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Implied Volatility of Leveraged ETF Options

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  • Tim Leung
  • Ronnie Sircar

Abstract

This paper studies the problem of understanding implied volatilities from options written on leveraged exchanged-traded funds (LETFs), with an emphasis on the relations between LETF options with different leverage ratios. We first examine from empirical data the implied volatility skews for LETF options based on the S&P 500. In order to enhance their comparison with non-leveraged ETFs, we introduce the concept of moneyness scaling and provide a new formula that links option implied volatilities between leveraged and unleveraged ETFs. Under a multiscale stochastic volatility framework, we apply asymptotic techniques to derive an approximation for both the LETF option price and implied volatility. The approximation formula reflects the role of the leverage ratio, and thus allows us to link implied volatilities of options on an ETF and its leveraged counterparts. We apply our result to quantify matches and mismatches in the level and slope of the implied volatility skews for various LETF options using data from the underlying ETF option prices. This reveals some apparent biases in the leverage implied by the market prices of different products, long and short with leverage ratios two times and three times.

Suggested Citation

  • Tim Leung & Ronnie Sircar, 2015. "Implied Volatility of Leveraged ETF Options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(2), pages 162-188, April.
    • Handle: RePEc:taf:apmtfi:v:22:y:2015:i:2:p:162-188
    DOI: 10.1080/1350486X.2014.975825
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    Citations

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    Cited by:

    1. Hyungbin Park, 2018. "Sensitivity analysis of long-term cash flows," Finance and Stochastics, Springer, vol. 22(4), pages 773-825, October.
    2. Tim Leung & Brian Ward, 2015. "The golden target: analyzing the tracking performance of leveraged gold ETFs," Studies in Economics and Finance, Emerald Group Publishing Limited, vol. 32(3), pages 278-297, August.
    3. Peter Carr & Roger Lee & Matthew Lorig, 2021. "Robust replication of volatility and hybrid derivatives on jump diffusions," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1394-1422, October.
    4. Tim Leung & Hyungbin Park, 2017. "LONG-TERM GROWTH RATE OF EXPECTED UTILITY FOR LEVERAGED ETFs: MARTINGALE EXTRACTION APPROACH," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(06), pages 1-33, September.
    5. Hongkai Cao & Rupak Chatterjee & Zhenyu Cui, 2019. "Options valuation and calibration for leveraged exchange-traded funds with Heston–Nandi and inverse Gaussian GARCH models," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 6(03), pages 1-37, September.
    6. Tim Leung & Hyungbin Park & Heejun Yeo, 2023. "Robust Long-Term Growth Rate of Expected Utility for Leveraged ETFs," Papers 2310.02084, arXiv.org.
    7. Sergey Nasekin & Wolfgang Karl Hardle, 2020. "Model-driven statistical arbitrage on LETF option markets," Papers 2009.09713, arXiv.org.
    8. Tim Leung & Brian Ward, 2018. "Dynamic Index Tracking and Risk Exposure Control Using Derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 25(2), pages 180-212, March.
    9. Tim Leung & Matthew Lorig & Andrea Pascucci, 2014. "Leveraged {ETF} implied volatilities from {ETF} dynamics," Papers 1404.6792, arXiv.org, revised Apr 2015.
    10. Pieter van Staden & Peter Forsyth & Yuying Li, 2024. "Smart leverage? Rethinking the role of Leveraged Exchange Traded Funds in constructing portfolios to beat a benchmark," Papers 2412.05431, arXiv.org.
    11. Tim Leung & Hyungbin Park & Heejun Yeo, 2024. "Robust long-term growth rate of expected utility for leveraged ETFs," Mathematics and Financial Economics, Springer, volume 18, number 5, February.
    12. Tim Leung & Matthew Lorig, 2016. "Optimal static quadratic hedging," Quantitative Finance, Taylor & Francis Journals, vol. 16(9), pages 1341-1355, September.

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