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

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


Introduction

Talk Outline

1 Introduction




5 Conclusions


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


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]


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


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


Framework for Bilateral Counterparty Risk

Talk Outline

1 Introduction




5 Conclusions



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



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 ]



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


Bilateral Counterparty Risk of Large Credit Derivatives Portfolio

Talk Outline

1 Introduction




5 Conclusions



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



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



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



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



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]



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

(N(1), . . . , N(K), N(I), N(C), N(c))

with constant intensity λj , j ∈ 1, . . . ,K , I,C , cN(j), j ∈ 1, . . . ,K , I,C: idiosyncratic component of jump riskN(c): systematic component of jump risk



Default Intensity for Portfolio

Default intensity process ξ(k)t , k ∈ 1, 2, . . . ,K, is a mean-revertingCEV process with jumps

ξ(k)t = ξ

(k)0 +

∫ t

0(αk − κkξ

(k)s )︸︷︷︸

mean reversion

ds +

∫ t

0σk(ξ(k)s )ρ︸︷︷︸volatility risk

dW (k)s + ck

N(k)t∑

i=1Y (k)

i︸︷︷︸jump risk

ξ(k)0 > 0

N(k)t = N(k)

t + N(c)t

where (Y (k)1 , . . . ,Y (k)

i , . . . ) are i.i.d. positive random variables



Default Intensity for I and C

Default intensity process ξ(j)t , j ∈ I,C, is CIR process with jumps

ξ(j)t = ξ

(j)0 +

∫ t

0(αj − κjξ

(j)s )︸︷︷︸

mean reversion

ds +

∫ t

0σj(ξ

(j)s )ρ︸︷︷︸

volatility risk

dW (j)s + cj

N(j)t∑

i=1Y (j)

i︸︷︷︸jump risk

ξ(j)0 > 0

N(j)t = N(j)

t + N(c)t

where (Y (j)1 , . . . ,Y (j)

i , . . . ) are i.i.d. positive random variables.



Default Times

Defined using doubly stochastic modelsGiven independent unit mean exponential r.v Θj ,j ∈ 1, . . . ,K , I,C, the default time of j-th name is

τj = inft ≥ 0;

∫ t

0ξ(j)s ds ≥ Θj



The Market Value of K CDS Portfolio

Consider K credit default swaps, whereSk : CDS spread of k th CDS contractLk : Loss given default of k th CDS contract

zk ∈ 1,−1 denotes long/short position on k-th CDSOn τ∗K > t, the market value of K CDSs portfolio is

ε(K)(t,T )=

∫ T

tD(t, s) Et

[ K∑k=1

zkSkH(k)s

]︸︷︷︸

spread payments till default

ds+

∫ T

tD(t, s)

∂

∂s Et

[ K∑k=1

zkLkH(k)s

]︸︷︷︸losses paid at default



Law of Large Numbers Approximation (K−→∞)

(1) Develop weak convergence analysis and recover the weak limit of sums

K∑k=1

1K akH

(k)t

where ak is a sequence of real numbers(2) Introduce a limit default time τ∗X associated with limit intensity

process(3) Provide explicit formula for market value of the K CDS portfolio

ε(K)(t,T ) ≈ 1τ∗X>tε

(K ,∗)(t,T )

where ε(K ,∗) is the law of large number portfolio market value



Step 1: Weak Convergence Analysis I

Define type parameter set related to the K -intensity processes

pk =(αk , κk , σk , ck , λk

)pk ∈ Op := R5

+

Define sequence of empirical measures

qK =1K

K∑k=1

δpk ηK =1K

K∑k=1

δY (k)1

φK =1K

K∑k=1

δξ(k)0

Define sequence of measure-valued processes

ν(K)t =

1K

K∑k=1

δ(pk ,Y

(k)1 ,ξ

(k)t )

H(k)t

(pk ,Y (k)

1 , ξ(k)t)∈ O

Assume that limits exist

q = limK−→∞

qK∈ P(Op) η = limK−→∞

ηK∈ P(R+) φ = limK−→∞

φK∈ P(R+)



Step 1: Weak Convergence Analysis II

For any smooth function f ∈ C∞(O) and ν ∈ S, define

ν(f ) :=

∫Of (p, y , x)ν(dp × dy × dx)

It holds that

ν(K)t (f ) =

1K

K∑k=1

f (pk ,Y (k)1 , ξ

(k)t ) H(k)

t t ≥ 0

For each K ∈ N, define the M-dimensional stochastic process

ν(K)t (f ) :=

(ν(K)t (f1), . . . , ν

(K)t (fM)

)t ≥ 0



Step 1: Weak Convergence Analysis III

For given smooth functions ϕ ∈ C∞(RM) and ν ∈ S, considerconvergence of martingale problem for

Φ(ν) = ϕ (ν(f ))

where ν(f ) = (ν(f1), . . . , ν(fM)) ∈ RM .

