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Numerical Solution of Heston-Hull-White Three-Dimensional PDE with a High Order FD Scheme

Author

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  • Malik Zaka Ullah

    (Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

Abstract

A new numerical method for tackling the three-dimensional Heston–Hull–White partial differential equation (PDE) is proposed. This PDE has an application in pricing options when not only the asset price and the volatility but also the risk-free rate of interest are coming from stochastic nature. To solve this time-dependent three-dimensional PDE as efficiently as possible, high order adaptive finite difference (FD) methods are applied for the application of method of lines. It is derived that the new estimates have fourth order of convergence on non-uniform grids. In addition, it is proved that the overall procedure is conditionally time-stable. The results are upheld via several numerical tests.

Suggested Citation

  • Malik Zaka Ullah, 2019. "Numerical Solution of Heston-Hull-White Three-Dimensional PDE with a High Order FD Scheme," Mathematics, MDPI, vol. 7(8), pages 1-13, August.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:8:p:704-:d:255023
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    References listed on IDEAS

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    1. Nusret Cakici & Sris Chatterjee & Ren-Raw Chen, 2019. "Default Risk and Cross Section of Returns," JRFM, MDPI, vol. 12(2), pages 1-15, June.
    2. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    3. Zhang Sumei & Zhao Jieqiong, 2017. "Efficient Simulation for Pricing Barrier Options with Two-Factor Stochastic Volatility and Stochastic Interest Rate," Mathematical Problems in Engineering, Hindawi, vol. 2017, pages 1-8, November.
    4. Ballestra, Luca Vincenzo & Cecere, Liliana, 2016. "A numerical method to estimate the parameters of the CEV model implied by American option prices: Evidence from NYSE," Chaos, Solitons & Fractals, Elsevier, vol. 88(C), pages 100-106.
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    Cited by:

    1. Tao Liu & Zixiao Zhao & Shiyi Ling & Heyang Chao & Hasan Fattahi Nafchi & Stanford Shateyi, 2024. "Efficient Scheme for the Economic Heston–Hull–White Problem Using Novel RBF-FD Coefficients Derived from Multiquadric Function Integrals," Mathematics, MDPI, vol. 12(14), pages 1-15, July.
    2. Tao Liu & Malik Zaka Ullah & Stanford Shateyi & Chao Liu & Yanxiong Yang, 2023. "An Efficient Localized RBF-FD Method to Simulate the Heston–Hull–White PDE in Finance," Mathematics, MDPI, vol. 11(4), pages 1-15, February.

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