Abstract
The Heston model is the most renowned stochastic volatility function in finance, but the calibration input parameters is a challenging task. This contest grows up because the instantaneous volatility is unobservable or market quotes are absent/unviable to agents, remaining the asset time-series as tangible. Moreover, these conditions are unfit to approach based on Maximum Likelihood Estimation or optimisation of the differences in Pricing models and quotes in the real market. Today, neural networks are a well-known tool with malleable and powerful features to map accurately any nonlinear continuous operator in complex systems. This work adopts the deep operator networks (DeepONets) to learn the Heston parameters based on observed time series. We perform simulations of trajectories following the Heston model with a truncated Euler discretization scheme and randomised inputs parameters. The five parameters are estimated by Stacked and Unstacked DeepONets and compared with GJR-GARCH and standard neural network with the Tukeys’ test. The results indicated the improvement in accuracy of the Unstacked model. However, the statistical test indicates some similarity between the Unstacked model and standard neural network.
Supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-CAPES, Fundação de Amparo à Pesquisa do Estado de Minas Gerais-FAPEMIG (grant TEC-APQ-00334/18) and Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq (grant 429639/2016-3).
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Notes
- 1.
The problem of parameter estimation for stochastic processes is not limited to the Heston model. This gap is shown in [21] with exemplification for interest rates.
- 2.
See [7, Chapter 6.4.1].
- 3.
See [11][Chapter 15, Sect. 11] for more details about implied volatilities.
- 4.
The GJR-GARCH model also captures the stylised facts in finance, like the GARCH model, but appends the relation between negative shocks returns at last observation as positive shocks. This asymmetry is known as the leverage effect.
- 5.
CIR is a short-form of Cox, Ingersoll, and Ross (the original authors), see [6, Chapter 2, pg. 15].
- 6.
Mrázek and Pospíšil [23] describes some discretization and truncation schemes with richness of details.
- 7.
These interval was choice to comprise the values observed in literature. However, the plenty combinations of parameters are feasible depending of the market considered.
- 8.
The price time-series is convert in return time-series, it is a common procedure in finance and can be interpreted as the standardisation step.
- 9.
Linear activation was considered in output node to \(\alpha \) and \(\rho \). The values inside brackets indicate the number of neurons in each layer. Other topologies were tested, but without significant gain.
- 10.
A Python library designed for scientific machine learning (https://github.com/lululxvi/deepxde).
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Leite, I.M.S., Yamim, J.D.M., da Fonseca, L.G. (2021). The DeepONets for Finance: An Approach to Calibrate the Heston Model. In: Marreiros, G., Melo, F.S., Lau, N., Lopes Cardoso, H., Reis, L.P. (eds) Progress in Artificial Intelligence. EPIA 2021. Lecture Notes in Computer Science(), vol 12981. Springer, Cham. https://doi.org/10.1007/978-3-030-86230-5_28
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