Prove the relative compactness for ν(K)(f )K≥1 as stochasticprocesses with sample paths in DR([0,∞)).



Step 1: Weak Convergence Analysis IVTheorem (Limit Measure Theorem, Bo and Capponi(2013))

It holds that ν(K) ==⇒ ν as K−→∞, where the limit measure-valuedprocess ν = (νt ; t ≥ 0) is given by

νt(A×B×C)=

∫O1A×B(p, y)E

[exp(−∫ t

0Xs(p)ds

)1Xt(p)∈C

]q(dp)η(dy)φ(dx)

and the limit default intensity process X (p) = (Xt(p); t ≥ 0) satisfies:

Xt(p) = x +

∫ t

0(D(p) + α− κXs(p)) ds + σ

∫ t

0(Xs(p))ρ dWs

with p = (p, y , x) ∈ O, p = (α, κ, σ, c, λ) ∈ Op. The drift rate is given by

D(p) = y c (λ+ λc)



Step 2: The Limit Default time

Apply the limit measure theorem and get weak convergence for sumsof the form

1K

K∑k=1

akH(k)t ==⇒ a∗νt(O)

Introduce limit default time τ∗X associated to limit default intensityprocess

E

1τ∗X>t

∣∣∣∣ ∨k∈1,...,∞∪I,C

F (k)t

= F (t)

where

F (t) = E[∫OE[exp

(−∫ t

0Xs(p)ds

)]q(dp)η(dy)φ(dx)

]



Step 3: Explicit Formula for Portfolio Market Value I

Weak convergence implies convergence of expectations, hence onτ∗X > t,

Et

[1K

K∑k=1

H(k)s

]→ F (t, s),

with

F (t, s) := E[∫OE[exp

(−∫ s

tXu(p)du

)]q(dp)η(dy)φ0(dx)

].

On τ∗X > t, the default time of the limiting portfolio satisfies

Et[1τ∗

X>s]

= F (t, s), 0 ≤ t ≤ s



Step 3: Explicit Formula for Portfolio Market Value II

Approximate portfolio market value as

ε(K)(t,T ) ≈ 1τ∗X>tε

(K ,∗)(t,T )

where law of large number market value is

ε(K ,∗)(t,T )

K =−L∗z[1−D(t,T )F (t,T )

]+(S∗z +rL∗z)

∫ T

tD(t, s)F (t, s)ds



Semi-Explicit Expression for BCVA of Large Portfolio

Theorem (Law of Large Number BCVA)Define τ∗ = min(τ∗K , τI , τC ). On the event τ∗ > t, it holds that

BCVA(K ,∗)(t,T ) = CVA(K ,∗)(t,T )− DVA(K ,∗)(t,T )



Law of Large Numbers CVA and DVA

CVA(K ,∗)(t,T ) :=LC

∫ T

tD(t, tC )ε

(K ,∗)+ (tC ,T )F (t, tC )H1(t, tC , ξ(I)t , ξ

(C)t )dtC

DVA(K ,∗)(t,T ) :=LI

∫ T

tD(t, tI)ε(K ,∗)− (tI ,T )F (t, tI)H2(t, tI , ξ(I)t , ξ

(C)t )dtI

where H1 and H2 are given by

H1(t, tC , ξ(I)t , ξ(C)t ) :=E

exp(−∫ tC

t(ξ(I)s + ξ(C)

s )ds)ξ(C)tC

∣∣∣∣ ∨j∈I,C

F (j)t

H2(t, tI , ξ(I)t , ξ

(C)t ) :=E

exp(−∫ tI

t(ξ(I)s + ξ(C)

s )ds)ξ(I)tI

∣∣∣∣ ∨j∈I,C

F (j)t



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)


Numerical Analysis

Talk Outline

1 Introduction



4 Numerical AnalysisQuality of Exposure ApproximationEconomic Analysis

5 Conclusions


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.



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

)



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




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


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



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


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


Conclusions

Talk Outline

1 Introduction




5 Conclusions


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


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


http://arxiv.org/abs/1305.5575

Documents

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