A structural approach to pricing credit derivatives with … · 2015. 3. 2. · I the rm borrowed a...

CVA and DVA on a standard CDSModelling frameworkOne-dimensional caseTwo dimensional case

Three dimensional caseNumerical results

A structural approach to pricing credit derivatives

with counterparty adjustments

Alexander Lipton and Ioana SavescuBank of America Merrill Lynch

July 25, 2012

Alexander Lipton and Ioana Savescu Pricing CDSs with bilateral value adjustments 1 / 32



Outline

CVA and DVA on a standard CDS

Modelling framework

One-dimensional case

Two dimensional casePricing problem and domainGreen's function - eigenvalues expansion methodJoint survival probabilityApplication to CVA/DVA computation

Three dimensional casePricing problem and domainGreen's functionJoint survival probabilityApplication to the CVA/DVA computation

Numerical results

Conclusion




De�nition of a standard CDS and notations

De�nition of a CDS - the protection buyer (PB) agrees to pay a periodiccoupon c to a protection seller (PS) in exchange for a potential cash�owin the event of a default of the reference name (RN) of the swap beforethe maturity of the contract T

CLt = −E[∑

TicD (t,Ti )1{Ti≤τRN}∆T

∣∣∣Ft]DLt = E

[(1− RRN)D(t, τRN)1{t



De�nition of CVA/DVA

Counterparty credit risk: the risk of a party to a �nancial contractdefaulting prior to the contract's expiration and not ful�lling all of itsobligations.Credit Value Adjustment (CVA) � the additional cost associated withthe possibility of the counterparty's default

CVA = (1− RPS)E[1{τPS



Structural default framework - I

Original framework proposed by Merton:

I the �rm's value at is driven by a log-normal di�usion

I the �rm borrowed a zero-coupon bond with face value N andmaturity T

I defaults at time T if its value aT is less than the bond's face value N

Extensions:

I more complicated forms of debt

I the default event may be triggered continuously up to the debtmaturity

I making default barriers stochastic

I incorporating jumps into the �rm's value dynamics




Structural default framework - II

Firm's asset value dynamics:

dat = (%t − ζt − λtκ) atdt + σtatdWt +(e j − 1

)dNt

Default barrier � deterministic function of time: lt = l0Et , where

Et = exp(∫ t

0

(ru − ζu − λuκ− 12σ

2u

)du)and l0 = RL0

Simplifying assumption: vol constant in time We obtain:

at = l0Eteσxt

where the stochastic factor xt has the following dynamics:

dxt = dWt +j

σdNt , x0 =

1

σln

(a0

l0

)We consider the case without jumps.For the multi-dimensional case, the Brownian motions are correlated:d〈W it ,W

jt 〉 = ρijdt.




References

Structural default framework

I Single-name pricing: Merton [1974], Black and Cox [1976], Kimet al. [1993], Longsta� and Schwartz [1995], Leland and Toft [1996],Hyer et al. [1999], Avellaneda and Zhu [2001], Hull and White[2001], Zhou [2001b], Hilberink and Rogers [2002], Lipton [2002],Sepp [2004], Cariboni and Schoutens [2007]

I Extensions to two dimensions: He et al. [1998], Lipton [2001],Zhou [2001a], Patras [2006], Valuzis [2008]

I Applications to CVA/DVA computation: Lipton and Sepp[2009], Blanchet-Scaillet and Patras [2011]




One-dimensional case

Process yt associated with the RN of the CDSGreen's function:

G (τ, y0, y′) =

1√2πτ

(e−

(y′−y0)22τ − e−

(y′+y0)22τ

)Survival probability:

Q(t,T , y0) =

∫ ∞0

G (τ, y0, y′)dy ′ = 2N

(y0√T − t

)− 1

Price of a standard CDS:

V (t,T , y0) =CL(t,T , y0) + DL(t,T , y0)

=− (c + % (1− RRN))A(t,T , y0)

+ (1− RRN)[1− e−%(T−t)Q(t,T , y0)

].

