This is a collection of Python files which have been used in the article:
"Joint calibration to SPX and VIX options with signature-based models"
of Christa Cuchiero, Guido Gazzani, Janka Möller and Sara Svaluto-Ferro.
For citations:
MDPI and ACS Style
Cuchiero, C.; Gazzani, G.; Möller J.; Svaluto-Ferro, S. Joint calibration to SPX and VIX options with signature-based models.
@article{CGMS:23,
title={{Joint calibration of SPX and VIX options with signature-based models}},
author={Cuchiero, C. and Gazzani, G. and Möller, J. and Svaluto-Ferro, S.},
journal={Preprint arXiv:2301.13235},
year={2023}
}
The codes that compose the present repository rely strongly on the theory outlined in the article in section 4, 5 and 6. In particular the use of the polynomial processes' theory for the computation of the conditional expected signature of a polynomial process. Recall that data were purchased from OptionMetrics and therefore are not present in the current repository.
For an introduction to signature-based models in mathematical finance we address the reader to (forthcoming in SIAM Journal on Financial Mathematics):
Cuchiero, C.; Gazzani, G.; Svaluto-Ferro, S. Signature-based models: theory and calibration.
@article{CGS:22,
title={{Signature-based models: theory and calibration}},
author={Cuchiero, C. and Gazzani, G. and Svaluto-Ferro, S.},
journal={Preprint arXiv:2207.13136},
year={2022}
}
We reference additionally to the Github repository AffPolySig , where a more general implementation of the expected signature of a polynomial process can be found. For details on the theory we refer to
Cuchiero, C.; Svaluto-Ferro, S; Teichmann, J. Signature SDEs from an affine and polynomial perspective.
- Code for sampling: the Cholesky matrix for the VIX/VIX squared (see Remark 5.5)
- Code for sampling: the log-price in particular the matrix Q^0 and the regression basis \tilde{e}^{B} (Proposition 6.5, Equation 6.3)
- For the VIX squared both numerical integration and exact simulation are reported see Remark 5.4 in the paper.
- Code for calibration to option prices of SPX and VIX options.