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Hedging with physical or cash settlement under transient multiplicative price impact

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  • Dirk Becherer
  • Todor Bilarev

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 the 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 non-negativity of underlying asset prices while maintaining an arbitrage-free model. The basic (log-)linear example is a Black-Scholes model with relative price impact being proportional to the volume of shares traded, where the transience for impact on log-prices is being modelled like in Obizhaeva-Wang \cite{ObizhaevaWang13} for nominal prices. More generally, we allow for non-linear 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. Pricing equations under illiquidity extend no-arbitrage pricing a la Black-Scholes for complete markets in a non-paradoxical way (cf.\ {\c{C}}etin, Soner and Touzi \cite{CetinSonerTouzi10}) even without additional frictions, and can recover it in base cases.

Suggested Citation

  • Dirk Becherer & Todor Bilarev, 2018. "Hedging with physical or cash settlement under transient multiplicative price impact," Papers 1807.05917, arXiv.org, revised Jun 2023.
  • Handle: RePEc:arx:papers:1807.05917
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    References listed on IDEAS

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    Cited by:

    1. Bruno Bouchard & Xiaolu Tan, 2019. "Understanding the dual formulation for the hedging of path-dependent options with price impact," Papers 1912.03946, arXiv.org, revised Jan 2020.
    2. Bruno Bouchard & Xiaolu Tan, 2019. "Understanding the dual formulation for the hedging of path-dependent options with price impact," Working Papers hal-02398881, HAL.
    3. Bruno Bouchard & Xiaolu Tan, 2022. "Understanding the dual formulation for the hedging of path-dependent options with price impact," Post-Print hal-02398881, HAL.

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