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Risk Minimization with Incomplete Information in a Model for High‐Frequency Data

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  • Rüdiger Frey

Abstract

We study risk‐minimizing hedging‐strategies for derivatives in a model where the asset price follows a marked point process with stochastic jump‐intensity, which depends on some unobservable state‐variable process. This model reflects stylized facts that are typical for high frequency data. We assume that agents in our model are restricted to observing past asset prices. This poses some problems for the computation of risk‐minimizing hedging strategies as the current value of the state variable is unobservable for our agents. We overcome this difficulty by a two‐step procedure, which is based on a projection result of Schweizer and show that in our context the computation of risk‐minimizing strategies leads to a filtering problem that has received some attention in the nonlinear filtering literature.

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  • Rüdiger Frey, 2000. "Risk Minimization with Incomplete Information in a Model for High‐Frequency Data," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 215-225, April.
    • Handle: RePEc:bla:mathfi:v:10:y:2000:i:2:p:215-225
    DOI: 10.1111/1467-9965.00090
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    Cited by:

    1. Colaneri, Katia & Frey, Rüdiger, 2021. "Classical solutions of the backward PIDE for Markov modulated marked point processes and applications to CAT bonds," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 498-507.
    2. Peter Bank & Yan Dolinsky, 2020. "A Note on Utility Indifference Pricing with Delayed Information," Papers 2011.05023, arXiv.org, revised Mar 2021.
    3. Yan Dolinsky, 2023. "Delayed Semi-static Hedging in the Continuous Time Bachelier Model," Papers 2311.17270, arXiv.org, revised Dec 2023.
    4. Ceci, Claudia & Cretarola, Alessandra & Russo, Francesco, 2014. "BSDEs under partial information and financial applications," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2628-2653.
    5. Ceci, Claudia & Colaneri, Katia & Cretarola, Alessandra, 2014. "A benchmark approach to risk-minimization under partial information," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 129-146.
    6. Yan Dolinsky & Jonathan Zouari, 2017. "Market Delay and G-expectations," Papers 1709.09442, arXiv.org, revised Dec 2018.
    7. Dolinsky, Yan & Zouari, Jonathan, 2020. "Market delay and G-expectations," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 694-707.
    8. Sang-Hyeon Park & Kiseop Lee, 2020. "Hedging with Liquidity Risk under CEV Diffusion," Risks, MDPI, vol. 8(2), pages 1-12, June.
    9. Claudia Ceci & Anna Gerardi, 2011. "Utility indifference valuation for jump risky assets," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 34(2), pages 85-120, November.
    10. Mauricio Junca & Rafael Serrano, 2014. "Utility maximization in pure-jump models driven by marked point processes and nonlinear wealth dynamics," Papers 1411.1103, arXiv.org, revised Sep 2015.
    11. Tomoyuki Ichiba & Seyyed Mostafa Mousavi, 2017. "Option Pricing with Delayed Information," Papers 1707.01600, arXiv.org.
    12. Yan Dolinsky & Or Zuk, 2023. "Explicit Computations for Delayed Semistatic Hedging," Papers 2308.10550, arXiv.org.
    13. Yan Dolinsky & Or Zuk, 2023. "Exponential Utility Maximization in a Discrete Time Gaussian Framework," Papers 2305.18136, arXiv.org, revised Jun 2023.
    14. Kiseop Lee & Seongjoo Song, 2007. "Insiders' hedging in a jump diffusion model," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 537-545.
    15. Su, Xiaonan & Wang, Wensheng & Hwang, Kyo-Shin, 2012. "Risk-minimizing option pricing under a Markov-modulated jump-diffusion model with stochastic volatility," Statistics & Probability Letters, Elsevier, vol. 82(10), pages 1777-1785.
    16. Pilar Iglesias & Jaime San Martín & Soledad Torres & Frederi Viens, 2011. "Option pricing under a Gamma-modulated diffusion process," Annals of Finance, Springer, vol. 7(2), pages 199-219, May.

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