Robust XVA
Maxim Bichuch
Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, Maryland
Search for more papers by this authorCorresponding Author
Agostino Capponi
Industrial Engineering and Operations Research Department, Columbia University, New York City, New York
Correspondence
Agostino Capponi, Industrial Engineering and Operations Research Department, Mudd Engineering Building, 500 West 120th Street, Columbia University, New York City, NY 10027.
Email: ac3827@columbia.edu
Search for more papers by this authorStephan Sturm
Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts
Search for more papers by this authorMaxim Bichuch
Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, Maryland
Search for more papers by this authorCorresponding Author
Agostino Capponi
Industrial Engineering and Operations Research Department, Columbia University, New York City, New York
Correspondence
Agostino Capponi, Industrial Engineering and Operations Research Department, Mudd Engineering Building, 500 West 120th Street, Columbia University, New York City, NY 10027.
Email: ac3827@columbia.edu
Search for more papers by this authorStephan Sturm
Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts
Search for more papers by this authorData sharing is not applicable to this article as no new data were created or analyzed in this study.
Funding information:
NSF (DMS-1736414, DMS-1716145); Acheson J. Duncan Fund for the Advancement of Research in Statistics.
Abstract
We introduce an arbitrage-free framework for robust valuation adjustments. An investor trades a credit default swap portfolio with a risky counterparty, and hedges credit risk by taking a position in defaultable bonds. The investor does not know the exact return rate of her counterparty's bond, but she knows it lies within an uncertainty interval. We derive both upper and lower bounds for the XVA process of the portfolio, and show that these bounds may be recovered as solutions of nonlinear ordinary differential equations. The presence of collateralization and closeout payoffs leads to important differences with respect to classical credit risk valuation. The value of the super-replicating portfolio cannot be directly obtained by plugging one of the extremes of the uncertainty interval in the valuation equation, but rather depends on the relation between the XVA replicating portfolio and the closeout value throughout the life of the transaction. Our comparative statics analysis indicates that credit contagion has a nonlinear effect on the replication strategies and on the XVA.
REFERENCES
- Acharya, V., Amihud, Y., & Bharath, S. T. (2013). Liquidity risk of corporate bond returns: Conditional approach. Journal of Financial Economics, 110, 358–386.
- Albanese, C., & Andersen, L. (2014). Accounting for OTC Derivatives: Funding adjustments and the re-hypothecation option. Working Paper. Retrieved from https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2482955.
- Albanese, C., & Crépey, C. (2018). XVA analysis from the balance sheet. Retrieved from https://www.maths.univ-evry.fr/crepey/papers/KVA-FVA-I-REVISED.pdf.
- Albanese, C., & Crépey, S. (2019). Capital valuation adjustment and funding valuation adjustment. Working Paper. Retrieved from https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2745909.
- Altman, E., & Kishore, V. (1996). Almost everything you always wanted to know about recoveries on defaulted bonds. Financial Analysts Journal, 52, 57–64.
10.2469/faj.v52.n6.2040 Google Scholar
- Andersen, L., Duffie, D., & Song, Y. (2019). Funding value adjustments. Journal of Finance, 74, 145–192.
- Avellaneda, M., Levy, A., & Parás, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Applied Mathematical Finance, 2, 73–88.
10.1080/13504869500000005 Google Scholar
- Basel Committee on Banking Supervision and Board of the International Organization of Securities Commissions. (2013). Margin requirements for non-centrally cleared derivatives. Retrieved from https://www.bis.org/bcbs/publ/d317.pdf.
- BCBS/IOSCO. (2013). Second consultative document on margin requirements for noncentrally cleared derivatives (Technical report), Bank for International Settlements and International Organization of Securities Commissions.
- Becker, L. (2015). BAML takes $497m FVA loss, Risk. Retrieved from https://www.risk.net/risk-management/2390522/baml-takes-497m-fva-loss.
