WEAK ERROR RATES FOR OPTION PRICING UNDER LINEAR ROUGH VOLATILITY

dc.contributor.authorBayer, Christian
dc.contributor.authorHall, Eric Joseph
dc.contributor.authorTempone, Raul
dc.contributor.departmentApplied Mathematics and Computational Science Program
dc.contributor.departmentComputer, Electrical and Mathematical Science and Engineering (CEMSE) Division
dc.contributor.institutionWeierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany
dc.contributor.institutionMathematics Division, School of Science and Engineering, University of Dundee, Dundee, DD1 4HR, UK
dc.contributor.institutionChair of Mathematics for Uncertainty Quantification, RWTH Aachen University, Pontdriesch 14-16, 52062 Aachen, Germany
dc.date.accepted2022-11-01
dc.date.accessioned2023-01-22T12:21:00Z
dc.date.available2020-09-14T12:09:48Z
dc.date.available2023-01-22T12:21:00Z
dc.date.issued2023-01-19
dc.description.abstractIn quantitative finance, modeling the volatility structure of underlying assets is vital to pricing options. Rough stochastic volatility models, such as the rough Bergomi model [C. Bayer, P. K. Friz & J. Gatheral (2016) Pricing under rough volatility, Quantitative Finance 16 (6), 887–904, doi:10.1080/14697688.2015.1099717], seek to fit observed market data based on the observation that the log-realized variance behaves like a fractional Brownian motion with small Hurst parameter, [Formula: see text], over reasonable timescales. Both time series of asset prices and option-derived price data indicate that [Formula: see text] often takes values close to [Formula: see text] or less, i.e. rougher than Brownian motion. This change improves the fit to both option prices and time series of underlying asset prices while maintaining parsimoniousness. However, the non-Markovian nature of the driving fractional Brownian motion in rough volatility models poses severe challenges for theoretical and numerical analyses and for computational practice. While the explicit Euler method is known to converge to the solution of the rough Bergomi and similar models, its strong rate of convergence is only [Formula: see text]. We prove rate [Formula: see text] for the weak convergence of the Euler method for the rough Stein–Stein model, which treats the volatility as a linear function of the driving fractional Brownian motion, and, surprisingly, we prove rate one for the case of quadratic payoff functions. Indeed, the problem of weak convergence for rough volatility models is very subtle; we provide examples demonstrating the rate of convergence for payoff functions that are well approximated by second-order polynomials, as weighted by the law of the fractional Brownian motion, may be hard to distinguish from rate one empirically. Our proof uses Talay–Tubaro expansions and an affine Markovian representation of the underlying and is further supported by numerical experiments. These convergence results provide a first step toward deriving weak rates for the rough Bergomi model, which treats the volatility as a nonlinear function of the driving fractional Brownian motion.
dc.description.sponsorshipThis work was supported by the KAUST Office of Sponsored Research (OSR) under Award No. OSR2019-CRG8-4033 and the Alexander von Humboldt Foundation. R. Tempone is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering. A portion of this work was carried out while E. Hall was a Postdoctoral Research Scientist in the Chair of Mathematics for Uncertainty Quantification at RWTH Aachen University. C. Bayer gratefully acknowledges support by the German Research Council (DFG) via the Research Unit FOR 2402. We are grateful to Andreas Neuenkirch for pointing out the simpler proof in the case of quadratic payoffs presented in Section 4.2.
dc.eprint.versionPost-print
dc.identifier.arxivid2009.01219
dc.identifier.citationBAYER, C., HALL, E. J., & TEMPONE, R. (2023). WEAK ERROR RATES FOR OPTION PRICING UNDER LINEAR ROUGH VOLATILITY. International Journal of Theoretical and Applied Finance. https://doi.org/10.1142/s0219024922500297
dc.identifier.doi10.1142/s0219024922500297
dc.identifier.issn0219-0249
dc.identifier.issn1793-6322
dc.identifier.journalInternational Journal of Theoretical and Applied Finance
dc.identifier.urihttp://hdl.handle.net/10754/665128
dc.publisherWorld Scientific Pub Co Pte Ltd
dc.relation.urlhttps://www.worldscientific.com/doi/10.1142/S0219024922500297
dc.rightsThis is an accepted manuscript version of a paper before final publisher editing and formatting. Archived with thanks to World Scientific Pub Co Pte Ltd. The version of record is available from International Journal of Theoretical and Applied Finance.
dc.rights.embargodate2024-01-19
dc.titleWEAK ERROR RATES FOR OPTION PRICING UNDER LINEAR ROUGH VOLATILITY
dc.typeArticle
