Eduardo Abi JaberXiaoyuan, ShaunXiaoyuan, Li, Xuyang Lin
{"title":"Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models","authors":"Eduardo Abi JaberXiaoyuan, ShaunXiaoyuan, Li, Xuyang Lin","doi":"arxiv-2405.02170","DOIUrl":null,"url":null,"abstract":"We consider the Fourier-Laplace transforms of a broad class of polynomial\nOrnstein-Uhlenbeck (OU) volatility models, including the well-known\nStein-Stein, Sch\\\"obel-Zhu, one-factor Bergomi, and the recently introduced\nQuintic OU models motivated by the SPX-VIX joint calibration problem. We show\nthe connection between the joint Fourier-Laplace functional of the log-price\nand the integrated variance, and the solution of an infinite dimensional\nRiccati equation. Next, under some non-vanishing conditions of the\nFourier-Laplace transforms, we establish an existence result for such Riccati\nequation and we provide a discretized approximation of the joint characteristic\nfunctional that is exponentially entire. On the practical side, we develop a\nnumerical scheme to solve the stiff infinite dimensional Riccati equations and\ndemonstrate the efficiency and accuracy of the scheme for pricing SPX options\nand volatility swaps using Fourier and Laplace inversions, with specific\nexamples of the Quintic OU and the one-factor Bergomi models and their\ncalibration to real market data.","PeriodicalId":501294,"journal":{"name":"arXiv - QuantFin - Computational Finance","volume":"40 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - QuantFin - Computational Finance","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2405.02170","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the Fourier-Laplace transforms of a broad class of polynomial
Ornstein-Uhlenbeck (OU) volatility models, including the well-known
Stein-Stein, Sch\"obel-Zhu, one-factor Bergomi, and the recently introduced
Quintic OU models motivated by the SPX-VIX joint calibration problem. We show
the connection between the joint Fourier-Laplace functional of the log-price
and the integrated variance, and the solution of an infinite dimensional
Riccati equation. Next, under some non-vanishing conditions of the
Fourier-Laplace transforms, we establish an existence result for such Riccati
equation and we provide a discretized approximation of the joint characteristic
functional that is exponentially entire. On the practical side, we develop a
numerical scheme to solve the stiff infinite dimensional Riccati equations and
demonstrate the efficiency and accuracy of the scheme for pricing SPX options
and volatility swaps using Fourier and Laplace inversions, with specific
examples of the Quintic OU and the one-factor Bergomi models and their
calibration to real market data.