where A(t,T , y0) denotes the annuity leg

A(t,T , y0) =∫ TtD(t, t ′)Q(t, t ′, y0)dt

′




Pricing problem and domainGreen's function - eigenvalues expansion methodJoint survival probabilityApplication to CVA/DVA computation

Pricing problem and domain

General pricing equation in the positive quadrant:

Vt +1

2Vxx +

1

2Vyy + ρxy Vxy − %V = 0

Changes of function/variables:

U(t,T , x , y) = e%(T−t)V (t,T , x , y)α(x , y) = x

β(x , y) =− 1ρ̄xy

(ρxyx − y)r =√α2 + β2

ϕ =$ + arctan

(−αβ

)

21,,tU

,0,tU

Figure: The new domain in which thePDE has to be solved in after thechange of variables.

Final form of the pricing equation: Ut +12

(Urr +

1rUr +

1r2Uϕϕ

)= 0





Green's function - eigenvalues expansion method

Green's function solves the forward equation:

Gτ −1

2

(Gr ′r ′ +

1

r ′Gr ′ +

1

r ′2Gϕ′ϕ′

)= 0

Initial condition:

G (0, r ′, ϕ′) =1

r0δ(r ′ − r0)δ(ϕ′ − ϕ0),

Boundary conditions:

G (τ, r ′, 0) = 0, G (τ, r ′, $) = 0, G (τ, 0, ϕ′) = 0, G (τ, r ′, ϕ′) −−−−→r ′→∞

0.

Solution obtained through the eigenvalues expansion method:

G (τ, r0, r′, ϕ0, ϕ

′) =2e−

r′2+r20

2τ

$τ

∞∑n=1

Iνn

(r ′r0

τ

)sin (νnϕ

′) sin (νnϕ0).

where νn =nπ$ .





32

10

-1-2

-301

23

45

60.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

beta alpha

Green's Function

Figure: Green's function (x0 = 0.1, y0 = 0.1, σx = 10%, σy = 10%,ρxy = 70%, T = 1 year).





Joint survival probability

Q(t,T , x , y) � the joint survival probability of issuers x and y to a �xedmaturity T

Qt +1

2Qxx +

1

2Qyy + ρxyQxy = 0,

with �nal condition Q(T ,T , x , y) = 1 and boundary conditionsQ(t,T , x , 0) = 0 and Q(t,T , 0, y) = 0

Solution:

Q(τ,r0, ϕ0)=∞∑k=0

4 sin (ν2k+1ϕ0)

(2k + 1)π

(r202τ

)ν2k+12

Γ(1+

ν2k+1

2

)Γ (1+ ν2k+1)

1F1

(ν2k+1

2 ,1+ ν2k+1,−r20

2τ

)

=2r0e

− r20

4τ

√2πτ

∞∑k=0

sin (ν2k+1ϕ0)

2k + 1

[I ν

2k+1−12

(r20

4τ

)+ I ν

2k+1+1

2

(r20

4τ

)]





Application to CVA computation

Pricing equation:

Vt +1

2Vxx +

1

2Vyy + ρxy Vxy − %V = 0,

with the �nal condition V (T ,T , x , y) = 0I If the RN of the CDS contract defaults �rst: since the PS has not

defaulted it will be able to honour the payment: V (t,T , x , 0) = 0.I If the PS defaults �rst: it will no longer be able to honour its

payments and hence the shortfall for the PB will be a fraction of theoutstanding PV of the CDS:

V (t,T , 0, y) = (1− RPS)VCDS (t,T , y)+ .I If the PS is risk free there is no shortfall: V (t,T ,∞, y) = 0.I If the CDS reference name is virtually risk-free we do not care what

happens to the PS: V (t,T , x ,∞) = 0.

