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Is the realized volatility good for option pricing during the recent financial crisis?

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Abstract

The contributions of this paper are threefold. The first contribution is the proposed logarithmic HAR (log-HAR) option-pricing model, which is more convenient compared with other option pricing models associated with realized volatility in terms of simpler estimation procedure. The second contribution is the test of the empirical implications of heterogeneous autoregressive model of the realized volatility (HAR)-type models in the S&P 500 index options market with comparison of the non-linear asymmetric GARCH option-pricing model, which is the best model in pricing options among generalized autoregressive conditional heteroskedastic-type models. The third contribution is the empirical analysis based on options traded from July 3, 2007 to December 31, 2008, a period covering a recent financial crisis. Overall, the HAR-type models successfully predict out-of-sample option prices because they are based on realized volatilities, which are closer to the expected volatility in financial markets. However, mixed results exist between the log-HAR and the heterogeneous auto-regressive gamma models in pricing options because the former is better than the latter in times of turmoil, whereas it is worse during the rather stable periods.

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Notes

  1. Such as, underpricing of out-of-the-money options (Gultekin et al. 1982), underpricing of options on low-volatility securities (Black and Scholes 1972; Gultekin et al. 1982; Whaley 1982), and underpricing of short-maturity option (Whaley 1982).

  2. See for example, Eisenberg and Jarrow (1994), Lee et al. (2004, 2005).

  3. Ding et al. (1993), Baillie et al. (1996) and Bollerslev and Mikkelsen (1999) suggest that the autocorrelation of volatility decreases with a hyperbolic rate and it could be displayed by long-memory processes to be a more adequate representation for the conditional variance of returns.

  4. More empirical evidence of NGARCH option pricing models are provided as follows. First, in terms of Hang-Seng index options, the empirical performance of NGARCH option pricing model outperforms two ad hoc versions of the BS model (Duan and Zhang 2001). Comparing stochastic volatility model on the FTSE 100 index, Lehar et al. (2002) conclude the empirical results to out-of-sample option pricing performance of FTSE 100 index option show that NGARCH dominates both stochastic volatility and the benchmark BS model.

  5. For example, HARG includes non-linear optimization procedure, while our log-HAR option price model only requires the ordinary-least-square method.

  6. See, for example, Bakshi et al. (1997), Heston and Nandi (2000), and Corsi et al. (2010).

  7. Tick data database sources intraday option data from the Chicago board options exchange (CBOE), and then provides historical tick-by-tick options data for all listed US equity and index options contracts reported by the options price reporting authority (OPRA) back to July 2, 2004. All data is recorded in a timestamp to the second prior to July 1, 2008, and to the millisecond after July 1, 2008.

  8. In Corsi et al. (2010), the authors gather option data from OptionMetrics, and thus ignore the nonsynchroneity problem, which could generate bias empirical analysis in evaluating options.

  9. See, for example, Bakshi et al. (1997) and Yuan and Zhang (2003).

  10. See, for example, Heston and Nandi (2000).

  11. See Santa-Clara and Yan (2010).

  12. In line with Corsi (2009), for every time of implementing estimation, he selected a data set that contains 1,000 observations in estimating parameters.

  13. To be clear, this paper denotes the square root of the realized variance as realized volatility.

  14. Corsi et al. (2010) study SPX from January 5, 2000 to December 31, 2004.

  15. See “Appendix”.

  16. The OLS method is also used in Corsi (2009), and Andersen et al. (2007).

  17. The word “for” means forecast.

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Authors and Affiliations

  1. Department of Information and Finance Management, Graduate Institute of Finance, National Chiao Tung University, Management Building 1, 1001 Ta Hsueh Road, Hsinchu, 300, Taiwan, ROC

    Yow-Jen Jou & Chih-Wei Wang

  2. Department of Business Administration and Quantitative Methods, Universidad Carlos III de Madrid, c/Madrid 126, 28903, Getafe, Madrid, Spain

    Chih-Wei Wang & Wan-Chien Chiu

Authors
  1. Yow-Jen Jou
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  2. Chih-Wei Wang
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  3. Wan-Chien Chiu
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Correspondence to Yow-Jen Jou.

Additional information

This paper has been accepted for presentation at the 3rd international conference on business in Asia (iCBA) and the 17th annual conference on Pacific Basin finance, economics, accounting and management (PBFEAM) which will be held on July 1–2, 2009.

Appendix

Appendix

Assuming r t  = 0 (for computational convenience), the SDF satisfies the following framework:

$$ M_{t,t + 1} = \exp \left( { - \nu_{0} - \nu_{1} RV_{t + 1} - \nu_{2} \left( {RV} \right)_{t} - \nu_{3} \left( {\sum\limits_{i = 1}^{4} {RV_{t - i} } } \right) - \nu_{4} \left( {\sum\limits_{i = 1}^{21} {RV_{t - i} } } \right) - \nu_{5} y_{t + 1} - \nu_{6} L_{t} } \right) $$
(19)

It complies with the no arbitrage conditions if the following implicit parameter-restrictions are satisfied:

$$ \begin{aligned} \nu_{0} & = - \delta \log \left( {1 + c\left( {\frac{1}{2}\gamma^{2} + \nu_{1} - \frac{1}{8}} \right)} \right) \\ \nu_{2} & = - c\beta_{1} \frac{{\frac{1}{2}\gamma^{2} - \frac{1}{8} + \nu_{1} }}{{c\left( {\gamma^{2} /2 - 1/8 - \nu_{1} } \right) + 1}} \\ \nu_{3} & = - c\beta_{2} \frac{{\frac{1}{2}\gamma^{2} - \frac{1}{8} + \nu_{1} }}{{c\left( {\gamma^{2} /2 - 1/8 - \nu_{1} } \right) + 1}} \\ \nu_{4} & = - c\beta_{3} \frac{{\frac{1}{2}\gamma^{2} - \frac{1}{8} + \nu_{1} }}{{c\left( {\gamma^{2} /2 - 1/8 - \nu_{1} } \right) + 1}} \\ \nu_{6} & = - c\beta_{4} \frac{{\frac{1}{2}\gamma^{2} - \frac{1}{8} + \nu_{1} }}{{c\left( {\gamma^{2} /2 - 1/8 - \nu_{1} } \right) + 1}} \\ \nu_{5} & = \gamma + \frac{1}{2} \\ \end{aligned} $$
(20)

where \( \nu_{1} \) remains a free parameter.

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Jou, YJ., Wang, CW. & Chiu, WC. Is the realized volatility good for option pricing during the recent financial crisis?. Rev Quant Finan Acc 40, 171–188 (2013). https://doi.org/10.1007/s11156-012-0285-0

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