Arbitrage-Free Pricing of XVA Motivation Model Hedging Agostino - - PowerPoint PPT Presentation

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Arbitrage-Free Pricing of XVA

Agostino Capponi

Columbia University joint work with Maxim Bichuch (WPI) and Stephan Sturm (WPI)

IAQF/Thalesians Seminar Series

New York, September 21, 2015

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The LIBOR-OIS Spread

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The LIBOR-OIS Spread

Consequences

Widening of spreads is due to counterparty credit risk LIBOR cannot be considered a risk-free rate any longer One cannot assume the existence of a universal risk-free rate r

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The LIBOR-OIS Spread

Consequences

Widening of spreads is due to counterparty credit risk LIBOR cannot be considered a risk-free rate any longer One cannot assume the existence of a universal risk-free rate r Rates at which derivatives traders borrow and lend unsecured cash differ How to price and hedge derivatives in presence of funding spread and counterparty risk?

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The LIBOR-OIS Spread

Consequences

Widening of spreads is due to counterparty credit risk LIBOR cannot be considered a risk-free rate any longer One cannot assume the existence of a universal risk-free rate r Rates at which derivatives traders borrow and lend unsecured cash differ How to price and hedge derivatives in presence of funding spread and counterparty risk? 2013: Many banks (Barclays, JPM, BoA,...) introduce XVA desks

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples 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), Cr´ epey (2011, 2013), Cr´ epey, Bielecki and Brigo (2014)

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Main Contributions

Develop a framework to characterize the total valuation adjustment (XVA) of a European style claim on a stock in presence of

counterparty credit risk funding spread

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Main Contributions

Develop a framework to characterize the total valuation adjustment (XVA) of a European style claim on a stock in presence of

counterparty credit risk funding spread

Derive a nonlinear backward stochastic differential equation (BSDE) associated with the replicating portfolios

• f long and short positions in the claim.

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XVA Pricing

• A. Capponi

Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Main Contributions

Develop a framework to characterize the total valuation adjustment (XVA) of a European style claim on a stock in presence of

counterparty credit risk funding spread

Derive a nonlinear backward stochastic differential equation (BSDE) associated with the replicating portfolios

• f long and short positions in the claim.

Develop an explicit representation of XVA in case of symmetric rates, but in presence of counterparty risk

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples 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 (PI

t , PC t ): underwritten by

investor/counterparty and used to hedge default risk. Trading does not happen in the repo market

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Stock Short-Selling

Trader Treasury Desk (1) (6) Stock Market (5) (4) Repo Market (2) (3) r

r

Figure: Security driven repo activity: Solid lines are purchases/sales, dashed lines borrowing/lending, dotted lines interest due; blue lines are cash, red lines are stock.

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Stock Purchasing

Trader Treasury 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 lines are cash, red lines are stock.

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Market Model

The market model (II)

We consider the dynamics dSt ✏ µSt dt σSt dWt dPI

t ✏ µIPI t dt ✁ PI t✁ d1

ltτI ↕t✉ ✏ ♣µI ✁ hIqPI

t dt ✁ PI t✁ d̟I t

dPC

t ✏ µCPC t dt ✁ PC t✁ d1

ltτC ↕t✉ ✏ ♣µC ✁ hCqPC

t dt ✁ PC t✁ d̟C t

for independent default times τI, τC with constant default intensities hI, hC and martingales ̟I, ̟C

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Market Model

The market model (III)

Can we guarantee that there are no arbitrage opportunities in the market?

↕ ✁ ↕ ✁ ➔ ➔ ↕ ↕ ✁

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Market Model

The market model (III)

Can we guarantee that there are no arbitrage opportunities in the market? As we only model from the point of the trader, we can

• nly conclude this from her perspective. . .

↕ ✁ ↕ ✁ ➔ ➔ ↕ ↕ ✁

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Market Model

The market model (III)

Can we guarantee that there are no arbitrage opportunities in the market? As we only model from the point of the trader, we can

• nly 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

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Market Model

The market model (III)

Can we guarantee that there are no arbitrage opportunities in the market?