V CVA(t,T, r0, ϕ0)=−1−RPS

2

T∫t

∞∫0

D(t, t ′)Gϕ(t′− t, r , $)V CDS

(t ′,T , ρxy r

)1rdrdt ′




Pricing problem and domainGreen's functionJoint survival probabilityApplication to the CVA/DVA computation

Pricing problem and domain

Pricing equation in the positive octant

Vt +1

2Vxx +

1

2Vyy +

1

2Vzz + ρxyVxy + ρxzVxz + ρyzVyz − %V = 0

Changes of variables:α(x , y , z) =x

β(x , y , z) = 1ρxy(−ρxyx + y)

γ(x , y , z) = 1ρxyχ[(ρxyρyz − ρxz) x + (ρxyρxz − ρyz) y + ρ2xyz

],

α =r sin θ sinϕ

β =r sin θ cosϕ

γ =r cos θ

←→

r =√α2 + β2 + γ2

θ = arccos(γr

)ϕ =arctan

(α

β

)Alexander Lipton and Ioana Savescu Pricing CDSs with bilateral value adjustments 14 / 32




Final form of the pricing equation

Ut +1

2

[1

r

∂2

∂r2(rU) +

1

r2

(1

sin2 θUϕϕ +

1

sin θ

∂

∂θ(sin θUθ)

)]= 0

I r > 0

I 0 ≤ ϕ ≤ $, where$ = arccos (−ρxy )

I the possible range of values for θdepends on ϕ

ϕ (ω) =arccos(

1−ρxyω√1−2ρxyω+ω2

)

Θ (ω) =arccos

− ρyz−ρxzρxy+ω(ρxz−ρyzρxy )√ρxy(ρ2xz−2ω(ρxy−ρxzρyz )+ω2ρ2yz)

Figure: Domain after thechange in coordinates forρxy = 20%, ρxz = 0%,ρyz = 30%

αβ

γ

φ

θ





Green's function

Green's function satis�es the forward equation:

Gτ −1

2

[1

r ′∂2

∂r ′2(r ′G ) +

1

r ′2

(1

sin2 θ′Gϕϕ′ +

1

sin θ′∂

∂θ′(sin θ′Gθ′)

)]= 0,

with inital condition:

G (0, r ′, ϕ′, θ′) =1

r20 sin θ0δ (r ′ − r0) δ (ϕ′ − ϕ0) δ (θ′ − θ0) ,

and boundary conditions:

G (τ, r ′, 0, θ′) = G (τ, r ′, $, θ′) = G (τ, r ′, ϕ′, 0) = 0,

G (τ, r ′, ϕ′,Θ(ϕ′)) = G (τ, 0, ϕ′, θ′) = 0, G (τ, r ′, ϕ′, θ′) −−−−→r ′→∞

0.





Solving the forward equation

I solve using eigenvalues expansion method

I separation of variables: G (τ, r ′, ϕ′, θ′) = g(τ, r ′)Ψ(ϕ′, θ′).

I Radial part (similar to the two dimensional case):

g(τ, r ′) =e−

r′2+r20

2τ

τ√r ′r0

I√Λ2+1/4

(r ′r0

τ

).

I For the angular part

1

sin2 θ′Ψϕ′ϕ′ +

1

sin θ′∂

∂θ′(sin θ′Ψθ′) = −Λ2Ψ,

Ψ(0, θ′) = 0, Ψ($, θ′) = 0, Ψ(ϕ′, 0) = 0, Ψ(ϕ′,Θ(ϕ′)) = 0.





Solving the angular part of the forward equation

I we solve this using the �nite element method (FEM)I we map the domain onto the (ϕ, θ) plane

ϕ

θ

$C1

C2C4

C3

Ω

Figure:ρxy = 0.8, ρxz = 0.5, ρyz = 0.3

ϕ

θ

$C1

C2

C4

C3

Ω

Figure:ρxy = 0.8, ρxz = 0.05, ρyz = 0.6

I Variational (weak) formulation:∫Ω

1

sin θ′Ψϕ′Ψ

′ϕ′dΩ +

∫Ω

sin θ′Ψθ′Ψ′θ′dΩ = Λ

2

∫Ω

ΨΨ′ sin θ′dΩ





Construction of the grid

I triangular mesh

I can build both uniform and adaptive meshes

I an iterative algorithm is used

I after each iteration, the Delaunay triangulation method is used

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

(a) First iteration mesh

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

(b) Mesh after 100 iterations

Figure: ρxy = 80%, ρxz = 20%, ρyz = 50% (mesh uses 1800 points).





