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XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Arbitrage-Free Pricing of XVA
Agostino CapponiColumbia University
joint work with Maxim Bichuch (WPI) and Stephan Sturm (WPI)
IAQF/Thalesians Seminar Series
New York, September 21, 2015
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The LIBOR-OIS Spread
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The LIBOR-OIS Spread
Consequences
Widening of spreads is due to counterparty credit riskLIBOR cannot be considered a risk-free rate any longerOne cannot assume the existence of a universal risk-freerate r
Rates at which derivatives traders borrow and lendunsecured cash differHow to price and hedge derivatives in presence of fundingspread and counterparty risk?
2013: Many banks (Barclays, JPM, BoA,...) introduceXVA desks
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The LIBOR-OIS Spread
Consequences
Widening of spreads is due to counterparty credit riskLIBOR cannot be considered a risk-free rate any longerOne cannot assume the existence of a universal risk-freerate r
Rates at which derivatives traders borrow and lendunsecured cash differHow to price and hedge derivatives in presence of fundingspread and counterparty risk?
2013: Many banks (Barclays, JPM, BoA,...) introduceXVA desks
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The LIBOR-OIS Spread
Consequences
Widening of spreads is due to counterparty credit riskLIBOR cannot be considered a risk-free rate any longerOne cannot assume the existence of a universal risk-freerate r
Rates at which derivatives traders borrow and lendunsecured cash differHow to price and hedge derivatives in presence of fundingspread and counterparty risk?
2013: Many banks (Barclays, JPM, BoA,...) introduceXVA desks
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Literature
Practitioner literature: Piterbarg (2010, 2012), Burgard &Kjaer (2010, 2011), Mercurio (2013)
(Corporate) Finance literature: Hull & White (2012, 2013)
Financial Mathematics literature: Bielecki & Rutkowski(2013), Brigo (2014), Crepey (2011, 2013), Crepey,Bielecki and Brigo (2014)
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Main Contributions
Develop a framework to characterize the total valuationadjustment (XVA) of a European style claim on a stock inpresence of
counterparty credit riskfunding spread
Derive a nonlinear backward stochastic differentialequation (BSDE) associated with the replicating portfoliosof long and short positions in the claim.
Develop an explicit representation of XVA in case ofsymmetric rates, but in presence of counterparty risk
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Main Contributions
Develop a framework to characterize the total valuationadjustment (XVA) of a European style claim on a stock inpresence of
counterparty credit riskfunding spread
Derive a nonlinear backward stochastic differentialequation (BSDE) associated with the replicating portfoliosof long and short positions in the claim.
Develop an explicit representation of XVA in case ofsymmetric rates, but in presence of counterparty risk
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Main Contributions
Develop a framework to characterize the total valuationadjustment (XVA) of a European style claim on a stock inpresence of
counterparty credit riskfunding spread
Derive a nonlinear backward stochastic differentialequation (BSDE) associated with the replicating portfoliosof long and short positions in the claim.
Develop an explicit representation of XVA in case ofsymmetric rates, but in presence of counterparty risk
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Market Model
The market model (I)
Treasury desk: borrowing and lending at rates r�f , r�f ,respectively
Stock (St): used to the hedge market risk of transaction.Trading happens through repo market at rates r�r , r�r(Duffie (1996))
Risky bonds (P It , PC
t ): underwritten byinvestor/counterparty and used to hedge default risk.Trading does not happen in the repo market
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Stock Short-Selling
TraderTreasury Desk
(1)
(6)
Stock Market
(5) (4)
Repo Market
(2)
(3)
r�r
Figure: Security driven repo activity: Solid lines arepurchases/sales, dashed lines borrowing/lending, dotted lines interestdue; blue lines are cash, red lines are stock.
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Stock Purchasing
TraderTreasury Desk
(1)
(6)
Stock Market
(2) (3)
Repo Market
(4)
(5)
r�r
Figure: Cash driven repo activity: Solid lines are purchases/sales,dashed lines borrowing/lending, dotted lines interest due; blue linesare cash, red lines are stock.
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Market Model
The market model (II)
We consider the dynamics
dSt � µSt dt � σSt dWt
dP It � µIP
It dt � P I
t� d1ltτI¤tu
� pµI � hI qPIt dt � P I
t� d$It
dPCt � µCP
Ct dt � PC
t� d1ltτC¤tu
� pµC � hC qPCt dt � PC
t� d$Ct
for independent default times τI , τC with constant defaultintensities hI , hC and martingales $I , $C
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Market Model
The market model (III)
Can we guarantee that there are no arbitrage opportunitiesin the market?
