PDEs for pricing interest rate derivatives under the new generalized forward market model (FMM). (English) Zbl 07894791
Summary: In this article we derive partial differential equations (PDEs) for pricing interest rate derivatives under the generalized Forward Market Model (FMM) recently presented by A. Lyashenko and F. Mercurio in [“LIBOR replacement: a modelling framework for in-arrears term rates”, Risk 57-62 (2019)] to model the dynamics of the Risk Free Rates (RFRs) that are replacing the traditional IBOR rates in the financial industry. Moreover, for the numerical solution of the proposed PDEs formulation, we develop some adaptations of the finite differences methods developed in [J. G. López-Salas et al., SIAM J. Sci. Comput. 43, No. 1, B30–B54 (2021; Zbl 1456.65073)] that are very suitable to treat the presence of spatial mixed derivatives. This work is the first article in the literature where PDE methods are used to value RFR derivatives. Additionally, Monte Carlo-based methods will be designed and the results are compared with those obtained by the numerical solution of PDEs.
MSC:
| 91-XX | Game theory, economics, finance, and other social and behavioral sciences |
| 35-XX | Partial differential equations |
Keywords:
IBOR replacement; generalized forward market model; forward rates; finite differences; AMFR-W methodsCitations:
Zbl 1456.65073References:
| [1] | Lyashenko, A.; Mercurio, F., LIBOR replacement: a modelling framework for in-arrears term rates, Risk, 57-62, June 2019 |
| [2] | López-Salas, J. G.; Pérez-Rodríguez, S.; Vázquez, C., AMFR-W numerical methods for solving high dimensional SABR/LIBOR PDE models, SIAM J. Sci. Comput., 43, B30-B54, 2021 · Zbl 1456.65073 |
| [3] | Fabrizi, M.; Huan, X.; Parbonetti, A., When LIBOR becomes LIEBOR: reputational penalties and bank contagion, Financ. Rev., 56, 1, 157-178, 2021 |
| [4] | Huan, X.; Previts, G. J.; Parbonetti, A., Understanding the LIBOR scandal: the historical, the ethical, and the technological, J. Bank. Regul., 24, 403-419, 2023 |
| [5] | McConnell, P., Systemic operational risk: the LIBOR manipulation scandal, J. Oper. Risk, 8, 3, 59-99, 2013 |
| [6] | Gandhi, P.; Golez, B.; Jackwerth, J. C.; Plazzi, A., Financial market misconduct and public enforcement: the case of libor manipulation, Manag. Sci., 65, 11, 5268-5289, 2013 |
| [7] | Duffie, D., Notes on LIBOR conversion, Available at |
| [8] | Huan, W.; Todorov, K., The post-Libor world: a global view from the BIS derivatives statistics, BIS Q. Rev., 2022 |
| [9] | Brace, A.; Gatarek, D.; Musiela, M., The market model of interest rate dynamics, Math. Finance, 7, 2, 127-155, 1997 · Zbl 0884.90008 |
| [10] | Jamshidian, F., LIBOR and swap market models and measures, Finance Stoch., 1, 293-330, 1997 · Zbl 0888.60038 |
| [11] | Brigo, D.; Mercurio, F., Interest Rate Models - Theory and Practice. With Smile, Inflation and Credit, 2007, Springer |
| [12] | Pietersz, R., PDE pricing for BGM, February 28, 2002, Available at SSRN: |
| [13] | Pascucci, A.; Suárez, M.; Vázquez, C., Mathematical analysis and numerical methods for a partial differential equation model governing a ratchet cap pricing problem, Math. Models Methods Appl. Sci., 21, 1479-1498, 2011 · Zbl 1222.91061 |
| [14] | Suárez, M.; Vázquez, C., A numerical method for pricing spread options on LIBOR rates with a PDE model, Math. Comput. Model., 52, 1074-1080, 2010 · Zbl 1205.91170 |
| [15] | Suárez, M.; Vázquez, C., Numerical solution of a PDE model for a rachet-cap pricing with BGM interest rate dynamic, Appl. Math. Comput., 218, 5217-5230, 2012 · Zbl 1239.91174 |
| [16] | Mercurio, F.; Morini, M., No-arbitrage dynamics for a tractable SABR term structure libor model, (Modeling Interest Rates: Advances in Derivatives Pricing, 2009, Risk Books) |
| [17] | Rebonato, R.; White, R., Linking caplets and swaptions prices in the LMM-SABR model, J. Comput. Finance, 13, 2, 19-45, 2009 · Zbl 1181.91317 |
| [18] | Rebonato, R., The Q-measure dynamics of forward rates, Annu. Rev. Financ. Econ., 15, 493-522, 2023 |
| [19] | Brace, A., Engineering BGM, Chapman & Hall/CRC Financial Mathematics Series, 2008 · Zbl 1149.91032 |
| [20] | López-Salas, J. G.; Vázquez, C., PDE formulation of some SABR/LIBOR market models and its numerical solution with a sparse grid combination technique, Comput. Math. Appl., 75, 1616-1634, 2018 · Zbl 1409.91277 |
| [21] | Lyashenko, A.; Mercurio, F., LIBOR replacement II: completing the generalized Forward Market Model, Risk, 61-66, July 2020 |
| [22] | González-Pinto, S.; Hairer, E.; Hernández-Abreu, D.; Pérez-Rodríguez, S., AMF-type W-methods for parabolic problems with mixed derivatives, SIAM J. Sci. Comput., 40, 5, A2905-A2929, 2018 · Zbl 1397.65159 |
| [23] | González-Pinto, S.; Hernández-Abreu, D.; Pérez-Rodríguez, S., AMFR-W-methods for parabolic problems with mixed derivates. Applications to the Heston model, J. Comput. Appl. Math., 387, 1-20, 2021 · Zbl 1458.65107 |
| [24] | Mikosh, T., Elementary Stochastic Calculus with Finance in View, 1998, World Scientific · Zbl 0934.60002 |
| [25] | Andresen, L.; Piterbarg, V., Interest Rate Modeling, 2010, Atlantic Financial Press |
| [26] | Glasserman, P.; Zhao, X., Arbitrage-free interpolation of lognormal forward LIBOR and swap rate models, Finance Stoch., 4, 35-68, 2000 · Zbl 0947.60050 |
| [27] | Oksendal, B., Stochastic Differential Equations. An Introduction with Applications, Universitext, 1998, Springer · Zbl 0897.60056 |
| [28] | in ’t Hout, K., Numerical Partial Differential Equations in Finance Explained, 2017, Palgrave Macmillan: Palgrave Macmillan London |
| [29] | Wilmott, P.; Dewynne, J.; Howison, S., Option Pricing. Mathematical Models and Computation, 1993, Oxford Financial Press |
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.