0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5


0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5

(b) Mesh after 100 iterationsFigure: ρxy = 80%, ρxz = 50%, ρyz = 50% (mesh uses 1500 points).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(b) Mesh after 100 iterationsFigure: ρxy = 20%, ρxz = −10%, ρyz = −60% (mesh uses 1600 points).





Applying FEM

I dimension of the space in which we search for the solution =number of free points in the mesh (n); (Φi )1≤i≤n basis for this space

I linear basis functions on each triangle

I the variational formulation is approximated by a linear system:

KΨ = Λ2MΨ

where we denote by K = (Kij)1≤i,j≤n the sti�ness matrix, and by

M = (Mij)1≤i,j≤n the mass matrix:

Kij =

∫Ω

(A∇Φj) · ∇ΦidΩ,

Mij =

∫Ω

sin θ′ ΦiΦjdΩ,

with the matrix A given by A =

(1

sin θ′ 00 sin θ′

).





Eigenvalues and eigenvectors (ρxy = 0%, ρxz = 0%, ρyz = 0% � 1500 point mesh)

(a) Eigenvector 1: Λ21

= 12.0 (b) Eigenvector 2: Λ22

= 30.2 (c) Eigenvector 3: Λ23

= 30.2

(d) Eigenvector 6: Λ26

= 56.8 (e) Eigenvector 10: Λ210

= 92.4 (f) Eigenvector 20: Λ220

= 189.2

Figure





Eigenvalues and eigenvectors (ρxy = 20%, ρxz =−10%, ρyz =−60% � 1600 pts mesh)

(a) Eigenvector 1: Λ21

= 21.5 (b) Eigenvector 2: Λ22

= 42.2 (c) Eigenvector 3: Λ23

= 63.8

(d) Eigenvector 5: Λ25

= 96 (e) Eigenvector 7: Λ27

= 129.5 (f) Eigenvector 12: Λ212

= 200.3

FigureAlexander Lipton and Ioana Savescu Pricing CDSs with bilateral value adjustments 23 / 32




Green's function

Eigenfunction expansion for Green's function:

G (τ, r ′, ϕ′, θ′) =∞∑n=1

Cngn(τ, r′)Ψn(ϕ

′, θ′)

Coe�cients Cn - can be computed by imposing the initial condition:

G (0, r ′, ϕ′, θ′) =1

r20 sin θ0δ(r ′ − r0)δ(ϕ′ − ϕ0)δ(θ′ − θ0),

and we obtain Cn = Ψn(ϕ0, θ0).

Final formula for Green's function:

G (τ, r0, r′, ϕ0, ϕ

′, θ0, θ′) =

e−r′2+r2

0

2τ

τ√r ′r0

∞∑n=1

I√Λ2n+

1

4

(r ′r0

τ

)Ψn(ϕ0, θ0)Ψn(ϕ

′, θ′)





Joint survival probability

Q(t,T , x , y , z): the joint survival probability of issuers x , y and z to a�xed maturity T Satis�es the backward pricing equation:

Qt +1

2Qxx +

1

2Qyy +

1

2Qzz + ρxyQxy + ρxzQxz + ρyzQyz = 0,

with �nal condition Q(T ,T , x , y , z) = 1 and zero boundary conditions.Solution using Green's function:

Q(τ, r0, ϕ0, θ0) =

∫ ∞0

∫ $0

∫ Θ(ϕ)0

G (τ, r0, r′, ϕ0, ϕ

′, θ0, θ′) r ′2 sin θ′ dθ′dϕ′dr ′

=∞∑n=1

(r20

2τ

) νn2− 1

4 Γ( νn2 +5

4 )Γ(νn+1) 1

F1

(2νn−1

4 , νn + 1,−r20

2τ

)×Ψn(ϕ0, θ0)

∫∫Ω

Ψn(ϕ′, θ′) sin θ′dϕ′dθ′





Pricing problem for CVA/DVA

Pricing equation:

