Deep calibration with random grids

Abstract

We propose a neural network-based approach to calibrating stochastic volatility models, which combines the pioneering grid approach by Horvath et al. [Deep learning volatility: A deep neural network perspective on pricing and calibration in (rough) volatility models. Quant. Finance, 2021, 21(1), 11–27]. with the pointwise two-stage calibration of Bayer and Stemper [Deep calibration of rough stochastic volatility models. Working Paper, arXiv:1810.03399, 2018] and Liu et al. [A neural network-based framework for financial model calibration. J. Math. Ind., 2019, 9(1), 1–28]. Our methodology inherits robustness from the former while not suffering from the need for interpolation/extrapolation techniques, a clear advantage ensured by the pointwise approach. The crucial point to the entire procedure is the generation of implied volatility surfaces on random grids, which one dispenses to the network in the training phase. We support the validity of our calibration technique with several empirical and Monte Carlo experiments for the rough Bergomi and Heston models under a simple but effective parametrization of the forward variance curve. The approach paves the way for valuable applications in financial engineering—for instance, pricing under local stochastic volatility models—and extensions to the fast-growing field of path-dependent volatility models.

Keywords:

• Neural network pricing and calibration
• Rough volatilty
• Forward variance curve

Open Scholarship

This article has earned the Center for Open Science badge for Open Materials. The materials are openly accessible at https://github.com/fabioBaschetti/random-grid-NN-calib.

Acknowledgments

We warmly thank Vola Dynamics LLC for making their tools available for academic purposes. Many explorations in the paper would not have been possible without their support.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Interested readers can contact Vola Dynamics LCC inquiring about an agreement for academic purposes or buy the software by visiting the following webpage: https://voladynamics.com.

2 We did not try any optimization on the network structure and therefore simpler architectures could provide satisfactory results as well.

3 Proper training clearly depends on the quality of the samples one has produced during the generation phase. We do not enter the technicalities behind Monte Carlo estimation of rBergomi option prices (for which we refer to Bennedsen et al. 2017, McCrickerd and Pakkanen 2018), but stress that those parameter combinations that produced prices for which Black-Scholes inversion fails need to be discarded.

4 Recall that we have included very short maturities in the adaptive grid and ensured the forward variance curve changes level over the same buckets.

5 Variance swaps can be priced in terms of an infinite log-strip of out of the money European options as explained in Gatheral (2011). The idea is that one comes up with a parametrization of the volatility surface and uses it to extend the market to a continuum of option prices to integrate over. Once variance swaps are priced on the market maturities, the forward variance curve can be readily computed by differentiation.

6 We indeed observe the calibrated forward variance curve to be negative at very short times (only) if no such short maturities as a week or so are included in the volatility surface, in which case the surface would be of reduced practical interest (no price for typical hedging options, no clue as to the exploding behavior of the ATM skew and so on). In any case, a non-linearly constrained optimization may be set up to avoid problems of this sort.

7 This is made possible by the tools from Vola Dynamics.