↕ ✁ ↕ ✁ ➔ ➔ ↕ ↕ ✁

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Market Model

The market model (III)

Can we guarantee that there are no arbitrage opportunities in the market? As we only model from the point of the trader, we can

• nly conclude this from her perspective. . .

↕ ✁ ↕ ✁ ➔ ➔ ↕ ↕ ✁

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Market Model

The market model (III)

Can we guarantee that there are no arbitrage opportunities in the market? As we only model from the point of the trader, we can

• nly 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

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Collateralization

Collateral is used to secure the derivatives deal

• ✁

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Collateralization

Collateral is used to secure the derivatives deal

• ✁

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Collateralization

Collateral is used to secure the derivatives deal

Collateral is provided in form of cash (80%)

• ✁

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Motivation Model Hedging Arbitrage Theory Explicit Examples 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%)

• ✁

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Motivation Model Hedging Arbitrage Theory Explicit Examples 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 . The

collateral taker pays interests at rate r ✁

c .

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Market Model

Trader Treasury Desk r

f

r✁

f

Cash Stock & Repo Market Stock r✁

r

r

r

Bond Market Bonds PI, PC Counterparty Collateral r

c

r✁

c

Figure: Solid lines are purchases/sales, dashed lines borrowing/lending, dotted lines interest due; blue lines are cash, red lines stock purchases for cash and black lines bond purchases for cash.

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Closeout Payments and Valuation

The closeout value of the claim is decided by a valuation agent (either party or third party) in accordance with market practices (ISDA)

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Closeout Payments and Valuation

The closeout value of the claim is decided by a valuation agent (either party or third party) in accordance with market practices (ISDA) The valuation agent determines collateral requirements and closeout value by calculating the Black-Scholes price

• f the transaction

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Closeout Payments and Valuation

The closeout value of the claim is decided by a valuation agent (either party or third party) in accordance with market practices (ISDA) The valuation agent determines collateral requirements and closeout value by calculating the Black-Scholes price

• f the transaction

Such a valuation is associated with a publicly known interest rate rD

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Closeout Payments and Valuation

The closeout value of the claim is decided by a valuation agent (either party or third party) in accordance with market practices (ISDA) The valuation agent determines collateral requirements and closeout value by calculating the Black-Scholes price

• f the transaction

Such a valuation is associated with a publicly known interest rate rD We can then introduce a valuation measure Q under which rD-discounted prices are Q martingales.

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XVA Pricing

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Closeout Payments and Valuation

The closeout value of the claim is decided by a valuation agent (either party or third party) in accordance with market practices (ISDA) The valuation agent determines collateral requirements and closeout value by calculating the Black-Scholes price

• f the transaction

Such a valuation is associated with a publicly known interest rate rD We can then introduce a valuation measure Q under which rD-discounted prices are Q martingales. The XVA will be computed under Q

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Collateral and Close-Out Valuation

Collateral is a percentage α of the price of the contract Ct ✏ α1 ltτI ❫τC →t✉EQ✑ e✁rD♣T✁tqΦ♣STq ✞ ✞ ✞ Ft ✙ :✏ α1 ltτI ❫τC →t✉ ˆ V ♣t, Stq ✏ ❫ ❫

♣ q ✏ ♣ q ✏ ♣ q

t ➔ ✉ ✁ ✁ t ➔ ✉

• ✏

✁

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Collateral and Close-Out Valuation

Collateral is a percentage α of the price of the contract Ct ✏ α1 ltτI ❫τC →t✉EQ✑ e✁rD♣T✁tqΦ♣STq ✞ ✞ ✞ Ft ✙ :✏ α1 ltτI ❫τC →t✉ ˆ V ♣t, Stq Set τ ✏ τI ❫ τC ❫ T. The close-out payment is

