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A reflection principle for a random walk with implications for volatility estimation using extreme values of asset prices

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  • Kumar, Dilip
  • Maheswaran, S.

Abstract

In this paper, we derive a reflection principle for a random walk with the symmetric double exponential distribution. This allows us to come up with the closed form solution for the joint probability of the running maximum and the terminal value of the random walk. Based on this new theoretical result, we propose an extreme value estimator for the variance of the random walk that is not just approximately unbiased but exactly so. In simulations, we find that this estimator continues to be unbiased even when intraday mean reversion is present, as captured by the Binomial Markov Random Walk model. On the empirical side, we find that this estimator works well in a variety of global stock indices, including the S&P 500 Index, in the sense of being unbiased relative to the “usual” estimator, i.e., the sample variance of the daily returns.

Suggested Citation

  • Kumar, Dilip & Maheswaran, S., 2014. "A reflection principle for a random walk with implications for volatility estimation using extreme values of asset prices," Economic Modelling, Elsevier, vol. 38(C), pages 33-44.
    • Handle: RePEc:eee:ecmode:v:38:y:2014:i:c:p:33-44
    DOI: 10.1016/j.econmod.2013.11.045
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    References listed on IDEAS

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    Cited by:

    1. Zargar, Faisal Nazir & Kumar, Dilip, 2020. "Modeling unbiased extreme value volatility estimator in presence of heterogeneity and jumps: A study with economic significance analysis," International Review of Economics & Finance, Elsevier, vol. 67(C), pages 25-41.
    2. Dilip Kumar, 2020. "Value-at-Risk in the Presence of Structural Breaks Using Unbiased Extreme Value Volatility Estimator," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 18(3), pages 587-610, September.
    3. Kumar, Dilip & Maheswaran, S., 2014. "Modeling and forecasting the additive bias corrected extreme value volatility estimator," International Review of Financial Analysis, Elsevier, vol. 34(C), pages 166-176.
    4. Muneer Shaik & S. Maheswaran, 2020. "A new unbiased additive robust volatility estimation using extreme values of asset prices," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 34(3), pages 313-347, September.
    5. Dilip Kumar, 2016. "Estimating and forecasting value-at-risk using the unbiased extreme value volatility estimator," Proceedings of Economics and Finance Conferences 3205528, International Institute of Social and Economic Sciences.
    6. Muneer Shaik & S. Maheswaran, 2019. "Robust Volatility Estimation with and Without the Drift Parameter," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 17(1), pages 57-91, March.
    7. Zargar, Faisal Nazir & Kumar, Dilip, 2020. "Heterogeneous market hypothesis approach for modeling unbiased extreme value volatility estimator in presence of leverage effect: An individual stock level study with economic significance analysis," The Quarterly Review of Economics and Finance, Elsevier, vol. 77(C), pages 271-285.
    8. Dilip Kumar, 2019. "Structural Breaks in Volatility Transmission from Developed Markets to Major Asian Emerging Markets," Journal of Emerging Market Finance, Institute for Financial Management and Research, vol. 18(2), pages 172-209, August.
    9. Vortelinos, Dimitrios I., 2015. "The Greek equity market in European equity portfolios," Economic Modelling, Elsevier, vol. 49(C), pages 144-153.
    10. Dilip Kumar, 2018. "Modeling and Forecasting Unbiased Extreme Value Volatility Estimator in Presence of Leverage Effect," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 16(2), pages 313-335, June.

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    More about this item

    Keywords

    Volatility estimation; Bias correction; Random walk effect; Binomial Markov Random Walk (BMRW) model;
    All these keywords.

    JEL classification:

    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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