Affine multiple yield curve models. (English) Zbl 1411.91589
Summary: We provide a general and tractable framework under which all multiple yield curve modeling approaches based on affine processes, be it short rate, Libor market, or Heath-Jarrow-Morton modeling, can be consolidated. We model a numéraire process and multiplicative spreads between Libor rates and simply compounded overnight indexed swap rates as functions of an underlying affine process. Besides allowing for ordered spreads and an exact fit to the initially observed term structures, this general framework leads to tractable valuation formulas for caplets and swaptions and embeds all existing multicurve affine models. The proposed approach also gives rise to new developments, such as a short rate type model driven by a Wishart process, for which we derive a closed-form pricing formula for caplets. The empirical performance of two specifications of our framework is illustrated by calibration to market data.
MSC:
| 91G30 | Interest rates, asset pricing, etc. (stochastic models) |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
Keywords:
affine processes; forward rate agreement; Libor rate; multiple yield curves; multiplicative spreadReferences:
| [1] | Alfonsi, A., & Ahdida, A. (2013). Exact and high‐order discretization schemes for Wishart processes and their affine extensions. Annals of Applied Probability, 23(3), 1025-1073. · Zbl 1269.65003 |
| [2] | Ametrano, F. M., & Bianchetti, M. (2013). Everything you always wanted to know about multiple interest rate curve bootstrapping but were afraid to ask. Preprint. Retrieved from https://doi.org/ssrn.com/abstract=2219548 · doi:https://doi.org/ssrn.com/abstract=2219548 |
| [3] | Biagini, F., Gnoatto, A., & Härtel, M. (2018). Long‐term yield in an affine HJM framework on \(S_d^+\). Applied Mathematics and Optimization, 77, 405-441. · Zbl 1416.91383 |
| [4] | Bianchetti, M. (2010). Two curves, one price. Risk Magazine, 74-80. Retrieved from https://doi.org/10.2139/ssrn.1334356 · doi:10.2139/ssrn.1334356 |
| [5] | Bianchetti, M. (2011). Interest rates after the credit crunch. Presentation, Scuola Superiore Sant’Anna, Pisa. |
| [6] | Bormetti, G., Brigo, D., Francischello, M., & Pallavicini, A. (2016). Impact of multiple‐curve dynamics in credit valuation adjustments. In K.Glau (ed.), Z.Grbac (ed.), M.Scherer (ed.), & R.Zagst (ed.) (Eds.), Innovations in derivatives markets (pp. 251-266). Berlin, Germany: Springer. · Zbl 1398.91566 |
| [7] | Brigo, D., & Mercurio, F. (2001). A deterministic‐shift extension of analytically‐tractable and time‐homogeneous short‐rate models. Finance and Stochastics, 5(3), 369-387. · Zbl 0978.91032 |
| [8] | Bruti‐Liberati, N., Nikitopoulos‐Sklibosios, C., & Platen, E. (2010). Real‐world jump‐diffusion term structure models. Quantitative Finance, 10(1), 23-37. · Zbl 1202.91332 |
| [9] | Caldana, R., Fusai, G., & Gambaro, A. M. (2017). Approximate pricing of swaptions in affine and quadratic models. Quantitative Finance, 17(9), 1325-1345. · Zbl 1402.91781 |
| [10] | Caldana, R., Fusai, G., Gnoatto, A., & Grasselli, M. (2016). General closed‐form basket option pricing bounds. Quantitative Finance, 16(4), 535-554. · Zbl 1468.91162 |
| [11] | Carr, P., & Madan, D. B. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance, 2(4), 61-73. |
| [12] | Castano‐Martinez, A., & Lopez‐Blazquez, F. (2005). Distribution of a sum of weighted noncentral chi‐square variables. TEST, 14(2), 397-415. · Zbl 1087.62027 |
| [13] | Cheridito, P., Filipović, D., & Kimmel, R. L. (2010). A note on the Dai‐Singleton canonical representation of affine term structure models. Mathematical Finance, 20(3), 509-519. · Zbl 1194.91185 |
| [14] | Collin‐Dufresne, P., & Solnik, B. (2001). On the term structure of default premia in the swap and libor markets. Journal of Finance, 56(3), 1095-1115. |
| [15] | Crépey, S., Grbac, Z., Ngor, N., & Skovmand, D. (2015). A Lévy HJM multiple‐curve model with application to CVA computation. Quantitative Finance, 15(3), 401-419. · Zbl 1398.91573 |