θτ♣ ˆ V q ✏ θτ♣C, ˆ V q :✏ ˆ V ♣τ, Sτq 1 ltτC ➔τI ✉LCY ✁ ✁ 1 ltτI ➔τC ✉LIY ,

where Y :✏ ˆ Vτ ✁ Cτ is the residual value of the claim at default

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Wealth Process

It is useful to distinguish between legal and actual wealth process

Legal wealth Vt ✏ ξtSt ξI

tPI t ξC t PC t ψrf t Brf t ψtBrr t Ct,

Actual wealth V C

t ✏ ξtSt ξI tPI t ξC t PC t ψrf t Brf t ψtBrr t ✏ Vt ✁ Ct,

(with Brf

t funding account Brr t sec lending account and ξt,

ξI

t, ξC t , ψrf t , ψt number of shares holding)

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Wealth Dynamics

The dynamics of the wealth is given by dVt ✏ ✁ r

f

• ξf

t Brf t

✟ ✁ r✁

f

• ξf

t Brf t

✟✁ ♣rD ✁ r✁

r q

• ξtSt

✟ ✁ ♣rD ✁ r

r q

• ξtSt

✟✁ rDξI

tPI t rDξC t PC t

✠ dt ✁ r✁

c

• ψc

t Brc t

✟dt r

c

• ψc

t Brc t

✟✁ dt

• ♣☎ ☎ ☎ q

❧♦ ♦♠♦ ♦♥

martingales

with Brf

t funding account, Brc t collateral account, ξt, and

ψt number of shares in the securities and various accounts

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Arbitrage Pricing

Definition A price P P R, of a derivative security with terminal payoff ξ P σ♣St; t ↕ Tq is called hedger’s arbitrage-free, if for all γ P R buying γ securities for the price γP and hedging in the market with an admissible strategy does not create hedger’s arbitrage.

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Replicating Wealth

V

t ♣γq: wealth process when replicating the claim

γΦ♣STq, γ → 0. This means hedging the position after selling γ securities with terminal payoff Φ♣STq.

• ✁V ✁

t ♣γq

✟ : wealth process when replicating the claim ✁γΦ♣STq, γ → 0. This means hedging the position after buying γ securities with terminal payoff Φ♣STq.

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

BSDE formulation

Set f t, v, z, zI, zC; ˆ V ✟ ✏ ✁ ✁ r

f

• v zI zC ✁ α ˆ

Vt ✟ ✁ r✁

f

• v zI zC ✁ α ˆ

Vt ✟✁ ♣rD ✁ r✁

r q 1

σz ✁ ♣rD ✁ r

r q 1

σz✁ ✁ rDzI ✁ rDzC r

c α ˆ

Vt ✁ ♣r✁

c ✁ r c q

• α ˆ

Vt ✟✁✠ f ✁ t, v, z, zI, zC; ˆ V ✟ ✏ ✁f t, ✁v, ✁z, ✁zI, ✁zC; ✁ ˆ Vt ✟

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

BSDE formulation

The BSDEs ✩ ✬ ✫ ✬ ✪ ✁dV

t ♣γq

✏ f t, V

t , Z t , Z I, t

, Z C,

t

; ˆ V ✟ dt ✁ Z

t dW Q t ✁ Z I, t

d̟I,Q

t

✁ Z C,

t

d̟C,Q

t

V

τ ♣γq

✏ γ ✁ θτ♣ ˆ V q1 ltτ➔T✉ Φ♣STq1 ltτ✏T✉ ✠ ✩ ✬ ✫ ✬ ✪ ✁dV ✁

t ♣γq

✏ f ✁ t, V ✁

t , Z ✁ t , Z I,✁ t

, Z C,✁

t

; ˆ V ✟ dt ✁ Z ✁

t dW Q t ✁ Z I,✁ t

d̟I,Q

t

✁ Z C,✁

t

d̟C,Q

t

V ✁

τ ♣γq

✏ γ ✁ θτ♣ ˆ V q1 ltτ➔T✉ Φ♣STq1 ltτ✏T✉ ✠ describe the wealth dynamics for buying/selling γ options Existence and uniqueness of solution can be guaranteed

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

BSDE Relations

The two BSDEs are intrinsically related: ♣V ✁

t , Z ✁ t , Z I,✁ t

, Z C,✁

t

q is a solution to the data

• f ✁, θτ♣ ˆ

V q, Φ♣STq ✟ iff ♣✁V ✁

t , ✁Z ✁ t , ✁Z I,✁ t

, ✁Z C,✁

t

q is a solution to the data

• f , θτ♣✁ ˆ

V q, ✁Φ♣STq ✟

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

No arbitrage

Theorem Let Φ be a function of polynomial growth. Assume that If V ✁

0 ↕ V 0 , then all prices in the closed interval

rπinf ✏ V ✁

0 , V 0 ✏ πsups are free of hedger’s arbitrage.