As we only model from the point of the trader, we canonly conclude this from her perspective. . .
Proposition
No-arbitrage conditions:Necessary: r�r ¤ r�f , r�f ¤ r�f , r�f µI , r�f µC .Sufficient: Necessary plus r�r ¤ r�f ¤ r�r
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Market Model
The market model (III)
Can we guarantee that there are no arbitrage opportunitiesin the market?
As we only model from the point of the trader, we canonly conclude this from her perspective. . .
Proposition
No-arbitrage conditions:Necessary: r�r ¤ r�f , r�f ¤ r�f , r�f µI , r�f µC .Sufficient: Necessary plus r�r ¤ r�f ¤ r�r
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Market Model
The market model (III)
Can we guarantee that there are no arbitrage opportunitiesin the market?
As we only model from the point of the trader, we canonly conclude this from her perspective. . .
Proposition
No-arbitrage conditions:Necessary: r�r ¤ r�f , r�f ¤ r�f , r�f µI , r�f µC .Sufficient: Necessary plus r�r ¤ r�f ¤ r�r
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Market Model
The market model (III)
Can we guarantee that there are no arbitrage opportunitiesin the market?
As we only model from the point of the trader, we canonly conclude this from her perspective. . .
Proposition
No-arbitrage conditions:Necessary: r�r ¤ r�f , r�f ¤ r�f , r�f µI , r�f µC .Sufficient: Necessary plus r�r ¤ r�f ¤ r�r
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Market Model
The market model (III)
Can we guarantee that there are no arbitrage opportunitiesin the market?
As we only model from the point of the trader, we canonly conclude this from her perspective. . .
Proposition
No-arbitrage conditions:Necessary: r�r ¤ r�f , r�f ¤ r�f , r�f µI , r�f µC .Sufficient: Necessary plus r�r ¤ r�f ¤ r�r
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Market Model
The market model (III)
Can we guarantee that there are no arbitrage opportunitiesin the market?
As we only model from the point of the trader, we canonly conclude this from her perspective. . .
Proposition
No-arbitrage conditions:Necessary: r�r ¤ r�f , r�f ¤ r�f , r�f µI , r�f µC .Sufficient: Necessary plus r�r ¤ r�f ¤ r�r
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Collateralization
Collateral is used to secure the derivatives deal
Collateral is provided in form of cash (80%)
Collateral can be reinvested (rehypothecated) (96%)
The collateral provider receives interests at rate r�c . Thecollateral taker pays interests at rate r�c .
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Collateralization
Collateral is used to secure the derivatives deal
Collateral is provided in form of cash (80%)
Collateral can be reinvested (rehypothecated) (96%)
The collateral provider receives interests at rate r�c . Thecollateral taker pays interests at rate r�c .
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Collateralization
Collateral is used to secure the derivatives deal
Collateral is provided in form of cash (80%)
Collateral can be reinvested (rehypothecated) (96%)
The collateral provider receives interests at rate r�c . Thecollateral taker pays interests at rate r�c .
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Collateralization
Collateral is used to secure the derivatives deal
Collateral is provided in form of cash (80%)
Collateral can be reinvested (rehypothecated) (96%)
The collateral provider receives interests at rate r�c . Thecollateral taker pays interests at rate r�c .
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Collateralization
Collateral is used to secure the derivatives deal
Collateral is provided in form of cash (80%)
Collateral can be reinvested (rehypothecated) (96%)
The collateral provider receives interests at rate r�c . Thecollateral taker pays interests at rate r�c .
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Market Model
TraderTreasury Desk
r�f
r�f
Cash
Stock &Repo Market
Stockr�r r�r
Bond MarketBonds P I , PC
Counterparty
Collateral
r�c r�c
Figure: Solid lines are purchases/sales, dashed linesborrowing/lending, dotted lines interest due; blue lines are cash, redlines stock purchases for cash and black lines bond purchases for cash.
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Closeout Payments and Valuation
The closeout value of the claim is decided by a valuationagent (either party or third party) in accordance withmarket practices (ISDA)
The valuation agent determines collateral requirementsand closeout value by calculating the Black-Scholes priceof the transaction
Such a valuation is associated with a publicly knowninterest rate rD
We can then introduce a valuation measure Q underwhich rD-discounted prices are Q martingales.
