IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1506.02074.html
   My bibliography  Save this paper

Optimal Static Quadratic Hedging

Author

Listed:
  • Tim Leung
  • Matthew Lorig

Abstract

We propose a flexible framework for hedging a contingent claim by holding static positions in vanilla European calls, puts, bonds, and forwards. A model-free expression is derived for the optimal static hedging strategy that minimizes the expected squared hedging error subject to a cost constraint. The optimal hedge involves computing a number of expectations that reflect the dependence among the contingent claim and the hedging assets. We provide a general method for approximating these expectations analytically in a general Markov diffusion market. To illustrate the versatility of our approach, we present several numerical examples, including hedging path-dependent options and options written on a correlated asset.

Suggested Citation

  • Tim Leung & Matthew Lorig, 2015. "Optimal Static Quadratic Hedging," Papers 1506.02074, arXiv.org, revised Nov 2015.
    • Handle: RePEc:arx:papers:1506.02074
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1506.02074
    File Function: Latest version
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Claude Bardos & Raphaël Douady & Andrei Fursikov, 2002. "Static Hedging Of Barrier Options With A Smile: An Inverse Problem," Post-Print hal-01477102, HAL.
    2. David Hobson & Peter Laurence & Tai-Ho Wang, 2005. "Static-arbitrage upper bounds for the prices of basket options," Quantitative Finance, Taylor & Francis Journals, vol. 5(4), pages 329-342.
    3. Peter Carr & Liuren Wu, 2014. "Static Hedging of Standard Options," Journal of Financial Econometrics, Oxford University Press, vol. 12(1), pages 3-46.
    4. repec:bla:jfinan:v:53:y:1998:i:3:p:1165-1190 is not listed on IDEAS
    5. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    6. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    7. Gabriel G. Drimus, 2012. "Options on realized variance by transform methods: a non-affine stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 12(11), pages 1679-1694, November.
    8. 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.
    9. Tim Leung & Ronnie Sircar, 2015. "Implied Volatility of Leveraged ETF Options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(2), pages 162-188, April.
    10. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2017. "Explicit Implied Volatilities For Multifactor Local-Stochastic Volatility Models," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 926-960, July.
    11. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Analytical expansions for parabolic equations," Papers 1312.3314, arXiv.org, revised Nov 2014.
    12. Pagliarani, Stefano & Pascucci, Andrea, 2011. "Analytical approximation of the transition density in a local volatility model," MPRA Paper 31107, University Library of Munich, Germany.
    13. Tim Leung & Matthew Lorig & Andrea Pascucci, 2014. "Leveraged {ETF} implied volatilities from {ETF} dynamics," Papers 1404.6792, arXiv.org, revised Apr 2015.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Navratil, Robert & Taylor, Stephen & Vecer, Jan, 2022. "On the utility maximization of the discrepancy between a perceived and market implied risk neutral distribution," European Journal of Operational Research, Elsevier, vol. 302(3), pages 1215-1229.
    2. 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.
    3. Georgios I. Papayiannis, 2022. "Static Hedging of Freight Risk under Model Uncertainty," Papers 2207.00862, arXiv.org.
    4. Peter Carr & Roger Lee & Matthew Lorig, 2015. "Robust replication of barrier-style claims on price and volatility," Papers 1508.00632, arXiv.org, revised Jan 2022.
    5. Fabien Le Floc’h, 2018. "Variance Swap Replication: Discrete or Continuous?," JRFM, MDPI, vol. 11(1), pages 1-15, February.
    6. Tim Leung & Brian Ward, 2020. "Tracking VIX with VIX Futures: Portfolio Construction and Performance," World Scientific Book Chapters, in: John B Guerard & William T Ziemba (ed.), HANDBOOK OF APPLIED INVESTMENT RESEARCH, chapter 21, pages 557-596, World Scientific Publishing Co. Pte. Ltd..
    7. Jun Deng & Bin Zou, 2020. "Quadratic Hedging for Sequential Claims with Random Weights in Discrete Time," Papers 2005.06015, arXiv.org, revised Dec 2020.
    8. Alvaro Cartea & Ryan Donnelly & Sebastian Jaimungal, 2019. "Hedging Non-Tradable Risks with Transaction Costs and Price Impact," Papers 1908.00054, arXiv.org, revised Mar 2020.
    9. Álvaro Cartea & Ryan Donnelly & Sebastian Jaimungal, 2020. "Hedging nontradable risks with transaction costs and price impact," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 833-868, July.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Matthew Lorig, 2014. "Indifference prices and implied volatilities," Papers 1412.5520, arXiv.org, revised Sep 2015.
    3. Weston Barger & Matthew Lorig, 2017. "Approximate pricing of European and Barrier claims in a local-stochastic volatility setting," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(02n03), pages 1-31, June.
    4. Stefano Pagliarani & Andrea Pascucci, 2017. "The exact Taylor formula of the implied volatility," Finance and Stochastics, Springer, vol. 21(3), pages 661-718, July.
    5. Weston Barger & Matthew Lorig, 2016. "Approximate pricing of European and Barrier claims in a local-stochastic volatility setting," Papers 1610.05728, arXiv.org, revised Apr 2017.
    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. Olesya Grishchenko & Xiao Han & Victor Nistor, 2018. "A volatility-of-volatility expansion of the option prices in the SABR stochastic volatility model," Papers 1812.09904, arXiv.org.
    8. Sergey Badikov & Mark H. A. Davis & Antoine Jacquier, 2018. "Perturbation analysis of sub/super hedging problems," Papers 1806.03543, arXiv.org, revised May 2021.
    9. Nteukam T., Oberlain & Planchet, Frédéric & Thérond, Pierre-E., 2011. "Optimal strategies for hedging portfolios of unit-linked life insurance contracts with minimum death guarantee," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 161-175, March.
    10. 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.
    11. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2014. "Asymptotics for $d$-dimensional L\'evy-type processes," Papers 1404.3153, arXiv.org, revised Nov 2014.
    12. Andrea Barletta & Elisa Nicolato & Stefano Pagliarani, 2019. "The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 928-966, July.
    13. Shuxin Guo & Qiang Liu, 2019. "The Black-Scholes-Merton dual equation," Papers 1912.10380, arXiv.org, revised May 2024.
    14. 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.
    15. Matthew Lorig & Ronnie Sircar, 2015. "Portfolio Optimization under Local-Stochastic Volatility: Coefficient Taylor Series Approximations & Implied Sharpe Ratio," Papers 1506.06180, arXiv.org.
    16. Sergey Nasekin & Wolfgang Karl Hardle, 2020. "Model-driven statistical arbitrage on LETF option markets," Papers 2009.09713, arXiv.org.
    17. Philipp Mayer & Natalie Packham & Wolfgang Schmidt, 2015. "Static hedging under maturity mismatch," Finance and Stochastics, Springer, vol. 19(3), pages 509-539, July.
    18. Wei Lin & Shenghong Li & Shane Chern, 2017. "Pricing VIX Derivatives With Free Stochastic Volatility Model," Papers 1703.06020, arXiv.org.
    19. Yan, Tingjin & Wong, Hoi Ying, 2020. "Open-loop equilibrium reinsurance-investment strategy under mean–variance criterion with stochastic volatility," Insurance: Mathematics and Economics, Elsevier, vol. 90(C), pages 105-119.
    20. Tim Leung & Matthew Lorig & Andrea Pascucci, 2014. "Leveraged {ETF} implied volatilities from {ETF} dynamics," Papers 1404.6792, arXiv.org, revised Apr 2015.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1506.02074. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.