Heat modulated affine stochastic volatility models for forward curve dynamics
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914953356640256 |
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| author | Karbach, Sven |
| author_facet | Karbach, Sven |
| contents | We present a function-valued stochastic volatility model designed to capture the continuous-time evolution of forward curves in fixed-income or commodity markets. The dynamics of the (logarithmic) forward curves are defined by a Heath-Jarrow-Morton-Musiela stochastic partial differential equation modulated by an instantaneous volatility process that describes the second-order moment structure of forwards with different time-to-maturity. We propose to model the operator-valued instantaneous covariance by an affine process on the cone of positive trace-class operators with drift given by the Lyapunov operator of the Laplacian. The so defined infinite-rank stochastic volatility model is analytically tractable due to its affine structure and allows to model maturity specific risk and volatility clustering in forward markets. Furthermore, we introduce a numerically feasible spectral Galerkin approximation of the associated operator-valued generalized Riccati equations and study the robustness of the model with respect to finite-rank approximations by providing explicit error bounds on the approximation error. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13070 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Heat modulated affine stochastic volatility models for forward curve dynamics Karbach, Sven Mathematical Finance Probability Pricing of Securities We present a function-valued stochastic volatility model designed to capture the continuous-time evolution of forward curves in fixed-income or commodity markets. The dynamics of the (logarithmic) forward curves are defined by a Heath-Jarrow-Morton-Musiela stochastic partial differential equation modulated by an instantaneous volatility process that describes the second-order moment structure of forwards with different time-to-maturity. We propose to model the operator-valued instantaneous covariance by an affine process on the cone of positive trace-class operators with drift given by the Lyapunov operator of the Laplacian. The so defined infinite-rank stochastic volatility model is analytically tractable due to its affine structure and allows to model maturity specific risk and volatility clustering in forward markets. Furthermore, we introduce a numerically feasible spectral Galerkin approximation of the associated operator-valued generalized Riccati equations and study the robustness of the model with respect to finite-rank approximations by providing explicit error bounds on the approximation error. |
| title | Heat modulated affine stochastic volatility models for forward curve dynamics |
| topic | Mathematical Finance Probability Pricing of Securities |
| url | https://arxiv.org/abs/2409.13070 |