| [16] | Crépey, S., Macrina, A., Nguyen, H., & Skovmand, D. (2016). Rational multi‐curve models with counterparty‐risk valuation adjustments. Quantitative Finance, 16(6), 847-866. · Zbl 1468.91180 |
| [17] | Cuchiero, C. (2011). Affine and polynomial processes (PhD thesis). ETH Zurich, Zurich. |
| [18] | Cuchiero, C., Filipović, D., Mayerhofer, E., & Teichmann, J. (2011). Affine processes on positive semidefinite matrices. Annals of Applied Probability, 21(2), 397-463. · Zbl 1219.60068 |
| [19] | Cuchiero, C., Fontana, C., & Gnoatto, A. (2016). A general HJM framework for multiple yield curve modeling. Finance and Stochastics, 20(2), 267-320. · Zbl 1376.91166 |
| [20] | Cuchiero, C., Klein, I., & Teichmann, J. (2016). A new perspective on the fundamental theorem of asset pricing for large financial markets. Theory of Probability and its Applications, 60(4), 561-579. · Zbl 1352.91035 |
| [21] | Cuchiero, C., & Teichmann, J. (2013). Path properties and regularity of affine processes on general state spaces. In C.Donati‐Martin (ed.), A.Lejay (ed.), & A.Rouault (ed.) (Eds.), Séminaire de probabilités XLV, Lecture Notes in Mathematics (Vol. 2078 , pp. 201-244). Berlin, Germany: Springer. · Zbl 1287.60091 |
| [22] | DaFonseca, J., Gnoatto, A., & Grasselli, M. (2013). A flexible matrix Libor model with smiles. Journal of Economic Dynamics and Control, 37(4), 774-793. · Zbl 1348.91267 |
| [23] | Döberlein, F., & Schweizer, M. (2001). On savings accounts in semimartingale term structure models. Stochastic Analysis and Its Applications, 19(4), 605-626. · Zbl 1017.91042 |
| [24] | Duffie, D., Filipović, D., & Schachermayer, W. (2003). Affine processes and applications in finance. Annals of Applied Probability, 13(3), 984-1053. · Zbl 1048.60059 |
| [25] | Filipović, D. (2005). Time‐inhomogeneous affine processes. Stochastic Processes and their Applications, 115(4), 639-659. · Zbl 1079.60068 |
| [26] | Filipović, D., Larsson, M., & Trolle, A. B. (2017). Linear‐rational term structure models. Journal of Finance, 72(2), 655-704. |
| [27] | Filipović, D., & Trolle, A. B. (2013). The term structure of interbank risk. Journal of Financial Economics, 109(3), 707-733. |
| [28] | Flesaker, B., & Hughston, L. P. (1996). Positive interest. Risk, 9(1), 46-49. |
| [29] | Fries, C. (2015). Finmath lib v1.3.0. Retrieved from http://www.finmath.net |
| [30] | Gallitschke, J., Seifried, S., & Seifried, F. (2017). Post‐crisis interest rates: Xibor mechanics and basis spreads. Journal of Banking & Finance, 78, 142-152. |
| [31] | Gnoatto, A. (2012). The Wishart short rate model. International Journal of Theoretical and Applied Finance, 15(8). Retrieved from https://doi.org/10.1142/S0219024912500562 · Zbl 1260.91247 · doi:10.1142/S0219024912500562 |
| [32] | Grasselli, M., & Miglietta, G. (2016). A flexible spot multiple‐curve model. Quantitative Finance, 6(10), 1465-1477. |
| [33] | Grbac, Z., Meneghello, L., & Runggaldier, W. J. (2016). Derivative pricing for a multi‐curve extension of the Gaussian, exponentially quadratic short rate model. In K.Glau (ed.), Z.Grbac (ed.), M. Scherer, & R.Zagst (ed.) (Eds.), Innovations in derivatives markets (pp. 191-226). Berlin, Germany: Springer. · Zbl 1398.91594 |
| [34] | Grbac, Z., Papapantoleon, A., Schoenmakers, J., & Skovmand, D. (2015). Affine LIBOR models with multiple curves: Theory, examples and calibration. SIAM Journal on Financial Mathematics, 6, 984-1025. · Zbl 1338.91143 |
| [35] | Grbac, Z., & Runggaldier, W. J. (2015). Interest rate modeling: Post‐crisis challenges and approaches. In SpringerBriefs in quantitative finance. Berlin, Germany: Springer. · Zbl 1418.91553 |
| [36] | Green, A. (2015). XVA: Credit, funding and capital valuation adjustments. Wiley finance. Chichester, England: John Wiley and Sons. |
| [37] | Henrard, M. (2010). The irony in the derivatives discounting part II: The crisis. Wilmott Journal, 2(6), 301-316. |
| [38] | Henrard, M. (2014). Interest rate modelling in the multi‐curve framework. Basingstoke, England: Palgrave Macmillan. |
| [39] | Jamshidian, F. (1989). An exact bond option pricing formula. Journal of Finance, 44(1), 205-209. |
