IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i6p589-d514138.html
   My bibliography  Save this article

Constructing C 0 -Semigroups via Picard Iterations and Generating Functions: An Application to a Black–Scholes Integro-Differential Operator

Author

Listed:
  • Marianito R. Rodrigo

    (School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia)

Abstract

An alternative approach is proposed for constructing a strongly continuous semigroup based on the classical method of successive approximations, or Picard iterations, together with generating functions. An application to a Black–Scholes integro-differential operator which arises in the pricing of European options under jump-diffusion dynamics is provided. The semigroup is expressed as the Mellin convolution of time-inhomogeneous jump and Black–Scholes kernel functions. Other applications to the heat and transport equations are also given. The connection of the proposed approach to the Adomian decomposition method is explored.

Suggested Citation

  • Marianito R. Rodrigo, 2021. "Constructing C 0 -Semigroups via Picard Iterations and Generating Functions: An Application to a Black–Scholes Integro-Differential Operator," Mathematics, MDPI, vol. 9(6), pages 1-15, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:6:p:589-:d:514138
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/6/589/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/6/589/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Jarrow, Robert A & Rosenfeld, Eric R, 1984. "Jump Risks and the Intertemporal Capital Asset Pricing Model," The Journal of Business, University of Chicago Press, vol. 57(3), pages 337-351, July.
    2. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    3. Brown, Stephen J & Dybvig, Philip H, 1986. "The Empirical Implications of the Cox, Ingersoll, Ross Theory of the Term Structure of Interest Rates," Journal of Finance, American Finance Association, vol. 41(3), pages 617-630, July.
    4. Ball, Clifford A & Torous, Walter N, 1985. "On Jumps in Common Stock Prices and Their Impact on Call Option Pricing," Journal of Finance, American Finance Association, vol. 40(1), pages 155-173, March.
    5. Marianito R. Rodrigo, 2020. "Pricing of Barrier Options on Underlying Assets with Jump-Diffusion Dynamics: A Mellin Transform Approach," Mathematics, MDPI, vol. 8(8), pages 1-20, August.
    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. Carolyn W. Chang, 1995. "A No-Arbitrage Martingale Analysis For Jump-Diffusion Valuation," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 18(3), pages 351-381, September.
    2. Marianito R. Rodrigo, 2020. "Pricing of Barrier Options on Underlying Assets with Jump-Diffusion Dynamics: A Mellin Transform Approach," Mathematics, MDPI, vol. 8(8), pages 1-20, August.
    3. Eric Djeutcha & Jules Sadefo-Kamdem & Louis Aimé Fono, 2021. "Mixed Modified Fractional Merton model of the bear spread Basket put option using the multidimensional Mellin transform," Working Papers hal-03330043, HAL.
    4. Khalaf, Lynda & Saphores, Jean-Daniel & Bilodeau, Jean-Francois, 2003. "Simulation-based exact jump tests in models with conditional heteroskedasticity," Journal of Economic Dynamics and Control, Elsevier, vol. 28(3), pages 531-553, December.
    5. Christensen, Kim & Oomen, Roel C.A. & Podolskij, Mark, 2014. "Fact or friction: Jumps at ultra high frequency," Journal of Financial Economics, Elsevier, vol. 114(3), pages 576-599.
    6. 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.
    7. Danielsson, Jon & Zigrand, Jean-Pierre, 2006. "On time-scaling of risk and the square-root-of-time rule," Journal of Banking & Finance, Elsevier, vol. 30(10), pages 2701-2713, October.
    8. Calvet, Laurent E. & Fisher, Adlai J., 2008. "Multifrequency jump-diffusions: An equilibrium approach," Journal of Mathematical Economics, Elsevier, vol. 44(2), pages 207-226, January.
    9. Mi-Hsiu Chiang & Chang-Yi Li & Son-Nan Chen, 2016. "Pricing currency options under double exponential jump diffusion in a Markov-modulated HJM economy," Review of Quantitative Finance and Accounting, Springer, vol. 46(3), pages 459-482, April.
    10. Oliver X. Li & Weiping Li, 2015. "Hedging jump risk, expected returns and risk premia in jump-diffusion economies," Quantitative Finance, Taylor & Francis Journals, vol. 15(5), pages 873-888, May.
    11. Tunaru, Radu & Zheng, Teng, 2017. "Parameter estimation risk in asset pricing and risk management: A Bayesian approach," International Review of Financial Analysis, Elsevier, vol. 53(C), pages 80-93.
    12. Torben G. Andersen & Luca Benzoni & Jesper Lund, 2002. "An Empirical Investigation of Continuous‐Time Equity Return Models," Journal of Finance, American Finance Association, vol. 57(3), pages 1239-1284, June.
    13. Mandeep S. Chahal & Jun Wang, 1997. "Jump Diffusion Processes and Emerging Bond and Stock Markets: An Investigation Using Daily Data," Multinational Finance Journal, Multinational Finance Journal, vol. 1(3), pages 169-197, September.
    14. Andrew Ziogas & Carl Chiarella, 2003. "McKean’s Method applied to American Call Options on Jump-Diffusion Processes," Computing in Economics and Finance 2003 39, Society for Computational Economics.
    15. Patrick Asea & Mthuli Nube, 1997. "Heterogeneous Information Arrival and Option Pricing," UCLA Economics Working Papers 763, UCLA Department of Economics.
    16. Andrew Ziogas, 2005. "Pricing American Options Using Fourier Analysis," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2-2005, January-A.
    17. Leisen, Dietmar P. J., 1999. "The random-time binomial model," Journal of Economic Dynamics and Control, Elsevier, vol. 23(9-10), pages 1355-1386, September.
    18. Andrew Ziogas & Carl Chiarella, 2004. "Pricing American Options on Jump-Diffusion Processes using Fourier-Hermite Series Expansions," Computing in Economics and Finance 2004 177, Society for Computational Economics.
    19. Asea, Patrick K. & Ncube, Mthuli, 1998. "Heterogeneous information arrival and option pricing," Journal of Econometrics, Elsevier, vol. 83(1-2), pages 291-323.
    20. Carl Chiarella & Andrew Ziogas, 2009. "American Call Options Under Jump-Diffusion Processes - A Fourier Transform Approach," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(1), pages 37-79.

    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:9:y:2021:i:6:p:589-:d:514138. 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.