IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i20p3233-d1499426.html
   My bibliography  Save this article

Pricing of a Binary Option Under a Mixed Exponential Jump Diffusion Model

Author

Listed:
  • Yichen Lu

    (School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China)

  • Ruili Song

    (School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China)

Abstract

This paper focuses on the pricing problem of binary options under stochastic interest rates, stochastic volatility, and a mixed exponential jump diffusion model. Considering the negative interest rates in the market in recent years, this paper assumes that the stochastic interest rate follows the Hull–White (HW) model. In addition, we assume that the stochastic volatility follows the Heston volatility model, and the price of the underlying asset follows the jump diffusion model in which the jumps follow the mixed exponential jump model. Considering these factors comprehensively, the mixed exponential jump diffusion of the Heston–HW (abbreviated as MEJ-Heston–HW) model is established. Using the idea of measure transformation, the pricing formula of binary call options is derived by the martingale method, eigenfunction, and Fourier transform. Finally, the effects of the volatility term and the parameters of the mixed-exponential jump diffusion model on the option price in the O-U process are analyzed. In the numerical simulation, compared with the double exponential jump Heston–HW (abbreviated as DEJ-Heston–HW) model and the Heston–HW model, the mixed exponential jump model is an extension of the double exponential jump model, which can approximate any distribution in the sense of weak convergence, including arbitrary discrete distributions, normal distributions, and various thick-tailed distributions. Therefore, the MEJ-Heston–HW model adopted in this paper can better describe the price of the underlying asset.

Suggested Citation

  • Yichen Lu & Ruili Song, 2024. "Pricing of a Binary Option Under a Mixed Exponential Jump Diffusion Model," Mathematics, MDPI, vol. 12(20), pages 1-14, October.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:20:p:3233-:d:1499426
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/20/3233/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/20/3233/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. JosE Da Fonseca & Martino Grasselli & Claudio Tebaldi, 2008. "A multifactor volatility Heston model," Quantitative Finance, Taylor & Francis Journals, vol. 8(6), pages 591-604.
    2. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    3. Ning Cai & S. G. Kou, 2011. "Option Pricing Under a Mixed-Exponential Jump Diffusion Model," Management Science, INFORMS, vol. 57(11), pages 2067-2081, November.
    4. 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.
    5. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(4), pages 627-627, November.
    6. Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," The Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 573-592.
    7. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    8. He, Xin-Jiang & Zhu, Song-Ping, 2016. "An analytical approximation formula for European option pricing under a new stochastic volatility model with regime-switching," Journal of Economic Dynamics and Control, Elsevier, vol. 71(C), pages 77-85.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ravi Kashyap, 2022. "Options as Silver Bullets: Valuation of Term Loans, Inventory Management, Emissions Trading and Insurance Risk Mitigation using Option Theory," Annals of Operations Research, Springer, vol. 315(2), pages 1175-1215, August.
    2. Dong-Mei Zhu & Jiejun Lu & Wai-Ki Ching & Tak-Kuen Siu, 2019. "Option Pricing Under a Stochastic Interest Rate and Volatility Model with Hidden Markovian Regime-Switching," Computational Economics, Springer;Society for Computational Economics, vol. 53(2), pages 555-586, February.
    3. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    4. Kozarski, R., 2013. "Pricing and hedging in the VIX derivative market," Other publications TiSEM 221fefe0-241e-4914-b6bd-c, Tilburg University, School of Economics and Management.
    5. Duy Nguyen, 2018. "A hybrid Markov chain-tree valuation framework for stochastic volatility jump diffusion models," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 5(04), pages 1-30, December.
    6. Samuel Chege Maina, 2011. "Credit Risk Modelling in Markovian HJM Term Structure Class of Models with Stochastic Volatility," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2011, January-A.
    7. Suresh M. Sundaresan, 2000. "Continuous‐Time Methods in Finance: A Review and an Assessment," Journal of Finance, American Finance Association, vol. 55(4), pages 1569-1622, August.
    8. Duffie, Darrell, 2003. "Intertemporal asset pricing theory," Handbook of the Economics of Finance, in: G.M. Constantinides & M. Harris & R. M. Stulz (ed.), Handbook of the Economics of Finance, edition 1, volume 1, chapter 11, pages 639-742, Elsevier.
    9. repec:wyi:journl:002108 is not listed on IDEAS
    10. repec:uts:finphd:41 is not listed on IDEAS
    11. Cai, Lili & Swanson, Norman R., 2011. "In- and out-of-sample specification analysis of spot rate models: Further evidence for the period 1982-2008," Journal of Empirical Finance, Elsevier, vol. 18(4), pages 743-764, September.
    12. Alexander Lipton, 2024. "Hydrodynamics of Markets:Hidden Links Between Physics and Finance," Papers 2403.09761, arXiv.org.
    13. Qian Li & Li Wang, 2023. "Option pricing under jump diffusion model," Papers 2305.10678, arXiv.org.
    14. Mauricio Contreras G. & Roberto Ortiz H, 2021. "Three little arbitrage theorems," Papers 2104.10187, arXiv.org.
    15. Alexander Lipton, 2023. "Kelvin Waves, Klein-Kramers and Kolmogorov Equations, Path-Dependent Financial Instruments: Survey and New Results," Papers 2309.04547, arXiv.org.
    16. Ravi Kashyap, 2016. "Options as Silver Bullets: Valuation of Term Loans, Inventory Management, Emissions Trading and Insurance Risk Mitigation using Option Theory," Papers 1609.01274, arXiv.org, revised Mar 2022.
    17. Kang, Boda & Ziveyi, Jonathan, 2018. "Optimal surrender of guaranteed minimum maturity benefits under stochastic volatility and interest rates," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 43-56.
    18. Samuel Chege Maina, 2011. "Credit Risk Modelling in Markovian HJM Term Structure Class of Models with Stochastic Volatility," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 5, July-Dece.
    19. Blessing Taruvinga & Boda Kang & Christina Sklibosios Nikitopoulos, 2018. "Pricing American Options with Jumps in Asset and Volatility," Research Paper Series 394, Quantitative Finance Research Centre, University of Technology, Sydney.
    20. Cui, Zhenyu & Lars Kirkby, J. & Nguyen, Duy, 2019. "A general framework for time-changed Markov processes and applications," European Journal of Operational Research, Elsevier, vol. 273(2), pages 785-800.
    21. Mondher Bellalah, 2009. "Derivatives, Risk Management & Value," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 7175, August.
    22. Xin-Jiang He & Sha Lin, 2022. "An Analytical Approximation Formula for Barrier Option Prices Under the Heston Model," Computational Economics, Springer;Society for Computational Economics, vol. 60(4), pages 1413-1425, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:20:p:3233-:d:1499426. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.