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A Computational Approach to Hedging Credit Valuation Adjustment in a Jump-Diffusion Setting

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  • T. van der Zwaard
  • L. A. Grzelak
  • C. W. Oosterlee

Abstract

This study contributes to understanding Valuation Adjustments (xVA) by focussing on the dynamic hedging of Credit Valuation Adjustment (CVA), corresponding Profit & Loss (P&L) and the P&L explain. This is done in a Monte Carlo simulation setting, based on a theoretical hedging framework discussed in existing literature. We look at hedging CVA market risk for a portfolio with European options on a stock, first in a Black-Scholes setting, then in a Merton jump-diffusion setting. Furthermore, we analyze the trading business at a bank after including xVAs in pricing. We provide insights into the hedging of derivatives and their xVAs by analyzing and visualizing the cash-flows of a portfolio from a desk structure perspective. The case study shows that not charging CVA at trade inception results in an expected loss. Furthermore, hedging CVA market risk is crucial to end up with a stable trading strategy. In the Black-Scholes setting this can be done using the underlying stock, whereas in the Merton jump-diffusion setting we need to add extra options to the hedge portfolio to properly hedge the jump risk. In addition to the simulation, we derive analytical results that explain our observations from the numerical experiments. Understanding the hedging of CVA helps to deal with xVAs in a practical setting.

Suggested Citation

  • T. van der Zwaard & L. A. Grzelak & C. W. Oosterlee, 2020. "A Computational Approach to Hedging Credit Valuation Adjustment in a Jump-Diffusion Setting," Papers 2005.10504, arXiv.org, revised Sep 2020.
  • Handle: RePEc:arx:papers:2005.10504
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    References listed on IDEAS

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    1. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    2. Damiano Brigo & Andrea Pallavicini & Vasileios Papatheodorou, 2011. "Arbitrage-Free Valuation Of Bilateral Counterparty Risk For Interest-Rate Products: Impact Of Volatilities And Correlations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(06), pages 773-802.
    3. J. S. Kennedy & P. A. Forsyth & K. R. Vetzal, 2009. "Dynamic Hedging Under Jump Diffusion with Transaction Costs," Operations Research, INFORMS, vol. 57(3), pages 541-559, June.
    4. Andrea Pallavicini & Daniele Perini & Damiano Brigo, 2012. "Funding, Collateral and Hedging: uncovering the mechanics and the subtleties of funding valuation adjustments," Papers 1210.3811, arXiv.org, revised Dec 2012.
    5. Arregui, Iñigo & Salvador, Beatriz & Vázquez, Carlos, 2017. "PDE models and numerical methods for total value adjustment in European and American options with counterparty risk," Applied Mathematics and Computation, Elsevier, vol. 308(C), pages 31-53.
    6. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
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    9. Stéphane Crépey, 2015. "Bilateral Counterparty Risk Under Funding Constraints—Part Ii: Cva," Mathematical Finance, Wiley Blackwell, vol. 25(1), pages 23-50, January.
    10. Damiano Brigo & Agostino Capponi, 2008. "Bilateral counterparty risk valuation with stochastic dynamical models and application to Credit Default Swaps," Papers 0812.3705, arXiv.org, revised Nov 2009.
    11. Naik, Vasanttilak & Lee, Moon, 1990. "General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns," Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 493-521.
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    14. C. He & J. Kennedy & T. Coleman & P. Forsyth & Y. Li & K. Vetzal, 2006. "Calibration and hedging under jump diffusion," Review of Derivatives Research, Springer, vol. 9(1), pages 1-35, January.
    15. Emanuel Derman & Iraj Kani, 1998. "Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 1(01), pages 61-110.
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    Cited by:

    1. T. van der Zwaard & L. A. Grzelak & C. W. Oosterlee, 2022. "Relevance of Wrong-Way Risk in Funding Valuation Adjustments," Papers 2204.02680, arXiv.org, revised Jun 2022.
    2. T. van der Zwaard & L. A. Grzelak & C. W. Oosterlee, 2022. "Efficient Wrong-Way Risk Modelling for Funding Valuation Adjustments," Papers 2209.12222, arXiv.org, revised Oct 2022.

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