American Institute of Mathematical Sciences

January  2018, 3: 1 doi: 10.1186/s41546-017-0025-4

Continuous tenor extension of affine LIBOR models with multiple curves and applications to XVA

 1. Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece; 2. Institut fur Mathematische Stochastik, TU Dresden, 01062 Dresden, Germany

Received  January 31, 2017 Revised  November 26, 2017 Published  January 2018

Fund Project: RW acknowledges funding from the Excellence Initiative of the German Research Foundation (DFG) under grant ZUK 64. Financial support from the Europlace Institute of Finance project "Post-crisis models for interest rate markets" is gratefully acknowledged.

We consider the class of affine LIBOR models with multiple curves, which is an analytically tractable class of discrete tenor models that easily accommodates positive or negative interest rates and positive spreads. By introducing an interpolating function, we extend the affine LIBOR models to a continuous tenor and derive expressions for the instantaneous forward rate and the short rate. We show that the continuous tenor model is arbitrage-free, that the analytical tractability is retained under the spot martingale measure, and that under mild conditions an interpolating function can be found such that the extended model fits any initial forward curve. This allows us to compute value adjustments (i.e. XVAs) consistently, by solving the corresponding ‘pre-default’ BSDE. As an application, we compute the price and value adjustments for a basis swap, and study the model risk associated to different interpolating functions.
Citation: Antonis Papapantoleon, Robert Wardenga. Continuous tenor extension of affine LIBOR models with multiple curves and applications to XVA. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 1-. doi: 10.1186/s41546-017-0025-4
References:
 [1] Beveridge, C, Joshi, M:Interpolation schemes in the displaced-diffusion LIBOR market model. SIAM. J. Finan. Math. 3, 593-604 (2012) [2] Bichuch, M, Capponi, A, Sturm, S:Arbitrage-free XVA. Math.Finan. (2016). https://arxiv.org/abs/1608.02690 [3] Björk, T:Arbitrage Theory in Continuous Time, 3rd edition. Oxford University Press, Chichester (2009) [4] Brigo, D, Morini, M, Pallavicini, A:Counterparty Credit Risk, Collateral and Funding:with Pricing Cases for all Asset Classes. Wiley (2013) [5] Crépey, S:Bilateral Counterparty risk under funding constraints-Part I:Pricing. Math. Finan. 25, 1-22(2015a) [6] Crépey, S:Bilateral Counterparty risk under funding constraints-Part II:CVA. Math. Finan. 25, 23-50(2015b) [7] Crépey, S, Bielecki, TR:Counterparty Risk and Funding:A Tale of two Puzzles. Chapman & Hall/CRC Financial Mathematics Series. CRC Press, Boca Raton (2014). With an introductory dialogue by Damiano Brigo [8] Crépey, S, Grbac, Z, Nguyen, H-N:A multiple-curve HJM model of interbank risk. Math. Financ. Econ. 6, 155-190 (2012) [9] Crépey, S, Gerboud, R, Grbac, Z, Ngor, N:Counterparty risk and funding:The four wings of the TVA. Int. J. Theor. Appl. Financ. 16(1350006) (2013) [10] Crépey, S, Grbac, Z, Ngor, N, Skovmand, D:A Lévy HJM multiple-curve model with application to CVA computation. Quant. Financ. 15, 401-419 (2015) [11] Cuchiero, C, Fontana, C, Gnoatto, A:Affine multiple yield curve models. Preprint. arXiv:1603.00527(2016) [12] Duffie, D, Filipovic, D, Schachermayer, W:Affine processes and applications in finance. Ann. Appl. Probab. 13, 984-1053 (2003) [13] Filipović, D:Time-inhomogeneous affine processes. Stoch. Process. Appl. 115, 639-659 (2005) [14] Glau, K, Grbac, Z, Papapantoleon, A:A unified view of LIBOR models. In:Kallsen, J, Papapantoleon, A (eds.) Advanced Modelling in Mathematical Finance-In Honour of Ernst Eberlein, pp. 423-452. Springer, Cham (2016) [15] Grbac, Z, Runggaldier, WJ:Interest Rate Modeling:Post-Crisis Challenges and Approaches. Springer, Cham (2015) [16] Grbac, Z, Papapantoleon, A, Schoenmakers, J, Skovmand, D:Affine LIBOR models with multiple curves:theory, examples and calibration. SIAM. J. Financ. Math. 6, 984-1025 (2015) [17] Jacod, J, Shiryaev, AN:Limit Theorems for Stochastic Processes, 2nd edition. Springer, Berlin Heidelberg (2003) [18] Keller-Ressel, M:Affine Processes:Theory and Applications to Finance. PhD thesis, TU Vienna (2008) [19] Keller-Ressel, M:Affine LIBOR models with continuous tenor (2009). Unpublished manuscript Keller-Ressel, M, Papapantoleon, A, Teichmann, J:The affine LIBOR models. Math. Financ. 23, 627-658(2013) [20] Mercurio, F:Interest rates and the credit crunch:New formulas and market models. Preprint. SSRN/1332205 (2009) [21] Mercurio, F:A LIBOR market model with a stochastic basis. Risk. 84-89 (2010) [22] Musiela, M, Rutkowski, M:Continuous-time term structure models:forward measure approach. Financ. Stoch. 1, 261-291 (1997) [23] Musiela, M, Rutkowski, M::Martingale Methods in Financial Modelling, 2nd edition. Springer, Berlin Heidelberg (2005) [24] Papapantoleon, A:Old and new approaches to LIBOR modeling. Stat. Neerlandica. 64, 257-275 (2010)