The XVA will be computed under Q
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Closeout Payments and Valuation
The closeout value of the claim is decided by a valuationagent (either party or third party) in accordance withmarket practices (ISDA)
The valuation agent determines collateral requirementsand closeout value by calculating the Black-Scholes priceof the transaction
Such a valuation is associated with a publicly knowninterest rate rD
We can then introduce a valuation measure Q underwhich rD-discounted prices are Q martingales.
The XVA will be computed under Q
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Closeout Payments and Valuation
The closeout value of the claim is decided by a valuationagent (either party or third party) in accordance withmarket practices (ISDA)
The valuation agent determines collateral requirementsand closeout value by calculating the Black-Scholes priceof the transaction
Such a valuation is associated with a publicly knowninterest rate rD
We can then introduce a valuation measure Q underwhich rD-discounted prices are Q martingales.
The XVA will be computed under Q
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Closeout Payments and Valuation
The closeout value of the claim is decided by a valuationagent (either party or third party) in accordance withmarket practices (ISDA)
The valuation agent determines collateral requirementsand closeout value by calculating the Black-Scholes priceof the transaction
Such a valuation is associated with a publicly knowninterest rate rD
We can then introduce a valuation measure Q underwhich rD-discounted prices are Q martingales.
The XVA will be computed under Q
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Closeout Payments and Valuation
The closeout value of the claim is decided by a valuationagent (either party or third party) in accordance withmarket practices (ISDA)
The valuation agent determines collateral requirementsand closeout value by calculating the Black-Scholes priceof the transaction
Such a valuation is associated with a publicly knowninterest rate rD
We can then introduce a valuation measure Q underwhich rD-discounted prices are Q martingales.
The XVA will be computed under Q
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Collateral and Close-Out Valuation
Collateral is a percentage α of the price of the contract
Ct � α1ltτI^τC¡tuEQ�e�rDpT�tqΦpST q
���Ft
�
:� α1ltτI^τC¡tuV pt,Stq
Set τ � τI ^ τC ^ T . The close-out payment is
θτ pV q � θτ pC , V q
:� V pτ,Sτ q � 1ltτC τI uLCY� � 1ltτI τC uLIY
�,
where Y :� Vτ � Cτ is the residual value of the claim atdefault
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Collateral and Close-Out Valuation
Collateral is a percentage α of the price of the contract
Ct � α1ltτI^τC¡tuEQ�e�rDpT�tqΦpST q
���Ft
�
:� α1ltτI^τC¡tuV pt,Stq
Set τ � τI ^ τC ^ T . The close-out payment is
θτ pV q � θτ pC , V q
:� V pτ,Sτ q � 1ltτC τI uLCY� � 1ltτI τC uLIY
�,
where Y :� Vτ � Cτ is the residual value of the claim atdefault
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Wealth Process
It is useful to distinguish between legal and actual wealthprocess
Legal wealth
Vt � ξtSt � ξItPIt � ξCt P
Ct � ψrf
t Brft � ψtB
rrt � Ct ,
Actual wealth
V Ct � ξtSt � ξItP
It � ξCt P
Ct � ψrf
t Brft � ψtB
rrt � Vt � Ct ,
(with B rft funding account B rr
t sec lending account and ξt ,ξIt , ξCt , ψrf
t , ψt number of shares holding)
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Wealth Dynamics
The dynamics of the wealth is given by
dVt ��r�f�ξft B
rft
��� r�f
�ξft B
rft
��� prD � r�r q
�ξtSt
��� prD � r�r q
�ξtSt
��� rDξ
ItP
It � rDξ
Ct P
Ct
dt
� r�c�ψct B
rct
��dt � r�c
�ψct B
rct
��dt
� p� � � qloomoonmartingales
with B rft funding account, B rc
t collateral account, ξt , andψt number of shares in the securities and various accounts
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Arbitrage Pricing
Definition
A price P P R, of a derivative security with terminal payoffξ P σpSt ; t ¤ T q is called hedger’s arbitrage-free, if for all γ P Rbuying γ securities for the price γP and hedging in the marketwith an admissible strategy does not create hedger’s arbitrage.
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Replicating Wealth
V�t pγq: wealth process when replicating the claim
γΦpST q, γ ¡ 0. This means hedging the position afterselling γ securities with terminal payoff ΦpST q.��V�
t pγq�: wealth process when replicating the claim
�γΦpST q, γ ¡ 0. This means hedging the position afterbuying γ securities with terminal payoff ΦpST q.
