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Leveraged {ETF} implied volatilities from {ETF} dynamics

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  • Tim Leung
  • Matthew Lorig
  • Andrea Pascucci

Abstract

The growth of the exhange-traded fund (ETF) industry has given rise to the trading of options written on ETFs and their leveraged counterparts {(LETFs)}. We study the relationship between the ETF and LETF implied volatility surfaces when the underlying ETF is modeled by a general class of local-stochastic volatility models. A closed-form approximation for prices is derived for European-style options whose payoff depends on the terminal value of the ETF and/or LETF. Rigorous error bounds for this pricing approximation are established. A closed-form approximation for implied volatilities is also derived. We also discuss a scaling procedure for comparing implied volatilities across leverage ratios. The implied volatility expansions and scalings are tested in three well-known settings: CEV, Heston and SABR.

Suggested Citation

  • Tim Leung & Matthew Lorig & Andrea Pascucci, 2014. "Leveraged {ETF} implied volatilities from {ETF} dynamics," Papers 1404.6792, arXiv.org, revised Apr 2015.
    • Handle: RePEc:arx:papers:1404.6792
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    References listed on IDEAS

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    1. Fouque,Jean-Pierre & Papanicolaou,George & Sircar,Ronnie & Sølna,Knut, 2011. "Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives," Cambridge Books, Cambridge University Press, number 9780521843584, November.
    2. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 1997. "Empirical Performance of Alternative Option Pricing Models," Journal of Finance, American Finance Association, vol. 52(5), pages 2003-2049, December.
    3. Antoine Jacquier & Matthew Lorig, 2012. "The Smile of certain L\'evy-type Models," Papers 1207.1630, arXiv.org, revised Apr 2013.
    4. Kevin Guo & Tim Leung, 2016. "Understanding the Tracking Errors of Commodity Leveraged ETFs," Papers 1610.09404, arXiv.org.
    5. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "A family of density expansions for L\'evy-type processes," Papers 1312.7328, arXiv.org.
    6. Martin Forde & Antoine Jacquier, 2011. "Small-Time Asymptotics for an Uncorrelated Local-Stochastic Volatility Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 18(6), pages 517-535, April.
    7. 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.
    8. Tim Leung & Ronnie Sircar, 2015. "Implied Volatility of Leveraged ETF Options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(2), pages 162-188, April.
    9. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Analytical expansions for parabolic equations," Papers 1312.3314, arXiv.org, revised Nov 2014.
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    Citations

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    Cited by:

    1. Vasiliki A. Basdekidou, 2017. "The Leveraged ETF Inefficiency in Trending & Range-Bound Markets: An Application Case Study for a 3x Leveraged Gold Miners ETF," International Journal of Economics and Finance, Canadian Center of Science and Education, vol. 9(7), pages 1-13, July.
    2. 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.
    3. 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.
    4. 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.
    5. Tim Leung & Matthew Lorig, 2016. "Optimal static quadratic hedging," Quantitative Finance, Taylor & Francis Journals, vol. 16(9), pages 1341-1355, September.
    6. Vasiliki A. Basdekidou, 2017. "Green Entrepreneurship & Corporate Social Responsibility: Comparative and Correlative Performance Analysis," International Journal of Economics and Finance, Canadian Center of Science and Education, vol. 9(12), pages 1-12, December.
    7. 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.
    8. 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.
    9. Tim Leung & Hyungbin Park & Heejun Yeo, 2023. "Robust Long-Term Growth Rate of Expected Utility for Leveraged ETFs," Papers 2310.02084, arXiv.org.
    10. 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.
    11. Sergey Nasekin & Wolfgang Karl Hardle, 2020. "Model-driven statistical arbitrage on LETF option markets," Papers 2009.09713, arXiv.org.

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