Vt +1

2Vxx +

1

2Vyy +

1

2Vzz + ρxyVxy + ρxzVxz + ρyzVyz − %V = 0

Final condition: V (T ,T , x , y , z) = 0

Boundary conditions:

I If the PS defaults:

VCVA(t,T , 0, y , z) = (1− RPS)V (t,T , y)+

I If the PB defaults:

VDVA(t,T , x , y , 0) = (1− RPB)V (t,T , y)−





Pricing formulas for CVA/DVA

Pricing formula for CVA:

UCVA(t,T , r0, ϕ0, θ0)=1

2

T∫t

∞∫0

Θ(0)∫0

UCVA(t ′,T , r , 0, θ)Gϕ (t′ − t, r , 0, θ)

sin θdθdrdt ′,

Pricing formula for DVA:

UDVA(t,T , r0, ϕ0, θ0)=

− 12

T∫t

∞∫0

$∫0

sinΘ(ϕ)UDVA(t ′,T , r , ϕ,Θ (ϕ))Gθ(t′− t, r , ϕ,Θ (ϕ))dϕdrdt ′

+1

2

T∫t

∞∫0

∞∫0

UDVA(t ′,T , r , ϕ(ω),Θ(ω))Gϕ(t′− t, r , ϕ(ω),Θ(ω))

sinΘ (ω)Θω(ω)dωdrdt

′.




Numerical results for a CDS

I AIG: protection seller

I GE: reference name of the CDS

I UNICREDIT: protection buyer

Calibrated inputs (15th of December 2011):

Inputs AIG GE UNICREDIT

Initial value 0.0359 0.3035 0.1199σ 2.44% 10.45% 6.3%

Recovery 50% 40% 40%




PS, PB and RN are uncorrelated

BEC

0

50

100

150

200

250

300

- 1.0 2.0 3.0 4.0 5.0Time to maturity (years)

BE

C (b

ps)

Standard CDSPS riskyPB riskyBoth risky

(a) Break-even coupon

Change in BEC (bps)

-30

-25

-20

-15

-10

-5

0

5

10

15

20

- 1.0 2.0 3.0 4.0 5.0

Time to maturity (years)

BE

C (b

ps)

PS riskyPB riskyBoth risky

(b) Change in BEC (compared to thestandard CDS with non-risky counter-parts)

Figure: Impact of counterparty adjustments on the break-even coupon of aCDS: ρxy = 0%, ρxz = 0%, ρyz = 0%.




PS is highly correlated to the RN

BEC

0

50

100

150

200

250

300


BE

C (b

ps)



Change in BEC (bps)

-120

-100

-80

-60

-40

-20

0

20

- 1.0 2.0 3.0 4.0 5.0


BE

C (b

ps)



Figure: Impact of counterparty adjustments on the break-even coupon of aCDS: ρxy = 80%, ρxz = 50%, ρyz = 30%.




PB is highly anti-correlated to the RN

BEC

0

50

100

150

200

250

300

350


BE

C (b

ps)



Change in BEC (bps)

-50

-40

-30

-20

-10

0

10

20

30

40

- 1.0 2.0 3.0 4.0 5.0


BE

C (b

ps)



Figure: Impact of counterparty adjustments on the break-even coupon of aCDS: ρxy = 20%, ρxz = −10%, ρyz = −60%.




Conclusion

I bilateral CVA/DVA for a standard single name default swap

I 3D extension of the structural default framework

I semi-analytical expression for Green's function

I application to computing joint survival probabilities and CVA/DVAfor a CDS




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CVA and DVA on a standard CDSModelling frameworkOne-dimensional caseTwo dimensional casePricing problem and domainGreen's function - eigenvalues expansion methodJoint survival probabilityApplication to CVA/DVA computation

Three dimensional casePricing problem and domainGreen's functionJoint survival probabilityApplication to the CVA/DVA computation

Numerical results

A structural approach to pricing credit derivatives with … · 2015. 3. 2. · I the rm borrowed a...

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