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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

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    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Transient price impact; Multiplicative impact; Hedging; Option settlement; Resilience; Viscosity solution; Geometric dynamical programming; Effective coordinates;
    All these keywords.

    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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