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Hedging with physical or cash settlement under transient multiplicative price impact

Author

Listed:
  • Dirk Becherer

    (Humboldt Universität zu Berlin)

  • Todor Bilarev

    (FactSet Research Systems Inc.)

Abstract

We solve the superhedging problem for European options in an illiquid extension of the Black–Scholes model, in which transactions have transient price impact and the costs and strategies for hedging are affected by physical or cash settlement requirements at maturity. Our analysis is based on a convenient choice of reduced effective coordinates of magnitudes at liquidation for geometric dynamic programming. The price impact is transient over time and multiplicative, ensuring nonnegativity of underlying asset prices while maintaining an arbitrage-free model. The basic (log-)linear example is a Black–Scholes model with a relative price impact proportional to the volume of shares traded, where the transience for impact on log-prices is modelled like in Obizhaeva and Wang (J. Financ. Mark. 16:1–32, 2013) for nominal prices. More generally, we allow nonlinear price impact and resilience functions. The viscosity solutions describing the minimal superhedging price are governed by the transient character of the price impact and by the physical or cash settlement specifications. The pricing equations under illiquidity extend no-arbitrage pricing à la Black–Scholes for complete markets in a non-paradoxical way (cf. Çetin et al. (Finance Stoch. 14:317–341, 2010)) even without additional frictions, and can recover it in base cases.

Suggested Citation

  • Dirk Becherer & Todor Bilarev, 2024. "Hedging with physical or cash settlement under transient multiplicative price impact," Finance and Stochastics, Springer, vol. 28(2), pages 285-328, April.
    • Handle: RePEc:spr:finsto:v:28:y:2024:i:2:d:10.1007_s00780-024-00531-7
    DOI: 10.1007/s00780-024-00531-7
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    References listed on IDEAS

    as
    1. Dirk Becherer & Todor Bilarev & Peter Frentrup, 2018. "Optimal liquidation under stochastic liquidity," Finance and Stochastics, Springer, vol. 22(1), pages 39-68, January.
    2. Bruno Bouchard & Xiaolu Tan, 2022. "Understanding the dual formulation for the hedging of path-dependent options with price impact," Post-Print hal-02398881, HAL.
    3. Alfonsi Aurélien & Alexander Schied & Alla Slynko, 2012. "Order Book Resilience, Price Manipulation, and the Positive Portfolio Problem," Post-Print hal-00941333, HAL.
    4. Robert A. Jarrow, 2008. "Market Manipulation, Bubbles, Corners, and Short Squeezes," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 6, pages 105-130, World Scientific Publishing Co. Pte. Ltd..
    5. Bruno Bouchard & G Loeper & Y Zou, 2017. "Hedging of covered options with linear market impact and gamma constraint," Post-Print hal-01247523, HAL.
    6. Ulrich Horst & Evgueni Kivman, 2024. "Optimal trade execution under small market impact and portfolio liquidation with semimartingale strategies," Finance and Stochastics, Springer, vol. 28(3), pages 759-812, July.
    7. Gur Huberman & Werner Stanzl, 2004. "Price Manipulation and Quasi-Arbitrage," Econometrica, Econometric Society, vol. 72(4), pages 1247-1275, July.
    8. Peter Bank & Dietmar Baum, 2004. "Hedging and Portfolio Optimization in Financial Markets with a Large Trader," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 1-18, January.
    9. Umut Çetin & H. Soner & Nizar Touzi, 2010. "Option hedging for small investors under liquidity costs," Finance and Stochastics, Springer, vol. 14(3), pages 317-341, September.
    10. Soner, H. Mete & Cetin, Umut & Touzi, Nizar, 2010. "Option hedging for small investors under liquidity costs," LSE Research Online Documents on Economics 28992, London School of Economics and Political Science, LSE Library.
    11. Enzo Busseti & Fabrizio Lillo, 2012. "Calibration of optimal execution of financial transactions in the presence of transient market impact," Papers 1206.0682, arXiv.org.
    12. RØdiger Frey, 1998. "Perfect option hedging for a large trader," Finance and Stochastics, Springer, vol. 2(2), pages 115-141.
    13. Obizhaeva, Anna A. & Wang, Jiang, 2013. "Optimal trading strategy and supply/demand dynamics," Journal of Financial Markets, Elsevier, vol. 16(1), pages 1-32.
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    15. Bertsimas, Dimitris & Lo, Andrew W., 1998. "Optimal control of execution costs," Journal of Financial Markets, Elsevier, vol. 1(1), pages 1-50, April.
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    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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