PDE formulation of some SABR/LIBOR market models and its numerical solution with a sparse grid combination technique
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909276273901568 |
|---|---|
| author | López-Salas, J. G. Vázquez, C. |
| author_facet | López-Salas, J. G. Vázquez, C. |
| contents | SABR models have been used to incorporate stochastic volatility to LIBOR market models (LMM) in order to describe interest rate dynamics and price interest rate derivatives. From the numerical point of view, the pricing of derivatives with SABR/LIBOR market models (SABR/LMMs) is mainly carried out with Monte Carlo simulation. However, this approach could involve excessively long computational times. For first time in the literature, in the present paper we propose an alternative pricing based on partial differential equations (PDEs). Thus, we pose original PDE formulations associated to the SABR/LMMs proposed by Hagan \cite{haganSABRLIBOR}, Mercurio \& Morini \cite{mercurioMorini} and Rebonato \cite{rebonatoWhite}. Moreover, as the PDEs associated to these SABR/LMMs are high dimensional in space, traditional full grid methods (like standard finite differences or finite elements) are not able to price derivatives over more than three or four underlying interest rates. In order to overcome this curse of dimensionality, a sparse grid combination technique is proposed. A comparison between Monte Carlo simulation results and the ones obtained with the sparse grid technique illustrates the performance of the method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00563 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | PDE formulation of some SABR/LIBOR market models and its numerical solution with a sparse grid combination technique López-Salas, J. G. Vázquez, C. Analysis of PDEs SABR models have been used to incorporate stochastic volatility to LIBOR market models (LMM) in order to describe interest rate dynamics and price interest rate derivatives. From the numerical point of view, the pricing of derivatives with SABR/LIBOR market models (SABR/LMMs) is mainly carried out with Monte Carlo simulation. However, this approach could involve excessively long computational times. For first time in the literature, in the present paper we propose an alternative pricing based on partial differential equations (PDEs). Thus, we pose original PDE formulations associated to the SABR/LMMs proposed by Hagan \cite{haganSABRLIBOR}, Mercurio \& Morini \cite{mercurioMorini} and Rebonato \cite{rebonatoWhite}. Moreover, as the PDEs associated to these SABR/LMMs are high dimensional in space, traditional full grid methods (like standard finite differences or finite elements) are not able to price derivatives over more than three or four underlying interest rates. In order to overcome this curse of dimensionality, a sparse grid combination technique is proposed. A comparison between Monte Carlo simulation results and the ones obtained with the sparse grid technique illustrates the performance of the method. |
| title | PDE formulation of some SABR/LIBOR market models and its numerical solution with a sparse grid combination technique |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.00563 |