Weak error estimates for rough volatility models
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909690169917440 |
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| author | Friz, Peter K. Salkeld, William Wagenhofer, Thomas |
| author_facet | Friz, Peter K. Salkeld, William Wagenhofer, Thomas |
| contents | We consider a class of stochastic processes with rough stochastic volatility, examples of which include the rough Bergomi and rough Stein-Stein model, that have gained considerable importance in quantitative finance.
A basic question for such (non-Markovian) models concerns efficient numerical schemes. While strong rates are well understood (order $H$), we tackle here the intricate question of weak rates. Our main result asserts that the weak rate, for a reasonably large class of test function, is essentially of order $\min \{ 3H+\tfrac12, 1 \}$ where $H \in (0,1/2]$ is the Hurst parameter of the fractional Brownian motion that underlies the rough volatility process.
Interestingly, the phase transition at $H=1/6$ is related to the correlation between the two driving factors, and thus gives additional meaning to a quantity already of central importance in stochastic volatility modelling.Our results are complemented by a lower bound which show that the obtained weak rate is indeed optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_01591 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Weak error estimates for rough volatility models Friz, Peter K. Salkeld, William Wagenhofer, Thomas Computational Finance Probability 60L90 (Primary) 60G22, 91G20 (Secondary) We consider a class of stochastic processes with rough stochastic volatility, examples of which include the rough Bergomi and rough Stein-Stein model, that have gained considerable importance in quantitative finance. A basic question for such (non-Markovian) models concerns efficient numerical schemes. While strong rates are well understood (order $H$), we tackle here the intricate question of weak rates. Our main result asserts that the weak rate, for a reasonably large class of test function, is essentially of order $\min \{ 3H+\tfrac12, 1 \}$ where $H \in (0,1/2]$ is the Hurst parameter of the fractional Brownian motion that underlies the rough volatility process. Interestingly, the phase transition at $H=1/6$ is related to the correlation between the two driving factors, and thus gives additional meaning to a quantity already of central importance in stochastic volatility modelling.Our results are complemented by a lower bound which show that the obtained weak rate is indeed optimal. |
| title | Weak error estimates for rough volatility models |
| topic | Computational Finance Probability 60L90 (Primary) 60G22, 91G20 (Secondary) |
| url | https://arxiv.org/abs/2212.01591 |