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Pricing formula for European currency option and exchange option in a generalized jump mixed fractional Brownian motion with time-varying coefficients

Author

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  • Kim, Kyong-Hui
  • Yun, Sim
  • Kim, Nam-Ung
  • Ri, Ju-Hyuang

Abstract

In this paper, a new framework for pricing the European currency option is developed in the case where the spot exchange rate follows a generalized mixed fractional Brownian motion with jumps (hereafter GJMFBM). In addition we consider a general case that the coefficients of the model are time-varying. To capture the behaviors of exchange rate, the combination of Poisson jumps and generalized mixed fractional Brownian motion is introduced. To derive the pricing formula for some options, we firstly derive a generalized mixed fractional Girsanov theorem and some results regarding the quasi-conditional expectation that we will need for the rest of the paper. Then analytic pricing formulas for European currency option and exchange option are obtained using the equivalent martingale measure. Finally, through some numerical experiments and discussion we show that the GJMFBM model is different with the other previous ones.

Suggested Citation

  • Kim, Kyong-Hui & Yun, Sim & Kim, Nam-Ung & Ri, Ju-Hyuang, 2019. "Pricing formula for European currency option and exchange option in a generalized jump mixed fractional Brownian motion with time-varying coefficients," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 522(C), pages 215-231.
    • Handle: RePEc:eee:phsmap:v:522:y:2019:i:c:p:215-231
    DOI: 10.1016/j.physa.2019.01.145
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    References listed on IDEAS

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    Cited by:

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    2. Dastranj, Elham & Sahebi Fard, Hossein & Abdolbaghi, Abdolmajid & Reza Hejazi, S., 2020. "Power option pricing under the unstable conditions (Evidence of power option pricing under fractional Heston model in the Iran gold market)," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
    3. Ying Chang & Yiming Wang & Sumei Zhang, 2021. "Option Pricing under Double Heston Jump-Diffusion Model with Approximative Fractional Stochastic Volatility," Mathematics, MDPI, vol. 9(2), pages 1-10, January.
    4. Liu, Zhibin & Huang, Shan, 2021. "Carbon option price forecasting based on modified fractional Brownian motion optimized by GARCH model in carbon emission trading," The North American Journal of Economics and Finance, Elsevier, vol. 55(C).
    5. Anwer, Zaheer & Khan, Ashraf & Kabir Hassan, M. & Rashid, Mamunur, 2022. "Does the regional proximity lead to exchange rate spillover?," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 81(C).
    6. Axel A. Araneda, 2021. "Price modelling under generalized fractional Brownian motion," Papers 2108.12042, arXiv.org, revised Nov 2023.

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