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
BSDE formulation
Set
f ��t, v , z , z I , zC ; V
�� �
�r�f�v � z I � zC � αVt
��� r�f
�v � z I � zC � αVt
��� prD � r�r q
1
σz� � prD � r�r q
1
σz�
� rDzI � rDz
C
� r�c αVt � pr�c � r�c q�αVt
��
f ��t, v , z , z I , zC ; V
�� �f �
�t,�v ,�z ,�z I ,�zC ;�Vt
�
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
BSDE formulation
The BSDEs
$'&'%
�dV�t pγq � f �
�t,V�
t ,Z�t ,Z
I ,�t ,ZC ,�
t ; V�dt
� Z�t dWQ
t � Z I ,�t d$I ,Q
t � ZC ,�t d$C ,Q
t
V�τ pγq � γ
�θτ pV q1ltτ Tu � ΦpST q1ltτ�Tu
$'&'%
�dV�t pγq � f �
�t,V�
t ,Z�t ,Z
I ,�t ,ZC ,�
t ; V�dt
� Z�t dWQ
t � Z I ,�t d$I ,Q
t � ZC ,�t d$C ,Q
t
V�τ pγq � γ
�θτ pV q1ltτ Tu � ΦpST q1ltτ�Tu
describe the wealth dynamics for buying/selling γ options
Existence and uniqueness of solution can be guaranteed
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
BSDE Relations
The two BSDEs are intrinsically related:
pV�t ,Z
�t ,Z
I ,�t ,ZC ,�
t q
is a solution to the data
�f �, θτ pV q,ΦpST q
�
iffp�V�
t ,�Z�t ,�Z I ,�
t ,�ZC ,�t q
is a solution to the data
�f �, θτ p�V q,�ΦpST q
�
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
No arbitrage
Theorem
Let Φ be a function of polynomial growth. Assume that IfV�0 ¤ V�
0 , then all prices in the closed intervalrπinf � V�
0 ,V�0 � πsups are free of hedger’s arbitrage.
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
XVA
We define the total value adjustment XVAt as
Definition
The seller’s XVA is given as
XVAsellt � V�
t � V pt,Stq
and the buyer’s XVA as
XVAbuyt � V�
t � V pt,Stq.
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Extended Piterbarg Model
Extension of Piterbarg’s model
Allow for default of investor and counterpartyDefault risk is hedged by risky bondsMaintain Piterbarg’s assumption of symmetric rates:rf � r�f � r�f , rr � r�r � r�r , rc � r�c � r�cBSDE becomes linear and XVAsell
t � XVAbuyt
Note: If rf � rr � rc � rD we have no funding costs andrecover the classical CVA/DVA settingIn particular
XVAt � DVAt � CVAt
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Extended Piterbarg Model
Extension of Piterbarg’s model
Allow for default of investor and counterpartyDefault risk is hedged by risky bondsMaintain Piterbarg’s assumption of symmetric rates:rf � r�f � r�f , rr � r�r � r�r , rc � r�c � r�cBSDE becomes linear and XVAsell
t � XVAbuyt
Note: If rf � rr � rc � rD we have no funding costs andrecover the classical CVA/DVA settingIn particular
XVAt � DVAt � CVAt
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Extended Piterbarg Model
Decomposition allows for the nice interpretation of theXVA in terms of four separate contributing terms:
Default and collateralization free price under fundingconstraintsFunding-adjusted payout after default of the traderFunding-adjusted payout after counterparty’s defaultFunding costs of the collateralization procedure
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Extended Piterbarg Model
Price decomposition
erf tVt1ltτ¥tu � EQ��B rfT
��1ΦpST qΓ
Tt 1ltτ�Tu
���Gt
�
� EQ��B rfτI
��1lI V pτI ,SτI qΓ
τIt 1ltt τI τC^T ;V pτI ,SτI q¥0u
��B rfτI
��1V pτI ,SτI qΓ
τIt 1ltt τI τC^T ;V pτI ,SτI q 0u
���Gt
�
� EQ��B rfτC
��1lC�V pτC , SτC qΓ
τCt 1ltt τC τI^T ;V pτC ,SτC q 0u
��B rfτC
��1V pτC , SτC qΓ
τCt 1ltt τC τI^T ;V pτC ,SτC q¥0u
���Gt
�
� EQ�α�rf � rc
� » τt^τ
�B rfs
��1V ps, SsqΓ
st ds
���Gt
�.
with lI � 1 � p1 � αqLC and lc � 1 � p1 � αqLC .