- Bichuch, M., Capponi, A., & Sturm, S. (2018). Arbitrage-free XVA. Mathematical Finance, 28, 582–620.
- Bielecki, T., Crépey, S., Jeanblanc, M., & Zargari, B. (2012). Valuation and hedging of CDS counterparty exposure in a Markov chain Copula model. International Journal of Theoretical and Applied Finance, 15. https://doi.org/10.1142/97898144078920004.
10.1142/S0219024911006498 Google Scholar
- Bielecki, T., & Rutkowski, M. (2001). Credit risk: Modelling, valuation and hedging. New York, NY: Springer.
- Bielecki, T., & Rutkowski, M. (2015). Valuation and hedging of contracts with funding costs and collateralization. SIAM Journal on Financial Mathematics, 6, 594–655.
- Brigo, D., Capponi, A., & Pallavicini, A. (2014). Arbitrage-free bilateral counterparty risk valuation under collateralization and application to credit default swaps. Mathematical Finance, 24, 125–146.
- Brigo, D., & Pallavicini, A. (2014). Nonlinear consistent valuation of CCP cleared or CSA bilateral trades with initial margins under credit, funding and wrong-way risk. Journal of Financial Engineering, 1. https://doi.org/10.1142/S2345768614500019.
10.1142/S2345768614500019 Google Scholar
- Burgard, C., & Kjaer, M. (2011). Partial differential equation representations of derivatives with bilateral counterparty risk and funding costs. Journal of Credit Risk, 7(3), 1–19.
- Burgard, C., & Kjaer, M. (2013). Funding costs, funding strategies. Risk Magazine, 82–87. http://doi.org/10.2139/ssrn.2027195.
- Cameron, M. (2014). JP Morgan takes $1.5 billion FVA loss, Risk. Retrieved from https://www.risk.net/derivatives/2322843/jp-morgan-takes-15-billion-fva-loss.
- Collin-Dufresne, P., Goldstein, R., & Hugonnier, J. (2004). A general formula for pricing defaultable securities. Econometrica, 72, 1377–1407.
- Commodity Futures Trading Commission (CFTC). (2016). Margin requirements for uncleared swaps for swap dealers and major swap participants; final rule. Federal Register, 81, 3 (to be codified at 17 CFR Parts 23 and 140). Retrieved from http://www.cftc.gov/idc/groups/public/@lrfederalregister/documents/file/2015-32320a.pdf
- Crépey, S. (2015a). Bilateral counterparty risk under funding constraints—Part I: Pricing. Mathematical Finance, 25, 1–22.
- Crépey, S. (2015b). Bilateral counterparty risk under funding constraints—Part II: CVA. Mathematical Finance, 25, 23–50.
- Crépey, S., Bielecki, T. R., & Brigo, D. (2014). Counterparty risk and funding: A tale of two puzzles. Boca Raton, FL: Chapman and Hall/CRC.
10.1201/9781315373621 Google Scholar
- Crépey, S., & Song, S. (2015). BSDEs of counterparty risk. Stochastic Processes and their Applications, 125, 3023–3052.
- Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. Journal of the Royal Statistical Society Series B, 46(3), 353–388.
- Delbaen, F., & Schachermayer, W. (2006). The mathematics of arbitrage. Berlin: Springer Finance.
- Dełong, L. (2013). Backward stochastic differential equations with jumps and their actuarial and financial applications: BSDEs with jumps. London: Springer EAA Series.
10.1007/978-1-4471-5331-3 Google Scholar
- Denis, L., & Martini, C. (2006). A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Annals of Applied Probability, 16(2), 827–852.
- El Karoui, N., Jeanblanc, M., & Jiao, Y. (2015). Density approach in modeling successive default. SIAM Journal on Financial Mathematics, 6, 1–21.
- El Karoui, N., Jeanblanc-Picqué, M., & Shreve, S. E. (1998). Robustness of the Black and Scholes formula. Mathematical Finance, 8(2), 93–126.