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

XVA

We define the total value adjustment XVAt as Definition The seller’s XVA is given as XVAsell

t

✏ V

t ✁ ˆ

V ♣t, Stq and the buyer’s XVA as XVAbuy

t

✏ V ✁

t ✁ ˆ

V ♣t, Stq.

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Extended Piterbarg Model

Extension of Piterbarg’s model

Allow for default of investor and counterparty Default risk is hedged by risky bonds Maintain Piterbarg’s assumption of symmetric rates: rf ✏ r

f

✏ r ✁

f , rr ✏ r r ✏ r ✁ r , rc ✏ r c ✏ r ✁ c

BSDE becomes linear and XVAsell

t

✏ XVAbuy

t

✏ ✏ ✏ ✏ ✁

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Extended Piterbarg Model

Extension of Piterbarg’s model

Allow for default of investor and counterparty Default risk is hedged by risky bonds Maintain Piterbarg’s assumption of symmetric rates: rf ✏ r

f

✏ r ✁

f , rr ✏ r r ✏ r ✁ r , rc ✏ r c ✏ r ✁ c

BSDE becomes linear and XVAsell

t

✏ XVAbuy

t

Note: If rf ✏ rr ✏ rc ✏ rD we have no funding costs and recover the classical CVA/DVA setting In particular XVAt ✏ DVAt ✁ CVAt

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Extended Piterbarg Model

Decomposition allows for the nice interpretation of the XVA in terms of four separate contributing terms:

Default and collateralization free price under funding constraints Funding-adjusted payout after default of the trader Funding-adjusted payout after counterparty’s default Funding costs of the collateralization procedure

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Extended Piterbarg Model

Price decomposition erf tVt1 ltτ➙t✉ ✏ EQ✑ Brf

T

✟✁1Φ♣STqΓT

t 1

ltτ✏T✉ ✞ ✞ ✞ Gt ✙ EQ✑ Brf

τI

✟✁1lI ˆ V ♣τI, SτI qΓτI

t 1

ltt➔τI ➔τC ❫T; ˆ

V ♣τI ,SτI q➙0✉

• Brf

τI

✟✁1 ˆ V ♣τI, SτI qΓτI

t 1

ltt➔τI ➔τC ❫T; ˆ

V ♣τI ,SτI q➔0✉

✞ ✞ ✞ Gt ✙ EQ✑ Brf

τC

✟✁1lC ✟ ˆ V ♣τC, SτC qΓτC

t 1

ltt➔τC ➔τI ❫T; ˆ

V ♣τC ,SτC q➔0✉

• Brf

τC

✟✁1 ˆ V ♣τC, SτC qΓτC

t 1

ltt➔τC ➔τI ❫T; ˆ

V ♣τC ,SτC q➙0✉

✞ ✞ ✞ Gt ✙ EQ✑ α

• rf ✁ rc

✟ ➺ τ

t❫τ

• Brf

s

✟✁1 ˆ V ♣s, SsqΓs

t ds

✞ ✞ ✞ Gt ✙ . with lI ✏ 1 ✁ ♣1 ✁ αqLC and lc ✏ 1 ✁ ♣1 ✁ αqLC.