| [40] | Jamshidian, F. (1996). Bond, futures and option evaluation in the quadratic interest rate model. Applied Mathematical Finance, 3(2), 93-115. · Zbl 1097.91525 |
| [41] | Kallsen, J. (2006). A didactic note on affine stochastic volatility models. In Y.Kabanov (ed.), R.Lipster (ed.), & J.Stoyanov (ed.) (Eds.), From stochastic calculus to mathematical finance—The Shiryaev Festschrift (pp. 343-368). Berlin, Germany: Springer. · Zbl 1104.60024 |
| [42] | Kang, C., & Kang, W. (2013). Exact simulation of Wishart multidimensional stochastic volatility model. Preprint. Retrieved from http://arxiv.org/abs/1309.0557 |
| [43] | Keller‐Ressel, M. (2009a). Affine Libor models with continuous tenor. Unpublished manuscript. |
| [44] | Keller‐Ressel, M. (2009b). Affine processes—Theory and applications in finance (PhD thesis). Wien: Vienna University of Technology. |
| [45] | Keller‐Ressel, M., & Mayerhofer, E. (2015). Exponential moments of affine processes. Annals of Applied Probability, 25(2), 714-752. · Zbl 1332.60115 |
| [46] | Keller‐Ressel, M., Papapantoleon, A., & Teichmann, J. (2013). The affine LIBOR models. Mathematical Finance, 23(4), 627-658. · Zbl 1275.91140 |
| [47] | Keller‐Ressel, M., Schachermayer, W., & Teichmann, J. (2011). Affine processes are regular. Probability Theory and Related Fields, 151(3-4), 591-611. · Zbl 1235.60093 |
| [48] | Kenyon, C. (2010). Post‐shock short‐rate pricing. Risk, 83-87. Retrieved from https://doi.org/10.2139/ssrn.1558429 · doi:10.2139/ssrn.1558429 |
| [49] | Kijima, M., Tanaka, K., & Wong, T. (2009). A multi‐quality model of interest rates. Quantitative Finance, 9(2), 133-145. · Zbl 1158.91353 |
| [50] | Kim, D. H. (2014). Swaption pricing in affine and other models. Mathematical Finance, 24(4), 790-820. · Zbl 1314.91212 |
| [51] | Lee, R. (2004). Option pricing by transform methods: Extensions, unification and error control. Journal of Computational Finance, 7(3), 51-86. |
| [52] | Mercurio, F. (2013). A Libor market model with a stochastic basis. Risk, December, 96-101. |
| [53] | Mercurio, F., & Xie, Z. (2012). The basis goes stochastic. Risk, December, 78-83. |
| [54] | Moreni, N., & Pallavicini, A. (2014). Parsimonious HJM modelling for multiple yield‐curve dynamics. Quantitative Finance, 14(2), 199-210. · Zbl 1294.91181 |
| [55] | Morino, L., & Runggaldier, W. J. (2014). On multicurve models for the term structure. In R.Dieci (ed.), X.‐Z.He (ed.), & C.Hommes (ed.) (Eds.), Nonlinear economic dynamics and financial modelling: Essays in honour of carl chiarella. Berlin, Germany: Springer. · Zbl 1305.91013 |
| [56] | Nguyen, T., & Seifried, F. (2015a). The multi‐curve potential model. International Journal of Theoretical and Applied Finance, 18(7). Retrieved from https://papers.ssrn.com/sol3/papers.cfm?abstract id=2502374 · Zbl 1337.91120 |
| [57] | Nguyen, T. A., & Seifried, F. T. (2015b). The affine rational potential model. Preprint. Retrieved from http://ssrn.com/abstract=t=2707081 |
| [58] | Papapantoleon, A., & Wardenga, R. (2018). Continuous tenor extension of affine Libor models with multiple curves and applications to XVA. Probability, Uncertainty and Quantitative Risk, 3(1), 1-28. · Zbl 1432.91129 |
| [59] | Platen, E., & Heath, D. (2006). A benchmark approach to quantitative finance. Berlin, Germany: Springer. · Zbl 1104.91041 |
| [60] | Platen, E., & Tappe, S. (2015). Real‐world forward rate dynamics with affine realizations. Stochastic Analysis and Its Applications, 33(4), 573-608. · Zbl 1335.91094 |
| [61] | Rogers, L. C. G. (1997). The potential approach to the term structure of interest rates and foreign exchange rates. Mathematical Finance, 7(2), 157-176. · Zbl 0884.90046 |
| [62] | Singleton, K. J., & Umantsev, L. (2002). Pricing coupon‐bond options and swaptions in affine term structure models. Mathematical Finance, 12(4), 427-446. · Zbl 1047.91036 |
| [63] | Sokol, A. (2014). Long‐term portfolio simulation: For XVA, limits, liquidity and regulatory capital. London, England: Risk Books. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
