Mild to classical solutions for XVA equations under stochastic volatility. (English) Zbl 1536.91329
Summary: We extend the valuation of contingent claims in the presence of default, collateral, and funding to a random functional setting and characterize pre-default value processes by martingales. Pre-default value semimartingales can also be described by BSDEs with random path-dependent coefficients and martingales as drivers. En route, we relax conditions on the available market information and construct a broad class of default times. Moreover, under stochastic volatility, we characterize pre-default value processes via mild solutions to parabolic semilinear PDEs and give sufficient conditions for mild solutions to exist uniquely and to be classical.
MSC:
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 60G40 | Stopping times; optimal stopping problems; gambling theory |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 35K58 | Semilinear parabolic equations |
Keywords:
XVA; valuation; collateral; funding costs; default time; stochastic volatility; stochastic differential equation; mild solution; semilinear parabolic PDEReferences:
| [1] | Bauer, H., Measure and Integration Theory, , Walter de Gruyter & Co., Berlin, 2001. · Zbl 0985.28001 |
| [2] | Becherer, D. and Schweizer, M., Classical solutions to reaction-diffusion systems for hedging problems with interacting Itô and point processes, Ann. Appl. Probab., 15 (2005), pp. 1111-1144. · Zbl 1075.60080 |
| [3] | Biagini, F., Gnoatto, A., and Oliva, I., A unified approach to xVA with CSA discounting and initial margin, SIAM J. Financial Math., 12 (2021), pp. 1013-1053, doi:10.1137/20M1332153. · Zbl 1476.91140 |
| [4] | Bichuch, M., Capponi, A., and Sturm, S., Arbitrage-free XVA, Math. Finance, 28 (2018), pp. 582-620. · Zbl 1390.91276 |
| [5] | Bielecki, T. R., Cialenco, I., and Iyigunler, I., Collateralized CVA valuation with rating triggers and credit migrations, Int. J. Theor. Appl. Finance, 16 (2013), 1350009. · Zbl 1301.91052 |
| [6] | Bielecki, T. R. and Rutkowski, M., Credit Risk: Modeling, Valuation and Hedging, Springer Finance, Springer-Verlag, Berlin, 2004. |
| [7] | Biffis, E., Blake, D., Pitotti, L., and Sun, A., The cost of counterparty risk and collateralization in longevity swaps, J. Risk Insur., 83 (2016), pp. 387-419. |
| [8] | Black, F. and Scholes, M., The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), pp. 637-654. · Zbl 1092.91524 |
| [9] | Brigo, D., Buescu, C., Francischello, M., Pallavicini, A., and Rutkowski, M., Nonlinear valuation with XVAs: Two converging approaches, Mathematics, 10 (2022), 791. |
| [10] | Brigo, D., Capponi, A., and Pallavicini, A., Arbitrage-free bilateral counterparty risk valuation under collateralization and application to credit default swaps, Math. Finance, 24 (2014), pp. 125-146. · Zbl 1285.91137 |
| [11] | Brigo, D., Francischello, M., and Pallavicini, A., Analysis of nonlinear valuation equations under credit and funding effects, in Innovations in Derivatives Markets, Springer, Cham, 2016, pp. 37-52. · Zbl 1398.91635 |
| [12] | Brigo, D., Francischello, M., and Pallavicini, A., Nonlinear valuation under credit, funding, and margins: Existence, uniqueness, invariance, and disentanglement, European J. Oper. Res., 274 (2019), pp. 788-805. · Zbl 1406.91468 |
| [13] | Brigo, D. and Masetti, M., Risk neutral pricing of counterparty risk, in Counterparty Credit Risk Modelling: Risk Management, Pricing and Regulation, Risk Books, London, 2005. |
| [14] | Brigo, D., Morini, M., and Pallavicini, A., Counterparty Credit Risk, Collateral and Funding: With Pricing Cases for All Asset Classes, , John Wiley & Sons, New York, 2013. · Zbl 1288.91001 |
