is the opposite of the price of the sameportfolio to the counterparty
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 7 / 43
Framework for Bilateral Counterparty Risk
Talk Outline
1 Introduction
2 Framework for Bilateral Counterparty Risk
3 Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
4 Numerical Analysis
5 Conclusions
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 8 / 43
Framework for Bilateral Counterparty Risk
Notations and Definitions I
I: investor computing the BCVAτI : investor’s time of defaultLI : loss rate for investor’s claims
C : counterparty trading with investorτC : counterparty’s time of defaultLC : loss rate for counterparty’s claims
D(t, u): discount factor from u to t, u > tT : portfolio time horizonx+ := max(x , 0), x_ := max(−x , 0)
ε(t,T ): market value of portfolio at t
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 9 / 43
Framework for Bilateral Counterparty Risk
Notations and Definitions II
(Ω,G,Q)
Gt := Ft ∨Ht , withFt : market information except default eventsHt = HI
t ∨HCt : right-continuous filtration generated by default events
τI , τC are G-stopping timesEt [·] := EQ [·|Gt ]
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 10 / 43
Framework for Bilateral Counterparty Risk
Pricing Bilateral Counterparty Risk
Proposition (Brigo and Capponi (2008))The general counterparty risk pricing formula is given by
BCVA(t,T ) = LCEt[
1t<τC≤min(τI ,T )D(t, τC )ε+(τC ,T )]
−LIEt[
1t<τI≤min(τC ,T )D(t, τI)ε−(τI ,T )]
Inclusion of counterparty risk in the valuation of an otherwisedefault-free derivative =⇒ credit derivativeBCVA adjustment amounts to valuating two options on the portfoliomarket value at the earliest of investor and counterparty default time
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 11 / 43
Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
Talk Outline
1 Introduction
2 Framework for Bilateral Counterparty Risk
3 Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
4 Numerical Analysis
5 Conclusions
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 12 / 43
Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
Credit Default Swaps (CDS)
AIG
(Protection Seller)
Toyota
(Protection Buyer)
Credit Event
Payment
Notional per
Spread Premium
Notional: US $10MM
Term: 5 Years
Credit Event: Lehman default
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 13 / 43
Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
Why focus on CDSs?
Claimed to be the main source of default contagion because theyconstitute the most efficient way of transferring credit riskHighly liquid instruments with large outstanding notional amounts:nearly 57 trillions during crisisCDS is a credit derivative, even before accounting for counterpartydefault riskCredit sensitive securities: default events lead to large jumps invaluation
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 14 / 43
Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
BCVA for Large CDS Portfolios
Capture realistic market environments where institutions trade largeportfolios of (credit) derivativesAnalyze impact of idiosyncratic and systematic components of jumprisk on counterparty risk valuationEstimate the law of large number behavior for the market value of theportfolio
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 15 / 43
Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
Overview of Next Steps
Define correlated default intensity processes for:I,C : constant elasticity of variance (CEV) processes with jumpsK reference entities in the CDS portfolio: constant elasticity ofvariance (CEV) processes with jumps
Introduce doubly stochastic models for the default timesDevelop an asymptotic formula for the portfolio market valueObtain an explicit expression for the BCVA of the portfolio
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 16 / 43
Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
Notation and Definitions
τ∗K = mink∈1,...,K τk ; τk : default time of kth name
G(j)t = F (j)t ∨H
(j)t , j ∈ 1, . . . ,K , I,C
H(j)t = 1τj≤t, H
(j)t = 1− H(j)
t
H(j)t = σ(H(j)
s ; s ≤ t)
Et [·] := EQ[·|∨
j∈1,...,K∪I,C G(j)t]
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 17 / 43
Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
Market Information
Continuous component generated by K + 2-dimensional Brownianmotion
(W (1), . . . ,W (K),W (I),W (C))
Jump component generated by K + 3 independent Poisson processes
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 32 / 43
Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
Explicit Expression for BCVA of Large Portfolio
Assume ρ = 0.5, i.e. square root intensity processes for I,C . Then H1 andH2 admit the closed form representation as solution of generalized Riccatiequation:
H1(tC − t, ξ(I)t , ξ(C)t ) =
[h1(tC − t) + hC (tC − t)ξ
(C)t]
× exp(h1(tC − t) + hI(tC − t)ξ
(I)t + hC (tC − t)ξ
(C)t)
H2(tI − t, ξ(I)t , ξ(C)t ) =
[w1(tI − t) + wI(tI − t)ξ
(I)t]
× exp(w1(tI − t) + wI(tI − t)ξ
(I)t + wC (tI − t)ξ
(C)t)
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 33 / 43
Numerical Analysis
Talk Outline
1 Introduction
2 Framework for Bilateral Counterparty Risk
3 Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
4 Numerical AnalysisQuality of Exposure ApproximationEconomic Analysis
5 Conclusions
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 34 / 43
Numerical Analysis Quality of Exposure Approximation
Simulation Setup I
Compare the law of large approximation for market value of the KCDS portfolio with Monte-Carlo estimate.Fix parameters as:
Jump parameters γI = γC = 1.5, γIC = 0, cI = cC = 0.3, γ = 1.2,I/C losses: LI = LC = 0.4.Contractual CDS parameters: S∗
z = 0.02, L∗z = 0.4.
