xva pricing a. capponi arbitrage-free...
TRANSCRIPT
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Arbitrage-Free XVA
Agostino CapponiColumbia University
joint work with Maxim Bichuch and Stephan Sturm
Financial Engineering Practitioners Seminars
New York, January 25, 2016
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 fundingspreads and counterparty risk?
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 fundingspreads and counterparty risk?
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
XVA: Motivation and Challenges
Dealers need to account for the total costs of their trades
funding costs: financing the portfolio of liquid securitiesused to replicate the traded positioncollateral costs: funding and retrieving the collateralneeded to secure the deal with the counterpartycloseout costs: losses incurred if a premature liquidationneeds to be executed because the counterparty defaults
Swap quotes offered to clients should reflect the effect ofthese costs, referred to as XVA
Many banks (Barclays, JPM, BoA,...) have introduce XVAdesks
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
XVA: Motivation and Challenges
Dealers need to account for the total costs of their trades
funding costs: financing the portfolio of liquid securitiesused to replicate the traded positioncollateral costs: funding and retrieving the collateralneeded to secure the deal with the counterpartycloseout costs: losses incurred if a premature liquidationneeds to be executed because the counterparty defaults
Swap quotes offered to clients should reflect the effect ofthese costs, referred to as XVA
Many banks (Barclays, JPM, BoA,...) have introduce XVAdesks
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
XVA: Motivation and Challenges
Dealers need to account for the total costs of their trades
funding costs: financing the portfolio of liquid securitiesused to replicate the traded positioncollateral costs: funding and retrieving the collateralneeded to secure the deal with the counterpartycloseout costs: losses incurred if a premature liquidationneeds to be executed because the counterparty defaults
Swap quotes offered to clients should reflect the effect ofthese costs, referred to as XVA
Many banks (Barclays, JPM, BoA,...) have introduce XVAdesks
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
XVA: Motivation and Challenges
Dealers need to account for the total costs of their trades
funding costs: financing the portfolio of liquid securitiesused to replicate the traded positioncollateral costs: funding and retrieving the collateralneeded to secure the deal with the counterpartycloseout costs: losses incurred if a premature liquidationneeds to be executed because the counterparty defaults
Swap quotes offered to clients should reflect the effect ofthese costs, referred to as XVA
Many banks (Barclays, JPM, BoA,...) have introduce XVAdesks
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Related Literature
Practitioner literature: Piterbarg (2010, 2012), Burgard &Kjaer (2010, 2011), Mercurio (2013), Albanese (2015)
(Corporate) Finance literature: Andersen and Duffie(2016), 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
Derive stochastic differential equations tracking thereplicating portfolios of long and short positions in theclaim
Develop explicit representations of XVA and of thecorresponding hedging strategies
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
Derive stochastic differential equations tracking thereplicating portfolios of long and short positions in theclaim
Develop explicit representations of XVA and of thecorresponding hedging strategies
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
Derive stochastic differential equations tracking thereplicating portfolios of long and short positions in theclaim
Develop explicit representations of XVA and of thecorresponding hedging strategies
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 thetransaction. Trading happens through repo market at ratesr�r , r�r (Duffie (1996))
Risky bonds (P It , PC
t ): underwritten byinvestor/counterparty and used to hedge default risk.Bonds are not purchased/sold via 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 cost-free price of thetransaction
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 cost-free price of thetransaction
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 cost-free price of thetransaction
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 cost-free price of thetransaction
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 cost-free price of thetransaction
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 trader’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 trader’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
Replicating Wealth Equation
The BSDEs
$'&'%
�dV�t pγq � f �
�t,V�
t ,Z�t ,Z
I ,�t ,ZC ,�
t ; V�dt
� Z�t dWQt � Z I ,�
t d$I ,Qt � ZC ,�
t d$C ,Qt
V�τ pγq � γ
�θτ pV q1ltτ Tu � ΦpST q1ltτ�Tu
$'&'%
�dV�t pγq � f �
�t,V�
t ,Z�t ,Z
I ,�t ,ZC ,�
t ; V�dt
� Z�t dWQt � Z I ,�
t d$I ,Qt � ZC ,�
t d$C ,Qt
V�τ pγq � γ
�θτ pV q1ltτ Tu � ΦpST q1ltτ�Tu
describe the wealth dynamics for buying/selling γ options
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
No arbitrage
Theorem
Let Φ be a function of polynomial growth. If V�0 ¤ V�
0 , thenall prices in the closed interval rπinf � V�
0 ,V�0 � πsups are free
of trader’s arbitrage.
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Definition of XVA
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�cThe total costs of replicating long and short positionscoincide, and XVAsell
t � XVAbuyt
Note: If rf � rr � rc � rD we have no funding costs andrecover the classical CVA/DVA setting
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�cThe total costs of replicating long and short positionscoincide, and XVAsell
t � XVAbuyt
Note: If rf � rr � rc � rD we have no funding costs andrecover the classical CVA/DVA setting
XVAt � DVAt � CVAt
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
The Extended Piterbarg Model
Proposition (XVA decomposition)
Define η � hQI � hQC � 2rr � rf . On tτ ¡ tu, the total valuationadjustment is given by:
XVAt
Vt
��prr � rf q � αprf � rcq
1 � e�pη�rr qpT�tq
η � rrloooooooooooooooooooooooooomoooooooooooooooooooooooooonreplicating strategy and collateral costs
��rr � rf � hQC
LC
1 � e�pη�rr qpT�tq
η � rr
�p1 � αq1Vt 0
�looooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooon
CVA
��rr � rf � hQI
LI
1 � e�pη�rr qpT�tq
η � rr
�p1 � αq1Vt¡0
�loooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooon
DVA
:� Adj t
XVA Pricing
A. Capponi
Motivation
Model
Hedging
ArbitrageTheory
ExplicitExamples
PDE Repre-sentations
Conclusion
Hedging Strategies
ξt � Adj t∆t ,
ξIt �XVAt � LI p1 � αqpVtq
�
e�prD�hQI qpT�tq,
ξCt �XVAt � LC p1 � αqpVtq
�
e�prD�hQC qpT�tq.
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
100
rf
Pric
e C
ompo
nent
s (%
)
FundingDVA
0.08 0.1 0.12 0.14 0.16 0.18 0.220
30
40
50
60
70
80
rf
Pric
e C
ompo
nent
s (%
)
FundingDVA
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
The Impact of 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 use the semilinear PDE representation v tothe BSDE V for numerical analysis
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 1−5
0
5
10
15
20
25
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
Replicating strategies
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.075
0.08
0.085
0.09
0.095
0.1
0.105
0.11
0.115
α
Sto
ck s
hare
s
rf− = 0.08
rf− = 0.1
rf− = 0.15
rf− = 0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
0.065
αS
hare
s of
Tra
der
bond
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 XVA
Seller’s and buyer’s XVA characterized as the solution of anonlinear BSDEs with random terminal condition
Funding component of XVA is predominant, withDVA/CVA terms becoming material if trader/counterpartyare 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 XVA – Conditionallyaccepted in Mathematical Finance.
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. Short version inRisk.
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.