display.details.left<span><h5>Embargo End Date</h5>2024-01-19<br><br><h5>Type</h5>Article<br><br><h5>Authors</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.author=Bayer, Christian,equals">Bayer, Christian</a><br><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.author=Hall, Eric Joseph,equals">Hall, Eric Joseph</a><br><a href="https://repository.kaust.edu.sa/search?query=orcid.id:0000-0003-1967-4446&spc.sf=dc.date.issued&spc.sd=DESC">Tempone, Raul</a> <a href="https://orcid.org/0000-0003-1967-4446" target="_blank"><img src="https://repository.kaust.edu.sa/server/api/core/bitstreams/82a625b4-ed4b-40c8-865a-d6a5225a26a4/content" width="16" height="16"/></a><br><br><h5>KAUST Department</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.department=Applied Mathematics and Computational Science Program,equals">Applied Mathematics and Computational Science Program</a><br><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.department=Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division,equals">Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division</a><br><br><h5>KAUST Grant Number</h5>OSR2019-CRG8-4033<br><br><h5>Date</h5>2023-01-19</span>
display.details.right<span><h5>Abstract</h5>In quantitative finance, modeling the volatility structure of underlying assets is vital to pricing options. Rough stochastic volatility models, such as the rough Bergomi model [C. Bayer, P. K. Friz & J. Gatheral (2016) Pricing under rough volatility, Quantitative Finance 16 (6), 887–904, doi:10.1080/14697688.2015.1099717], seek to fit observed market data based on the observation that the log-realized variance behaves like a fractional Brownian motion with small Hurst parameter, [Formula: see text], over reasonable timescales. Both time series of asset prices and option-derived price data indicate that [Formula: see text] often takes values close to [Formula: see text] or less, i.e. rougher than Brownian motion. This change improves the fit to both option prices and time series of underlying asset prices while maintaining parsimoniousness. However, the non-Markovian nature of the driving fractional Brownian motion in rough volatility models poses severe challenges for theoretical and numerical analyses and for computational practice. While the explicit Euler method is known to converge to the solution of the rough Bergomi and similar models, its strong rate of convergence is only [Formula: see text]. We prove rate [Formula: see text] for the weak convergence of the Euler method for the rough Stein–Stein model, which treats the volatility as a linear function of the driving fractional Brownian motion, and, surprisingly, we prove rate one for the case of quadratic payoff functions. Indeed, the problem of weak convergence for rough volatility models is very subtle; we provide examples demonstrating the rate of convergence for payoff functions that are well approximated by second-order polynomials, as weighted by the law of the fractional Brownian motion, may be hard to distinguish from rate one empirically. Our proof uses Talay–Tubaro expansions and an affine Markovian representation of the underlying and is further supported by numerical experiments. These convergence results provide a first step toward deriving weak rates for the rough Bergomi model, which treats the volatility as a nonlinear function of the driving fractional Brownian motion.<br><br><h5>Citation</h5>BAYER, C., HALL, E. J., & TEMPONE, R. (2023). WEAK ERROR RATES FOR OPTION PRICING UNDER LINEAR ROUGH VOLATILITY. International Journal of Theoretical and Applied Finance. https://doi.org/10.1142/s0219024922500297<br><br><h5>Acknowledgements</h5>This work was supported by the KAUST Office of Sponsored Research (OSR) under Award No. OSR2019-CRG8-4033 and the Alexander von Humboldt Foundation. R. Tempone is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering. A portion of this work was carried out while E. Hall was a Postdoctoral Research Scientist in the Chair of Mathematics for Uncertainty Quantification at RWTH Aachen University. C. Bayer gratefully acknowledges support by the German Research Council (DFG) via the Research Unit FOR 2402. We are grateful to Andreas Neuenkirch for pointing out the simpler proof in the case of quadratic payoffs presented in Section 4.2.<br><br><h5>Publisher</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.publisher=World Scientific Pub Co Pte Ltd,equals">World Scientific Pub Co Pte Ltd</a><br><br><h5>Journal</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.journal=International Journal of Theoretical and Applied Finance,equals">International Journal of Theoretical and Applied Finance</a><br><br><h5>DOI</h5><a href="https://doi.org/10.1142/s0219024922500297">10.1142/s0219024922500297</a><br><br><h5>arXiv</h5><a href="https://arxiv.org/abs/2009.01219">2009.01219</a><br><br><h5>Additional Links</h5>https://www.worldscientific.com/doi/10.1142/S0219024922500297</span>
kaust.acknowledged.supportUnitOffice of Sponsored Research (OSR)
kaust.acknowledged.supportUnitSRI Center for Uncertainty Quantification in Computational Science and Engineering.
kaust.grant.numberOSR2019-CRG8-4033
kaust.personTempone, Raul
orcid.authorBayer, Christian
orcid.authorHall, Eric Joseph
orcid.authorTempone, Raul::0000-0003-1967-4446
orcid.id0000-0003-1967-4446
refterms.dateFOA2020-09-14T12:10:19Z
Files
Original bundle
Now showing 1 - 1 of 1
Loading...
Thumbnail Image
Name:
2009.01219v3.pdf
Size:
1.8 MB
Format:
Adobe Portable Document Format
Description:
Accepted Manuscript
License bundle
Now showing 1 - 1 of 1
Name:
license.txt
Size:
1.57 KB
Format:
Item-specific license agreed upon to submission
Description:

Version History

Now showing 1 - 2 of 2
VersionDateSummary
2*
2023-01-22 12:13:56
Published in journal
2020-09-14 12:09:48
* Selected version