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Unified Approach for the Affine and Non-affine Models: An Empirical Analysis on the S&P 500 Volatility Dynamics

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  • Shunwei Zhu

    (University of Shanghai for Science and Technology)

  • Bo Wang

    (University of Shanghai for Science and Technology)

Abstract

Being able to generate a volatility smile and adequately explain how it moves up and down in response to changes in risk, stochastic volatility models have replaced BS model. A single-factor volatility model can generate steep smiles or flat smiles at a given volatility level, but it cannot generate both for given parameters. In order to match the market implied volatility surface precisely, Grasselli introduced a 4/2 stochastic volatility model that includes the Heston model and the 3/2 model, performing as affine and non-affine model respectively. The present paper is intended to further investigate the 4/2 model, which falls into four parts. First, we apply Lewis’s fundamental transform approach instead of Grasselli’s method to deduce PDEs, which is intuitional and simple; Then, we use a result derived by Craddock and Lennox using Lie Symmetries theory for PDEs, and the results are more objective and reasonable; Finally, through adopting the data on S&P 500, we estimate the parameters of the 4/2 model; Furthermore, we investigate the 4/2 model along with the Heston model and the 3/2 model and compare their different performances. Our results illustrate that the 4/2 model outperforms the Heston and the 3/2 model for the fitting problem.

Suggested Citation

  • Shunwei Zhu & Bo Wang, 2019. "Unified Approach for the Affine and Non-affine Models: An Empirical Analysis on the S&P 500 Volatility Dynamics," Computational Economics, Springer;Society for Computational Economics, vol. 53(4), pages 1421-1442, April.
    • Handle: RePEc:kap:compec:v:53:y:2019:i:4:d:10.1007_s10614-018-9815-8
    DOI: 10.1007/s10614-018-9815-8
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    References listed on IDEAS

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    1. Mark Craddock & Eckhard Platen, 2003. "Symmetry Group Methods for Fundamental Solutions and Characteristic Functions," Research Paper Series 90, Quantitative Finance Research Centre, University of Technology, Sydney.
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    Cited by:

    1. Yumo Zhang, 2021. "Dynamic Optimal Mean-Variance Investment with Mispricing in the Family of 4/2 Stochastic Volatility Models," Mathematics, MDPI, vol. 9(18), pages 1-25, September.
    2. Li, Hongshan & Huang, Zhongyi, 2020. "An iterative splitting method for pricing European options under the Heston model☆," Applied Mathematics and Computation, Elsevier, vol. 387(C).
    3. Yumo Zhang, 2023. "Robust Optimal Investment Strategies for Mean-Variance Asset-Liability Management Under 4/2 Stochastic Volatility Models," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-32, March.
    4. Yumo Zhang, 2023. "Utility maximization in a stochastic affine interest rate and CIR risk premium framework: a BSDE approach," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 46(1), pages 97-128, June.

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