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References:
 [1] Beveridge, C, Joshi, M:Interpolation schemes in the displaced-diffusion LIBOR market model. SIAM. J. Finan. Math. 3, 593-604 (2012) [2] Bichuch, M, Capponi, A, Sturm, S:Arbitrage-free XVA. Math.Finan. (2016). https://arxiv.org/abs/1608.02690 [3] Björk, T:Arbitrage Theory in Continuous Time, 3rd edition. Oxford University Press, Chichester (2009) [4] Brigo, D, Morini, M, Pallavicini, A:Counterparty Credit Risk, Collateral and Funding:with Pricing Cases for all Asset Classes. Wiley (2013) [5] Crépey, S:Bilateral Counterparty risk under funding constraints-Part I:Pricing. Math. Finan. 25, 1-22(2015a) [6] Crépey, S:Bilateral Counterparty risk under funding constraints-Part II:CVA. Math. Finan. 25, 23-50(2015b) [7] Crépey, S, Bielecki, TR:Counterparty Risk and Funding:A Tale of two Puzzles. Chapman & Hall/CRC Financial Mathematics Series. CRC Press, Boca Raton (2014). With an introductory dialogue by Damiano Brigo [8] Crépey, S, Grbac, Z, Nguyen, H-N:A multiple-curve HJM model of interbank risk. Math. Financ. Econ. 6, 155-190 (2012) [9] Crépey, S, Gerboud, R, Grbac, Z, Ngor, N:Counterparty risk and funding:The four wings of the TVA. Int. J. Theor. Appl. Financ. 16(1350006) (2013) [10] Crépey, S, Grbac, Z, Ngor, N, Skovmand, D:A Lévy HJM multiple-curve model with application to CVA computation. Quant. Financ. 15, 401-419 (2015) [11] Cuchiero, C, Fontana, C, Gnoatto, A:Affine multiple yield curve models. Preprint. arXiv:1603.00527(2016) [12] Duffie, D, Filipovic, D, Schachermayer, W:Affine processes and applications in finance. Ann. Appl. Probab. 13, 984-1053 (2003) [13] Filipović, D:Time-inhomogeneous affine processes. Stoch. Process. Appl. 115, 639-659 (2005) [14] Glau, K, Grbac, Z, Papapantoleon, A:A unified view of LIBOR models. In:Kallsen, J, Papapantoleon, A (eds.) Advanced Modelling in Mathematical Finance-In Honour of Ernst Eberlein, pp. 423-452. Springer, Cham (2016) [15] Grbac, Z, Runggaldier, WJ:Interest Rate Modeling:Post-Crisis Challenges and Approaches. Springer, Cham (2015) [16] Grbac, Z, Papapantoleon, A, Schoenmakers, J, Skovmand, D:Affine LIBOR models with multiple curves:theory, examples and calibration. SIAM. J. Financ. Math. 6, 984-1025 (2015) [17] Jacod, J, Shiryaev, AN:Limit Theorems for Stochastic Processes, 2nd edition. Springer, Berlin Heidelberg (2003) [18] Keller-Ressel, M:Affine Processes:Theory and Applications to Finance. PhD thesis, TU Vienna (2008) [19] Keller-Ressel, M:Affine LIBOR models with continuous tenor (2009). Unpublished manuscript Keller-Ressel, M, Papapantoleon, A, Teichmann, J:The affine LIBOR models. Math. Financ. 23, 627-658(2013) [20] Mercurio, F:Interest rates and the credit crunch:New formulas and market models. Preprint. SSRN/1332205 (2009) [21] Mercurio, F:A LIBOR market model with a stochastic basis. Risk. 84-89 (2010) [22] Musiela, M, Rutkowski, M:Continuous-time term structure models:forward measure approach. Financ. Stoch. 1, 261-291 (1997) [23] Musiela, M, Rutkowski, M::Martingale Methods in Financial Modelling, 2nd edition. Springer, Berlin Heidelberg (2005) [24] Papapantoleon, A:Old and new approaches to LIBOR modeling. Stat. Neerlandica. 64, 257-275 (2010)
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