Nonparametric bias reduction of diffusion function in stochastic volatility models

Yunyan Wang; Shiguang Peng; Mingtian Tang; Yunyan Wang; Shiguang Peng; Mingtian Tang

doi:10.3934/math.2025729

AIMS Mathematics

2025, Volume 10, Issue 7: 16317-16333. doi: 10.3934/math.2025729

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Research article

Nonparametric bias reduction of diffusion function in stochastic volatility models

1.
School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China
2.
Key Laboratory of Low Dimensional Quantum Materials and Sensor Devices of Jiangxi Education Institutes, Ganzhou 341000, China

Received: 21 April 2025 Revised: 07 July 2025 Accepted: 09 July 2025 Published: 18 July 2025
MSC : 62G05, 62G20

This paper proposed a novel bias correction method based on nonparametric kernel estimator of the diffusion function in stochastic volatility models. In the case of fixed time span, the asymptotic bias of kernel estimation and the proposed nonparametric estimation of diffusion function have been developed. The results showed that conventional kernel-based estimators of the diffusion function suffer from nonvanishing discretization bias, primarily due to the local constant approximation in Nadaraya-Watson regression. We addressed this limitation by developing a dual-stage estimation method that incorporated nonparametric drift estimation into the diffusion function estimation procedure, thereby eliminating the dominant first-order bias term caused by discretization errors compared to traditional kernel estimation. Through rigorous theoretical analysis, the weak consistency and asymptotic normality of the new proposed estimator under mild regularity conditions were established. Furthermore, simulation studies and empirical analysis were provided to evaluate the finite sample performance of the proposed method. The proposed method offered a more robust tool for modeling volatility in financial time series, with significant applications in derivative pricing and risk management.
- stochastic volatility models,
- diffusion function,
- bias reduction,
- nonparametric estimator,
- consistency,
- asymptotic normality
Citation: Yunyan Wang, Shiguang Peng, Mingtian Tang. Nonparametric bias reduction of diffusion function in stochastic volatility models[J]. AIMS Mathematics, 2025, 10(7): 16317-16333. doi: 10.3934/math.2025729

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Abstract

This paper proposed a novel bias correction method based on nonparametric kernel estimator of the diffusion function in stochastic volatility models. In the case of fixed time span, the asymptotic bias of kernel estimation and the proposed nonparametric estimation of diffusion function have been developed. The results showed that conventional kernel-based estimators of the diffusion function suffer from nonvanishing discretization bias, primarily due to the local constant approximation in Nadaraya-Watson regression. We addressed this limitation by developing a dual-stage estimation method that incorporated nonparametric drift estimation into the diffusion function estimation procedure, thereby eliminating the dominant first-order bias term caused by discretization errors compared to traditional kernel estimation. Through rigorous theoretical analysis, the weak consistency and asymptotic normality of the new proposed estimator under mild regularity conditions were established. Furthermore, simulation studies and empirical analysis were provided to evaluate the finite sample performance of the proposed method. The proposed method offered a more robust tool for modeling volatility in financial time series, with significant applications in derivative pricing and risk management.

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