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Valuation and Hedging of Contracts with Funding Costs and Collateralization

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  • Tomasz R. Bielecki
  • Marek Rutkowski

Abstract

The research presented in this work is motivated by recent papers by Brigo et al. (2011), Burgard and Kjaer (2009), Cr\'epey (2012), Fujii and Takahashi (2010), Piterbarg (2010) and Pallavicini et al. (2012). Our goal is to provide a sound theoretical underpinning for some results presented in these papers by developing a unified framework for the non-linear approach to hedging and pricing of OTC financial contracts. We introduce a systematic approach to valuation and hedging in nonlinear markets, that is, in markets where cash flows of the financial contracts may depend on the hedging strategies. Our systematic approach allows to identify primary sources of and quantify various adjustment to valuation and hedging, primarily the funding and liquidity adjustment and credit risk adjustment. We propose a way to define no-arbitrage in such nonlinear markets, and we provide conditions that imply absence of arbitrage in some specific market trading models. Accordingly, we formulate a concept of no-arbitrage price, and we provide relevant (non-linear) BSDE that produces the no-arbitrage price in case when the contract's cash flows can be replicated.

Suggested Citation

  • Tomasz R. Bielecki & Marek Rutkowski, 2014. "Valuation and Hedging of Contracts with Funding Costs and Collateralization," Papers 1405.4079, arXiv.org, revised Dec 2014.
    • Handle: RePEc:arx:papers:1405.4079
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    References listed on IDEAS

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    1. T. R. Bielecki & S. Crépey & M. Jeanblanc & B. Zargari, 2012. "Valuation And Hedging Of Cds Counterparty Exposure In A Markov Copula Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(01), pages 1-39.
    2. Cyril Durand & Marek Rutkowski, 2013. "Cva Under Alternative Settlement Conventions And With Systemic Risk," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(07), pages 1-40.
    3. Bergman, Yaacov Z, 1995. "Option Pricing with Differential Interest Rates," The Review of Financial Studies, Society for Financial Studies, vol. 8(2), pages 475-500.
    4. Damiano Brigo & Agostino Capponi & Andrea Pallavicini & Vasileios Papatheodorou, 2011. "Collateral Margining in Arbitrage-Free Counterparty Valuation Adjustment including Re-Hypotecation and Netting," Papers 1101.3926, arXiv.org.
    5. Damiano Brigo & Agostino Capponi & Andrea Pallavicini, 2014. "Arbitrage-Free Bilateral Counterparty Risk Valuation Under Collateralization And Application To Credit Default Swaps," Mathematical Finance, Wiley Blackwell, vol. 24(1), pages 125-146, January.
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    Cited by:

    1. Han, Xingyu, 2018. "Pricing and hedging vulnerable option with funding costs and collateral," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 103-115.
    2. Yannick Armenti & Stéphane Crépey, 2017. "Central Clearing Valuation Adjustment," Working Papers hal-01169169, HAL.
    3. Maxim Bichuch & Agostino Capponi & Stephan Sturm, 2015. "Arbitrage-Free Pricing of XVA - Part II: PDE Representation and Numerical Analysis," Papers 1502.06106, arXiv.org, revised Aug 2016.
    4. Damiano Brigo & Marco Francischello & Andrea Pallavicini, 2015. "Invariance, existence and uniqueness of solutions of nonlinear valuation PDEs and FBSDEs inclusive of credit risk, collateral and funding costs," Papers 1506.00686, arXiv.org, revised Nov 2015.
    5. Damiano Brigo & Cristin Buescu & Andrea Pallavicini & Qing Liu, 2015. "A Note On The Self-Financing Condition For Funding, Collateral And Discounting," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(02), pages 1-10.
    6. Tianyang Nie & Marek Rutkowski, 2014. "Fair and profitable bilateral prices under funding costs and collateralization," Papers 1410.0448, arXiv.org, revised Dec 2014.
    7. Maxim Bichuch & Agostino Capponi & Stephan Sturm, 2015. "Arbitrage-Free Pricing of XVA -- Part I: Framework and Explicit Examples," Papers 1501.05893, arXiv.org, revised Aug 2016.
    8. Tianyang Nie & Marek Rutkowski, 2014. "Fair bilateral prices in Bergman's model," Papers 1410.0673, arXiv.org, revised Dec 2014.
    9. Stéphane Crépey & Shiqi Song, 2014. "Counterparty risk and funding: Immersion and beyond," Working Papers hal-00989062, HAL.
    10. Tianyang Nie & Marek Rutkowski, 2014. "A BSDE approach to fair bilateral pricing under endogenous collateralization," Papers 1412.2453, arXiv.org.
    11. Yannick Armenti & St'ephane Cr'epey, 2015. "Central Clearing Valuation Adjustment," Papers 1506.08595, arXiv.org, revised Feb 2017.

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