jointcalib_sigsde

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 to SPX and VIX options with signature-based model}},
  author={Cuchiero, C. and Gazzani, G. and M\"oller, J. and Svaluto-Ferro, S.},
  journal={Mathematical Finance},
pages={1–53},
  year={2024}
}

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,
  author = {Cuchiero, C. and Gazzani, G. and Svaluto-Ferro, S.},
title = {Signature-Based Models: Theory and Calibration},
journal = {SIAM Journal on Financial Mathematics},
volume = {14},
number = {3},
pages = {910-957},
year = {2023}
}

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.

Sampler for the log-price and the VIX squared

• 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.

Some comments on the sampler can be found in the paper.

Joint calibration to SPX and VIX options with constant parameters

• Code for calibration to option prices of SPX and VIX options.

Details of the calibration to option prices with signature-based models can be found in Section 7 of the paper.