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Derivatives Pricing with Market Impact and Limit Order Book

Author

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  • Taiga Saito

    (Graduate School of Economics, The University of Tokyo)

  • Akihiko Takahashi

    (Graduate School of Economics, The University of Tokyo)

Abstract

This paper investigates derivatives pricing under existence of liquidity costs and market impacts for the underlying asset in continuous time. Firstly, we formulate the charge for the liquidity cost and the market impact on the derivatives prices through a stochastic control problem that aims to maximize the mark-to-market value of the portfolio less the quadratic variation multiplied by a risk aversion parameter during the hedging period and the liquidation cost at maturity. Then, we obtain the derivatives price by reduction of this charge from the premium in the Bachelier model. Secondly, we consider a second order semilinear partial differential equation (PDE) of parabolic type associated with the control problem, which is analytically solved or approximated by an asymptotic expansion around a solution to an explicitly solvable nonlinear PDE. Finally, we present numerical examples of the pricing for a variance option and a European call option, and show comparative static analyses.

Suggested Citation

  • Taiga Saito & Akihiko Takahashi, 2017. "Derivatives Pricing with Market Impact and Limit Order Book," CARF F-Series CARF-F-385, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    • Handle: RePEc:cfi:fseres:cf385
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    Cited by:

    1. Taiga Saito & Akihiko Takahashi, 2019. "A novel approach to asset pricing with choice of probability measures," CARF F-Series CARF-F-471, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jan 2021.
    2. Taiga Saito & Akihiko Takahashi, 2021. "Portfolio Optimization with Choice of a Probability Measure," CIRJE F-Series CIRJE-F-1165, CIRJE, Faculty of Economics, University of Tokyo.
    3. Qing-Qing Yang & Wai-Ki Ching & Jiawen Gu & Tak-Kuen Siu, 2020. "Trading strategy with stochastic volatility in a limit order book market," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 43(1), pages 277-301, June.
    4. Taiga Saito & Akihiko Takahashi, 2018. "Online Supplement for "Stochastic Differential Game in High Frequency Market"," CIRJE F-Series CIRJE-F-1087, CIRJE, Faculty of Economics, University of Tokyo.
    5. Taiga Saito & Akihiko Takahashi, 2019. "A Novel Approach to Asset Pricing with Choice of Probability Measures," CIRJE F-Series CIRJE-F-1131, CIRJE, Faculty of Economics, University of Tokyo.
    6. He, Xin-Jiang & Pasricha, Puneet & Lin, Sha, 2024. "Analytically pricing European options in dynamic markets: Incorporating liquidity variations and economic cycles," Economic Modelling, Elsevier, vol. 139(C).
    7. Taiga Saito & Akihiko Takahashi, 2022. "Portfolio optimization with choice of a probability measure (forthcoming in proceedings of IEEE CIFEr 2022)," CARF F-Series CARF-F-534, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.

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