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On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models

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  • Jose Cruz
  • Daniel Sevcovic

Abstract

In this paper we focus on qualitative properties of solutions to a nonlocal nonlinear partial integro-differential equation (PIDE). Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of a solution in the scale of Bessel potential spaces. Our aim is to generalize known existence results for a wide class of L\'evy measures including with a strong singular kernel. As an application we consider a class of PIDEs arising in the financial mathematics. The classical linear Black-Scholes model relies on several restrictive assumptions such as liquidity and completeness of the market. Relaxing the complete market hypothesis and assuming a Levy stochastic process dynamics for the underlying stock price process we obtain a model for pricing options by means of a PIDE. We investigate a model for pricing call and put options on underlying assets following a Levy stochastic process with jumps. We prove existence and uniqueness of solutions to the penalized PIDE representing approximation of the linear complementarity problem arising in pricing American style of options under Levy stochastic processes. We also present numerical results and comparison of option prices for various Levy stochastic processes modelling underlying asset dynamics.

Suggested Citation

  • Jose Cruz & Daniel Sevcovic, 2020. "On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models," Papers 2003.03851, arXiv.org.
    • Handle: RePEc:arx:papers:2003.03851
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    References listed on IDEAS

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    9. Ionuţ Florescu & Maria Cristina Mariani & Granville Sewell, 2011. "Numerical solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Levy market," Quantitative Finance, Taylor & Francis Journals, vol. 14(8), pages 1445-1452, August.
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    11. Пигнастый, Олег & Koжевников, Георгий, 2019. "Распределенная Динамическая Pde-Модель Программного Управления Загрузкой Технологического Оборудования Производственной Линии [Distributed dynamic PDE-model of a program control by utilization of t," MPRA Paper 93278, University Library of Munich, Germany, revised 02 Feb 2019.
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    Cited by:

    1. Daniel Sevcovic & Cyril Izuchukwu Udeani, 2021. "Multidimensional linear and nonlinear partial integro-differential equation in Bessel potential spaces with applications in option pricing," Papers 2106.10498, arXiv.org.
    2. Daniel Ševčovič & Cyril Izuchukwu Udeani, 2021. "Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing," Mathematics, MDPI, vol. 9(13), pages 1-12, June.
    3. Jose Cruz & Maria Grossinho & Daniel Sevcovic & Cyril Izuchukwu Udeani, 2022. "Linear and Nonlinear Partial Integro-Differential Equations arising from Finance," Papers 2207.11568, arXiv.org.

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