K = 300.
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 35 / 43
Numerical Analysis Quality of Exposure Approximation
Simulation Setup II
Convergence rate given by
ξ(k)0 = x∗
(1 +
1k
)αk = α∗
(1 +
1k
)κk = κ∗
(1 +
1k
)σk = σ∗
(1 +
1k
)ck = c∗
(1 +
1k
)λk = λ∗
(1 +
1k
)Sk = S∗
(1 +
1k
)Lk = L∗
(1− 1
k
)
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 36 / 43
Numerical Analysis Quality of Exposure Approximation
Approximate vs Monte-Carlo Exposure without jumps
0 0.5 1 1.5 2 2.5 30
50
100
150
200
250
300
350
400
Time
Bas
is p
oint
s
Monte−CarloApproximation Formula
0 0.5 1 1.5 2 2.5 3−3000
−2500
−2000
−1500
−1000
−500
0
500
Time
Bas
is p
oint
s
Monte−CarloApproximation Formula
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 37 / 43
Numerical Analysis Quality of Exposure Approximation
Approximate vs Monte-Carlo Exposure with jumps
0 0.5 1 1.5 2 2.5 3−2500
−2000
−1500
−1000
−500
0
500
Time
Bas
is p
oint
s
Monte−CarloApproximation Formula
0 0.5 1 1.5 2 2.5 3−3500
−3000
−2500
−2000
−1500
−1000
−500
0
500
Time
Bas
is p
oint
s
Monte−CarloApproximation Formula
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 38 / 43
Numerical Analysis Economic Analysis
Impact of Portfolio Credit Risk Volatility
0 0.2 0.4 0.6 0.8 110
15
20
25
30
35
40
σ*
Bas
is p
oint
s
CVA vs σ*
λc = 0
λc = 0.1
λc = 0.2
λc = 0.4
Higher portfolio volatility =⇒ higher value of CVA embedded option=⇒ higher CVA adjustmentsHigher intensity of common jumps =⇒ reduced CVA adjustments dueto higher number of names in the portfolio defaulting before C
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 39 / 43
Numerical Analysis Economic Analysis
Impact of Systematic Jump Frequency
0 0.5 1 1.5 20
5
10
15
20
25
30
35
λc
Bas
is p
oint
s
CVA vs λc
σ* =0.01
σ*=0.1
σ*=0.5
σ*=0.7
0 0.5 1 1.5 20
10
20
30
40
50
60
70
λc
Bas
is p
oint
s
DVA vs λc
σ* =0.01
σ*=0.1
σ*=0.5
σ*=0.7
Low frequency of systematic jumps =⇒ small portfolio default risk=⇒ I measures positive on-default exposure to C =⇒ high CVA andlow DVAHigh frequency of systematic jumps =⇒ large portfolio default risk=⇒ C measures positive on-default exposure to I =⇒ high DVA andlow CVA
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 40 / 43
Conclusions
Talk Outline
1 Introduction
2 Framework for Bilateral Counterparty Risk
3 Bilateral Counterparty Risk of Large Credit Derivatives Portfolio
4 Numerical Analysis
5 Conclusions
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 41 / 43
Conclusions
Conclusions
Developed a closed form expression for bilateral CVA of a creditdefault swaps portfolio.Key insight: explicit characterization of portfolio exposure as theweak limit of measure-valued processesSuch processes are associated to survival indicators of referenceentities in underlying portfolio.Economic analysis suggests that counterparty risk is
increasing in credit risk volatility of underlying CDS portfoliohighly sensitive to systematic jump components of default intensity
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 42 / 43
Conclusions
Reference Papers
L. Bo and A. Capponi. Bilateral Credit Valuation Adjustment forLarge Credit Derivatives Portfolios. Finance and Stochastics, DOI10.1007/s00780-013-0217-4. Preprint available athttp://arxiv.org/abs/1305.5575.D. Brigo, A. Capponi, and A. Pallavicini. Arbitrage-free bilateralcounterparty risk valuation under collateralization and application tocredit default swaps. Mathematical Finance, Vol. 24, Issue 1, pp.125-146, 2014
Agostino Capponi ( Department of Applied Mathematics & Statistics Johns Hopkins University )Counterparty Risk Pricing Dublin 2013 43 / 43