| [15] | Brigo, D. and Pallavicini, A., Nonlinear consistent valuation of CCP cleared or CSA bilateral trades with initial margins under credit, funding and wrong-way risks, Int. J. Financ. Eng., 1 (2014), 1450001. |
| [16] | Cont, R. and Kalinin, A., On the support of solutions to stochastic differential equations with path-dependent coefficients, Stochastic Process. Appl., 130 (2020), pp. 2639-2674. · Zbl 1435.60045 |
| [17] | Cosso, A., Federico, S., Gozzi, F., Rosestolato, M., and Touzi, N., Path-dependent equations and viscosity solutions in infinite dimension, Ann. Probab., 46 (2018), pp. 126-174. · Zbl 1516.35192 |
| [18] | Crépey, S. and Bielecki, T. R., Counterparty Risk and Funding: A Tale of Two Puzzles, , CRC Press, Boca Raton, FL, 2014. · Zbl 1294.91005 |
| [19] | Crépey, S., Sabbagh, W., and Song, S., When capital is a funding source: The anticipated backward stochastic differential equations of X-value adjustments, SIAM J. Financial Math., 11 (2020), pp. 99-130, doi:10.1137/19M1242781. · Zbl 1443.91286 |
| [20] | Crépey, S. and Song, S., BSDEs of counterparty risk, Stochastic Process. Appl., 125 (2015), pp. 3023-3052. · Zbl 1317.60068 |
| [21] | Crépey, S. and Song, S., Counterparty risk and funding: Immersion and beyond, Finance Stoch., 20 (2016), pp. 901-930. · Zbl 1380.91139 |
| [22] | Karoui, N. El, Peng, S., and Quenez, M. C., Backward stochastic differential equations in finance, Math. Finance, 7 (1997), pp. 1-71. · Zbl 0884.90035 |
| [23] | Heston, S. L., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), pp. 327-343. · Zbl 1384.35131 |
| [24] | Kalinin, A., Markovian Integral Equations and Path-Dependent Partial Differential Equations, Doctoral thesis, University of Mannheim, Germany, 2017; available at https://madoc.bib.uni-mannheim.de/42417. |
| [25] | Kalinin, A., Markovian integral equations, Ann. Inst. Henri Poincaré Probab. Stat., 56 (2020), pp. 155-174. · Zbl 1465.60064 |
| [26] | Kalinin, A., Meyer-Brandis, T., and Proske, F., Stability, Uniqueness and Existence of Solutions to McKean-Vlasov SDEs: A Multidimensional Yamada-Watanabe Approach, preprint, arXiv:2107.07838, 2021. |
| [27] | Kalinin, A. and Schied, A., Mild and Viscosity Solutions to Semilinear Parabolic Path-Dependent PDEs, preprint, arXiv:1611.08318, 2018. |
| [28] | Lewis, A. L., Option Valuation under Stochastic Volatility, Finance Press, Newport Beach, CA, 2000. · Zbl 0937.91060 |
| [29] | Mishura, Y. and Posashkova, S., Positivity of solution of nonhomogeneous stochastic differential equation with non-Lipschitz diffusion, Theory Stoch. Process., 14 (2008), pp. 77-88. · Zbl 1224.60146 |
| [30] | Pallavicini, A., Perini, D., and Brigo, D., Funding Valuation Adjustment: A Consistent Framework Including CVA, DVA, Collateral, Netting Rules and Re-hypothecation, preprint, arXiv:1112.1521, 2011. |
| [31] | Pallavicini, A., Perini, D., and Brigo, D., Funding, Collateral and Hedging: Uncovering the Mechanics and the Subtleties of Funding Valuation Adjustments, preprint, arXiv:1210.3811, 2012. |
| [32] | Pardoux, E. and Peng, S., Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991), , Springer, Berlin, 1992, pp. 200-217. · Zbl 0766.60079 |
| [33] | Peskir, G., A change-of-variable formula with local time on curves, J. Theoret. Probab., 18 (2005), pp. 499-535. · Zbl 1085.60033 |
| [34] | Stroock, D. W. and Varadhan, S. R. S., Multidimensional Diffusion Processes, Classics Math., Springer-Verlag, Berlin, 2006. · Zbl 1103.60005 |
| [35] | Wong, B. and Heyde, C. C., On changes of measure in stochastic volatility models, J. Appl. Math. Stoch. Anal., (2006), 18130. · Zbl 1147.60321 |
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