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Extended Piterbarg Model
Explicit computation possible in terms of compound optionwith random maturity. In the case of a long position
XVAt
�
�B
rft
BrDt
�B
rDT
BrfT
eprf �rD�hQIqpT�tqeprf �rD�hQ
CqpT�tq
��1� p1� αqLI
�eprD�rf qt
�1�
rD � rf
hQI
hQI
λ� hQI
�� hQC
λ� hQC � hQI
�epλ�hQ
C�hQ
IqpT�tq � 1
�� 1� e�hQ
CpT�tqepλ�hQ
IqpT�tq
� eprD�rf qt�1�
rD � rf
hQC
hQC
λ� hQC
�� hQI
λ� hQC � hQI
�epλ�hQ
C�hQ
IqpT�tq � 1
�� 1� e�hQ
IpT�tqepλ�hQ
CqpT�tq
� αrf � rc
hQC � hQI � λ
�BrDt
Brft
�B
rDT
BrfT
eprf �rD�hQCqpT�tqeprf �rD�hQ
IqpT�tq
� 1
�V pt, Stq
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Extended Piterbarg Model
Direct computation leads to
XVAt � pA� 1qV pt,Stq,
where A � Vt
Vtis explicit
Hedging strategies are explicit and given by
ξt � A� VSpt, Stq,
ξit �A� V pt,Stq � θi pV pt,Stqq
P it
, i P tI ,Cu.
and
θC pvq :� v � LC pp1 � αqvq�,
θI pvq :� v � LI pp1 � αqvq�.
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Extended Piterbarg Model
0.08 0.1 0.12 0.14 0.16 0.18 0.20
10
20
30
40
50
60
70
80
90
rf
Pric
e C
ompo
nent
s (%
)
Pure fundingTrader defaultCounterparty defaultCollateralization
0.08 0.1 0.12 0.14 0.16 0.18 0.20
5
10
15
20
25
30
35
40
45
rf
Pric
e C
ompo
nent
s (%
)
Pure fundingTrader defaultCounterparty defaultCollateralization
Figure: Left graph: hQI � 0.15, hQC � 0.2. Right graph: hQI � 0.5,
hQC � 0.5.
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
XVA with Differential Rates
What if borrowing and lending rates differ?: r�f � r�f ,r�r � r�r , r�c � r�c
BSDE becomes nonlinear: V�t � V�
t . We have ano-arbitrage interval for prices
But, we can show that the semilinear PDE v correspondingto the BSDE V admits a unique classical solution
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Band and funding spreads
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 116
18
20
22
24
26
28
30
32
α
Rel
ativ
e X
VA
(%
)
rf− = 0.08
rf− = 0.1
rf− = 0.15
rf− = 0.2
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Conclusion
Developed an arbitrage-free valuation framework for XVAof an European style claim
Seller’s and buyer’s XVA characterized as the solution of anonlinear BSDEs with random terminal condition
Funding component of XVA is predominant, withCVA/DVA terms becoming relevant only if trader andcounterparty are very risky
The no-arbitrage band widens as funding spreads andcollateral levels increase
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
References
M. Bichuch, A. Capponi, and S. Sturm. Arbitrage-free pricing of XVA –Part I: Framework and explicit examples, 2015. Preprint available athttp://papers.ssrn.com/sol3/papers.cfm?abstract_id=2554600.
D. Brigo, and A. Capponi. Bilateral Counterparty risk with application toCDSs. Risk Magazine, March 2010.
D. Brigo, A. Capponi, and A. Pallavicini. Arbitrage-free bilateralcounterparty risk valuation under collateralization and application to creditdefault swaps. Mathematical Finance 24, 125–146, 2014.
L. Bo, and A. Capponi. Bilateral credit valuation adjustment for largecredit derivatives portfolios. Finance and Stochastics, 18, 431-482, 2014.
A. Capponi. Measuring portfolio counterparty risk. Creditflux, 2014.
A. Capponi. Pricing and Mitigation of Counterparty Credit Exposure. J.P.Fouque, J. Langsam, eds. Handbook of Systemic Risk. CambridgeUniversity Press, Cambridge, 2013.