- El Karoui, N., Peng, S., & Quenez, M.-C. (1997). Backward stochastic differential equations in finance. Mathematical Finance, 7, 1–71.
- European Commission. (2016). Commission Delegated Regulation (EU) 2016/2251 of 4 October 2016 supplementing Regulation (EU) No 648/2012 of the European Parliament and of the Council on OTC derivatives, central counterparties and trade repositories with regard to regulatory technical standards for risk-mitigation techniques for OTC derivative contracts not cleared by a central counterparty, 2016 C/2016/6329. (2016). Retrieved from http://eur-lex.europa.eu/legal-content/EN/ALL/?uri=CELEX:32016R2251&qid=1516465531629
- Fadina, T., & Schmidt, T. (2018). Ambiguity in defaultable term structure models. Retrieved from https://arxiv.org/pdf/1801.10498.pdf.
- Föllmer, H., & Schied, A. (2004). Stochastic finance: An introduction in discrete time (2nd ed.). Berlin: DeGruyter.
10.1515/9783110212075 Google Scholar
- Fouque, J. P., & Ning, N. (2018). Uncertain volatility models with stochastic bounds. SIAM Journal on Financial Mathematics, 9, 1175–1207.
- Frey, R., & Backhaus, J. (2008). Pricing and hedging of portfolio credit derivatives with interacting default intensities. International Journal of Theoretical and Applied Finance, 11, 611–634.
10.1142/S0219024908004956 Google Scholar
- Green, A., Kenyon, C., & Dennis, C. (2014). KVA: Capital valuation adjustment. Retrieved from https://arxiv.org/pdf/1405.0515.pdf
- Hartman, P. (2002). Ordinary differential equations (2nd ed.). Philadelphia, PA: Society for Industrial and Applied Mathematics.
10.1137/1.9780898719222 Google Scholar
- Hobson, D. G. (1998). Volatility misspecification, option pricing and superreplication via coupling. Annals of Applied Probability, 8(1), 193–205.
- ISDA Margin Survey. (2017). International Swaps and Derivatives Association. Retrieved from https://www.isda.org/a/VeiDE/margin-survey-final1.pdf.
- Jarrow, R., & Yu, F. (2001). Counterparty risk and the pricing of defaultable securities. Journal of Finance, 56, 1765–1799.
- Kwan, S. (1996). Firm-specific information and the correlation between individual stocks and bonds. Journal of Financial Economics, 40, 63–80.
- Linetsky, V. (2006). Pricing equity derivatives subject to bankruptcy. Mathematical Finance, 16(2), 255–282.
- Lipton, A. (2009). Credit value adjustments for credit default swaps via the structural default model. Journal of Credit Risk, 5, 127–150.
- Lyons, T. J. (1995). Uncertain volatility and the risk-free synthesis of derivatives. Applied Mathematical Finance, 2, 117–133.
10.1080/13504869500000007 Google Scholar
- Nie, T., & Rutkowski, M. (2016). A BSDE approach to fair bilateral pricing under endogenous collateralization. Finance and Stochastics, 20(4), 855–900.
- Nie, T., & Rutkowski, M. (2018). Fair bilateral pricing under funding costs and exogenous collateralization. Mathematical Finance, 18, 621–655.
- Pallavicini, A., Perini, D., & Brigo, D. (2012). Funding valuation adjustment: A consistent framework including CVA, DVA, collateral, netting rules and re-hypothecation. Working Paper, Imperial College London.
- Singh, M., & Aitken, J. (2010). The (sizable) role of rehypothecation in the shadow banking system. IMF Working Paper 10/172. Retrieved from https://www.imf.org/en/Publications/WP/Issues/2016/12/31/The-Sizable-Role-of-Rehypothecation-in-the-Shadow-Banking-System-24075.
- Yu, F. (2007). Correlated defaults in intensity-based models. Mathematical Finance, 17, 155–173.
