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Pricing European Options under Stochastic Volatility Models: Case of Five-Parameter Variance-Gamma Process

Author

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  • Aubain Hilaire Nzokem

    (Department of Mathematics & Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA)

Abstract

The paper builds a Variance-Gamma (VG) model with five parameters: location ( μ ), symmetry ( δ ), volatility ( σ ), shape ( α ), and scale ( θ ); and studies its application to the pricing of European options. The results of our analysis show that the five-parameter VG model is a stochastic volatility model with a Γ ( α , θ ) Ornstein–Uhlenbeck type process; the associated Lévy density of the VG model is a KoBoL family of order ν = 0 , intensity α , and steepness parameters δ σ 2 − δ 2 σ 4 + 2 θ σ 2 and δ σ 2 + δ 2 σ 4 + 2 θ σ 2 ; and the VG process converges asymptotically in distribution to a Lévy process driven by a normal distribution with mean ( μ + α θ δ ) and variance α ( θ 2 δ 2 + σ 2 θ ) . The data used for empirical analysis were obtained by fitting the five-parameter Variance-Gamma (VG) model to the underlying distribution of the daily SPY ETF data. Regarding the application of the five-parameter VG model, the twelve-point rule Composite Newton–Cotes Quadrature and Fractional Fast Fourier (FRFT) algorithms were implemented to compute the European option price. Compared to the Black–Scholes (BS) model, empirical evidence shows that the VG option price is underpriced for out-of-the-money (OTM) options and overpriced for in-the-money (ITM) options. Both models produce almost the same option pricing results for deep out-of-the-money (OTM) and deep-in-the-money (ITM) options.

Suggested Citation

  • Aubain Hilaire Nzokem, 2023. "Pricing European Options under Stochastic Volatility Models: Case of Five-Parameter Variance-Gamma Process," JRFM, MDPI, vol. 16(1), pages 1-28, January.
    • Handle: RePEc:gam:jjrfmx:v:16:y:2023:i:1:p:55-:d:1037927
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    References listed on IDEAS

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    1. Sharif Mozumder & Ghulam Sorwar & Kevin Dowd, 2015. "Revisiting variance gamma pricing: An application to S&P500 index options," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(02), pages 1-24.
    2. A. H. Nzokem & V. T. Montshiwa, 2022. "Fitting Generalized Tempered Stable distribution: Fractional Fourier Transform (FRFT) Approach," Papers 2205.00586, arXiv.org, revised Jun 2022.
    3. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    4. Ole E. Barndorff‐Nielsen & Neil Shephard, 2003. "Integrated OU Processes and Non‐Gaussian OU‐based Stochastic Volatility Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(2), pages 277-295, June.
    5. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
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    Cited by:

    1. A. H. Nzokem, 2023. "Bitcoin versus S&P 500 Index: Return and Risk Analysis," Papers 2310.02436, arXiv.org.
    2. A. H. Nzokem, 2023. "European Option Pricing Under Generalized Tempered Stable Process: Empirical Analysis," Papers 2304.06060, arXiv.org, revised Aug 2023.

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