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Optimal derivative liquidation timing under path-dependent risk penalties

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  • Tim Leung

    (Industrial Engineering and Operations Research Department, Columbia University, New York, NY 10027, United States)

  • Yoshihiro Shirai

    (Industrial Engineering and Operations Research Department, Columbia University, New York, NY 10027, United States)

Abstract

This paper studies the risk-adjusted optimal timing to liquidate an option at the prevailing market price. In addition to maximizing the expected discounted return from option sale, we incorporate a path-dependent risk penalty based on shortfall or quadratic variation of the option price up to the liquidation time. We establish the conditions under which it is optimal to immediately liquidate or hold the option position through expiration. Furthermore, we study the variational inequality associated with the optimal stopping problem, and prove the existence and uniqueness of a strong solution. A series of analytical and numerical results are provided to illustrate the nontrivial optimal liquidation strategies under geometric Brownian motion (GBM) and exponential Ornstein–Uhlenbeck models. We examine the combined effects of price dynamics and risk penalty on the sell and delay regions for various options. In addition, we obtain an explicit closed-form solution for the liquidation of a stock with quadratic penalty under the GBM model.

Suggested Citation

  • Tim Leung & Yoshihiro Shirai, 2015. "Optimal derivative liquidation timing under path-dependent risk penalties," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(01), pages 1-32.
    • Handle: RePEc:wsi:jfexxx:v:02:y:2015:i:01:n:s234576861550004x
    DOI: 10.1142/S234576861550004X
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    6. Tim Leung & Peng Liu, 2013. "An Optimal Timing Approach to Option Portfolio Risk Management," Palgrave Macmillan Books, in: Jonathan A. Batten & Peter MacKay & Niklas Wagner (ed.), Advances in Financial Risk Management, chapter 17, pages 391-404, Palgrave Macmillan.
    7. Tim Leung & Michael Ludkovski, 2010. "Optimal Timing to Purchase Options," Papers 1008.3650, arXiv.org, revised Apr 2011.
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    Cited by:

    1. Brian Bulthuis & Julio Concha & Tim Leung & Brian Ward, 2017. "Optimal execution of limit and market orders with trade director, speed limiter, and fill uncertainty," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(02n03), pages 1-29, June.
    2. Tim Leung & Jiao Li & Xin Li & Zheng Wang, 2016. "Speculative Futures Trading under Mean Reversion," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 23(4), pages 281-304, December.
    3. Tim Leung & Zheng Wang, 2019. "Optimal risk-averse timing of an asset sale: trending versus mean-reverting price dynamics," Annals of Finance, Springer, vol. 15(1), pages 1-28, March.
    4. Tim Leung & Xin Li, 2015. "Optimal Mean Reversion Trading With Transaction Costs And Stop-Loss Exit," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(03), pages 1-31.
    5. Jiao Li, 2016. "Trading VIX Futures under Mean Reversion with Regime Switching," Papers 1605.07945, arXiv.org, revised Jun 2016.
    6. Tim Leung & Zheng Wang, 2016. "Optimal Risk-Averse Timing of an Asset Sale: Trending vs Mean-Reverting Price Dynamics," Papers 1610.08143, arXiv.org.
    7. Tim Leung & Jiao Li & Xin Li, 2018. "Optimal Timing to Trade along a Randomized Brownian Bridge," IJFS, MDPI, vol. 6(3), pages 1-23, August.
    8. Jiao Li, 2016. "Trading VIX futures under mean reversion with regime switching," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(03), pages 1-20, September.

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