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Extended Piterbarg Model

Explicit computation possible in terms of compound option with random maturity. In the case of a long position

XVAt ✏ ✄ Brf

t

BrD

t

✂BrD

T

Brf

T

e♣rf ✁rD✁hQ

I q♣T✁tqe♣rf ✁rD✁hQ C q♣T✁tq

• 1 ✁ ♣1 ✁ αqLI

✟ e♣rD✁rf qt✁ 1 rD ✁ rf hQ

I

✠ hQ

I

λ ✁ hQ

I

☎ ✁ hQ

C

λ ✁ hQ

C ✁ hQ I

• e♣λ✁hQ

C ✁hQ I q♣T✁tq ✁ 1

✟ ✁ 1 e✁hQ

C ♣T✁tqe♣λ✁hQ I q♣T✁tq✠

e♣rD✁rf qt✁ 1 rD ✁ rf hQ

C

✠ hQ

C

λ ✁ hQ

C

☎ ✁ hQ

I

λ ✁ hQ

C ✁ hQ I

• e♣λ✁hQ

C ✁hQ I q♣T✁tq ✁ 1

✟ ✁ 1 e✁hQ

I ♣T✁tqe♣λ✁hQ C q♣T✁tq✠

α rf ✁ rc hQ

C hQ I ✁ λ

✁BrD

t

Brf

t

✁ BrD

T

Brf

T

e♣rf ✁rD✁hQ

C q♣T✁tqe♣rf ✁rD✁hQ I q♣T✁tq✠✡

✁ 1 ☛ ˆ V ♣t, Stq

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Extended Piterbarg Model

Direct computation leads to XVAt ✏ ♣A ✁ 1q ˆ V ♣t, Stq, where A ✏ Vt

ˆ Vt is explicit

Hedging strategies are explicit and given by ξt ✏ A ✂ ˆ VS♣t, Stq, ξi

t ✏ A ✂ ˆ

V ♣t, Stq ✁ θi♣ ˆ V ♣t, Stqq Pi

t

, i P tI, C✉. and θC♣ˆ vq :✏ ˆ v LC♣♣1 ✁ αqˆ vq✁, θI♣ˆ vq :✏ ˆ v ✁ LI♣♣1 ✁ αqˆ vq.

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Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

The Extended Piterbarg Model

0.08 0.1 0.12 0.14 0.16 0.18 0.2 10 20 30 40 50 60 70 80 90

rf Price Components (%) Pure funding Trader default Counterparty default Collateralization

0.08 0.1 0.12 0.14 0.16 0.18 0.2 5 10 15 20 25 30 35 40 45

rf Price Components (%) Pure funding Trader default Counterparty default Collateralization

Figure: Left graph: hQ

I ✏ 0.15, hQ C ✏ 0.2. Right graph: hQ I ✏ 0.5,

hQ

C ✏ 0.5.

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XVA Pricing

• A. Capponi

Motivation Model Hedging Arbitrage Theory Explicit Examples 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 a

no-arbitrage interval for prices But, we can show that the semilinear PDE v corresponding to the BSDE V admits a unique classical solution

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XVA Pricing

• A. Capponi

Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Band and funding spreads

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 16 18 20 22 24 26 28 30 32

α Relative XVA (%) rf

− = 0.08

rf

− = 0.1

rf

− = 0.15

rf

− = 0.2

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XVA Pricing

• A. Capponi

Motivation Model Hedging Arbitrage Theory Explicit Examples PDE Repre- sentations Conclusion

Conclusion

Developed an arbitrage-free valuation framework for XVA

• f an European style claim

Seller’s and buyer’s XVA characterized as the solution of a nonlinear BSDEs with random terminal condition Funding component of XVA is predominant, with CVA/DVA terms becoming relevant only if trader and counterparty are very risky The no-arbitrage band widens as funding spreads and collateral levels increase

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XVA Pricing

• A. Capponi

Motivation Model Hedging Arbitrage Theory Explicit Examples 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 at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2554600.

• D. Brigo, and A. Capponi. Bilateral Counterparty risk with application to
• CDSs. Risk Magazine, March 2010.
• D. Brigo, A. Capponi, and A. Pallavicini. Arbitrage-free bilateral

counterparty risk valuation under collateralization and application to credit default swaps. Mathematical Finance 24, 125–146, 2014.

• L. Bo, and A. Capponi. Bilateral credit valuation adjustment for large

credit 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. Cambridge University Press, Cambridge, 2013.