Arbitrage-Free Pricing of XVA...XVA Pricing A. Capponi Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre-sentations Conclusion Arbitrage-Free Pricing of XVA Agostino

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)

7th General Advanced Mathematical Methods in Finance

and Swissequote Conference 2015

September 7, 2015, Lausanne

XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion

The LIBOR-OIS Spread

XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion


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


Conclusion


Consequences




XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion


Consequences




XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


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


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


Conclusion

Main Contributions





XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion

Main Contributions





XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


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


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


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


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


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


Conclusion

The Market Model




Proposition


XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion

The Market Model




Proposition


XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion

The Market Model




Proposition


XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion

The Market Model




Proposition


XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion

The Market Model




Proposition


XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


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


Conclusion

Collateralization





XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion

Collateralization





XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion

Collateralization





XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion

Collateralization





XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


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


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


Conclusion







XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion







XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion







XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion







XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


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


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


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


�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


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


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


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


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


Conclusion

BSDE formulation

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

Existence and uniqueness of solution can be guaranteed

XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


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 hedger’s arbitrage.

XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


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


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


Conclusion


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


Conclusion


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 � αqLI and lc � 1 � p1 � αqLC .

XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


Conclusion


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


Conclusion


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


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


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


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 unless traderand his counterparty are very risky

The no-arbitrage band widens as funding spreads andcollateral levels increase

XVA Pricing

A. Capponi

Motivation

Model

Hedging

ArbitrageTheory

ExplicitExamples


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, 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.

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2554600

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Arbitrage-Free Pricing of XVA...XVA Pricing A. Capponi Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre-sentations Conclusion Arbitrage-Free Pricing of XVA Agostino