Rough volatility, path-dependent PDEs and weak rates of convergence
In the setting of stochastic Volterra equations, and in particular rough volatility models, we show that conditional expectations are the unique classical solutions to path-dependent PDEs. The latter arise from the functional It\^o formula developed by [Viens, F., & Zhang, J. (2019). A martingal...
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| creator | Bonesini, Ofelia Jacquier, Antoine Pannier, Alexandre |
| description | In the setting of stochastic Volterra equations, and in particular rough
volatility models, we show that conditional expectations are the unique
classical solutions to path-dependent PDEs. The latter arise from the
functional It\^o formula developed by [Viens, F., & Zhang, J. (2019). A
martingale approach for fractional Brownian motions and related path dependent
PDEs. Ann. Appl. Probab.]. We then leverage these tools to study weak rates of
convergence for discretised stochastic integrals of smooth functions of a
Riemann-Liouville fractional Brownian motion with Hurst parameter $H \in
(0,1/2)$. These integrals approximate log-stock prices in rough volatility
models. We obtain the optimal weak error rates of order~$1$ if the test
function is quadratic and of order~$(3H+1/2)\wedge 1$ if the test function is
five times differentiable; in particular these conditions are independent of
the value of~$H$. |
| doi_str_mv | 10.48550/arxiv.2304.03042 |
| format | Article |
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volatility models, we show that conditional expectations are the unique
classical solutions to path-dependent PDEs. The latter arise from the
functional It\^o formula developed by [Viens, F., & Zhang, J. (2019). A
martingale approach for fractional Brownian motions and related path dependent
PDEs. Ann. Appl. Probab.]. We then leverage these tools to study weak rates of
convergence for discretised stochastic integrals of smooth functions of a
Riemann-Liouville fractional Brownian motion with Hurst parameter $H \in
(0,1/2)$. These integrals approximate log-stock prices in rough volatility
models. We obtain the optimal weak error rates of order~$1$ if the test
function is quadratic and of order~$(3H+1/2)\wedge 1$ if the test function is
five times differentiable; in particular these conditions are independent of
the value of~$H$.</description><identifier>DOI: 10.48550/arxiv.2304.03042</identifier><language>eng</language><subject>Mathematics - Probability ; Quantitative Finance - Mathematical Finance</subject><creationdate>2023-04</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,777,882</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2304.03042$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2304.03042$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Bonesini, Ofelia</creatorcontrib><creatorcontrib>Jacquier, Antoine</creatorcontrib><creatorcontrib>Pannier, Alexandre</creatorcontrib><title>Rough volatility, path-dependent PDEs and weak rates of convergence</title><description>In the setting of stochastic Volterra equations, and in particular rough
volatility models, we show that conditional expectations are the unique
classical solutions to path-dependent PDEs. The latter arise from the
functional It\^o formula developed by [Viens, F., & Zhang, J. (2019). A
martingale approach for fractional Brownian motions and related path dependent
PDEs. Ann. Appl. Probab.]. We then leverage these tools to study weak rates of
convergence for discretised stochastic integrals of smooth functions of a
Riemann-Liouville fractional Brownian motion with Hurst parameter $H \in
(0,1/2)$. These integrals approximate log-stock prices in rough volatility
models. We obtain the optimal weak error rates of order~$1$ if the test
function is quadratic and of order~$(3H+1/2)\wedge 1$ if the test function is
five times differentiable; in particular these conditions are independent of
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volatility models, we show that conditional expectations are the unique
classical solutions to path-dependent PDEs. The latter arise from the
functional It\^o formula developed by [Viens, F., & Zhang, J. (2019). A
martingale approach for fractional Brownian motions and related path dependent
PDEs. Ann. Appl. Probab.]. We then leverage these tools to study weak rates of
convergence for discretised stochastic integrals of smooth functions of a
Riemann-Liouville fractional Brownian motion with Hurst parameter $H \in
(0,1/2)$. These integrals approximate log-stock prices in rough volatility
models. We obtain the optimal weak error rates of order~$1$ if the test
function is quadratic and of order~$(3H+1/2)\wedge 1$ if the test function is
five times differentiable; in particular these conditions are independent of
the value of~$H$.</abstract><doi>10.48550/arxiv.2304.03042</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Probability Quantitative Finance - Mathematical Finance |
| title | Rough volatility, path-dependent PDEs and weak rates of convergence |
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