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A remark on Lin and Chang's paper ‘Consistent modeling of S&P 500 and VIX derivatives’

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  • Cheng, Jun
  • Ibraimi, Meriton
  • Leippold, Markus
  • Zhang, Jin E.

Abstract

Lin and Chang (2009, 2010) establish a VIX futures and option pricing theory when modeling S&P 500 index by using a stochastic volatility process with asset return and volatility jumps. In this note, we prove that Lin and Chang's formula is not an exact solution of their pricing equation. More generally, we show that the characteristic function of their pricing equation cannot be exponentially affine, as proposed by them. Furthermore, their formula cannot serve as a reasonable approximation. Using the (Heston, 1993) model as a special case, we demonstrate that Lin and Chang formula misprices VIX futures and options in general and the error can become substantially large.

Suggested Citation

  • Cheng, Jun & Ibraimi, Meriton & Leippold, Markus & Zhang, Jin E., 2012. "A remark on Lin and Chang's paper ‘Consistent modeling of S&P 500 and VIX derivatives’," Journal of Economic Dynamics and Control, Elsevier, vol. 36(5), pages 708-715.
    • Handle: RePEc:eee:dyncon:v:36:y:2012:i:5:p:708-715
    DOI: 10.1016/j.jedc.2012.01.002
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    1. Yingzi Zhu & Jin E. Zhang, 2007. "Variance Term Structure And Vix Futures Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 10(01), pages 111-127.
    2. Lin, Yueh-Neng & Chang, Chien-Hung, 2010. "Consistent modeling of S&P 500 and VIX derivatives," Journal of Economic Dynamics and Control, Elsevier, vol. 34(11), pages 2302-2319, November.
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    4. Jin E. Zhang & Yuqin Huang, 2010. "The CBOE S&P 500 three‐month variance futures," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 30(1), pages 48-70, January.
    5. Konstantinidi, Eirini & Skiadopoulos, George, 2011. "Are VIX futures prices predictable? An empirical investigation," International Journal of Forecasting, Elsevier, vol. 27(2), pages 543-560.
    6. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    7. Hilal, Sawsan & Poon, Ser-Huang & Tawn, Jonathan, 2011. "Hedging the black swan: Conditional heteroskedasticity and tail dependence in S&P500 and VIX," Journal of Banking & Finance, Elsevier, vol. 35(9), pages 2374-2387, September.
    8. Jinghong Shu & Jin E. Zhang, 2012. "Causality in the VIX futures market," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 32(1), pages 24-46, January.
    9. Peng Cheng & Olivier Scaillet, 2007. "Linear‐Quadratic Jump‐Diffusion Modeling," Mathematical Finance, Wiley Blackwell, vol. 17(4), pages 575-598, October.
    10. Zhiguang Wang & Robert T. Daigler, 2011. "The performance of VIX option pricing models: Empirical evidence beyond simulation," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 31(3), pages 251-281, March.
    11. Dotsis, George & Psychoyios, Dimitris & Skiadopoulos, George, 2007. "An empirical comparison of continuous-time models of implied volatility indices," Journal of Banking & Finance, Elsevier, vol. 31(12), pages 3584-3603, December.
    12. Peter Carr & Roger Lee, 2009. "Volatility Derivatives," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 319-339, November.
    13. Song‐Ping Zhu & Guang‐Hua Lian, 2012. "An analytical formula for VIX futures and its applications," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 32(2), pages 166-190, February.
    14. Chen, Hsuan-Chi & Chung, San-Lin & Ho, Keng-Yu, 2011. "The diversification effects of volatility-related assets," Journal of Banking & Finance, Elsevier, vol. 35(5), pages 1179-1189, May.
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    Citations

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    Cited by:

    1. Takuji Arai, 2019. "Pricing And Hedging Of Vix Options For Barndorff-Nielsen And Shephard Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(08), pages 1-26, December.
    2. Jaegi Jeon & Geonwoo Kim & Jeonggyu Huh, 2021. "Consistent and efficient pricing of SPX and VIX options under multiscale stochastic volatility," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(5), pages 559-576, May.
    3. Takuji Arai, 2019. "Pricing and hedging of VIX options for Barndorff-Nielsen and Shephard models," Papers 1904.12260, arXiv.org.
    4. Jaegi Jeon & Geonwoo Kim & Jeonggyu Huh, 2019. "Consistent and Efficient Pricing of SPX and VIX Options under Multiscale Stochastic Volatility," Papers 1909.10187, arXiv.org.
    5. Ballestra, Luca Vincenzo & Pacelli, Graziella, 2013. "Pricing European and American options with two stochastic factors: A highly efficient radial basis function approach," Journal of Economic Dynamics and Control, Elsevier, vol. 37(6), pages 1142-1167.
    6. Lo, Chien-Ling & Shih, Pai-Ta & Wang, Yaw-Huei & Yu, Min-Teh, 2019. "VIX derivatives: Valuation models and empirical evidence," Pacific-Basin Finance Journal, Elsevier, vol. 53(C), pages 1-21.
    7. Xingguo Luo & Jin E. Zhang & Wenjun Zhang, 2019. "Instantaneous squared VIX and VIX derivatives," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(10), pages 1193-1213, October.

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    More about this item

    Keywords

    VIX option pricing; Affine jump diffusion; Characteristic function;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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