IDEAS home Printed from https://ideas.repec.org/p/pra/mprapa/21523.html
   My bibliography  Save this paper

Geometrical Considerations on Heston's Market Model

Author

Listed:
  • Dell'Era, Mario

Abstract

We propose to discuss a new technique to derive an good approximated solution for the price of a European call and put options, in a market model with stochastic volatility. In particular, the model that we have considered is the Heston's model. This allows arbitrary correlation between volatility and spot asset returns. We are able to write the price of European call and put, in the same form in which one can see in the Black-Scholes model. The solution technique is based upon coordinate transformations that reduce the initial PDE in a straightforward one-dimensional heat equation.

Suggested Citation

  • Dell'Era, Mario, 2010. "Geometrical Considerations on Heston's Market Model," MPRA Paper 21523, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:21523
    as

    Download full text from publisher

    File URL: https://mpra.ub.uni-muenchen.de/21523/1/MPRA_paper_21523.pdf
    File Function: original version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    2. Roger Lord & Remmert Koekkoek & Dick Van Dijk, 2010. "A comparison of biased simulation schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 177-194.
    3. Vladimir Piterbarg, 2005. "Stochastic Volatility Model with Time-dependent Skew," Applied Mathematical Finance, Taylor & Francis Journals, vol. 12(2), pages 147-185.
    4. Christian Kahl & Peter Jackel, 2006. "Fast strong approximation Monte Carlo schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 513-536.
    5. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    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. Dell'Era, Mario, 2010. "Geometrical Approximation method and stochastic volatility market models," MPRA Paper 22568, University Library of Munich, Germany.
    2. Dell'Era, Mario, 2010. "Vanilla Option Pricing on Stochastic Volatility market models," MPRA Paper 25645, University Library of Munich, Germany.
    3. Paul Glasserman & Kyoung-Kuk Kim, 2011. "Gamma expansion of the Heston stochastic volatility model," Finance and Stochastics, Springer, vol. 15(2), pages 267-296, June.
    4. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1, July-Dece.
    5. Mascagni Michael & Hin Lin-Yee, 2013. "Parallel pseudo-random number generators: A derivative pricing perspective with the Heston stochastic volatility model," Monte Carlo Methods and Applications, De Gruyter, vol. 19(2), pages 77-105, July.
    6. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2007.
    7. Wenbin Hu & Junzi Zhou, 2017. "Backward simulation methods for pricing American options under the CIR process," Quantitative Finance, Taylor & Francis Journals, vol. 17(11), pages 1683-1695, November.
    8. Bégin Jean-François & Bédard Mylène & Gaillardetz Patrice, 2015. "Simulating from the Heston model: A gamma approximation scheme," Monte Carlo Methods and Applications, De Gruyter, vol. 21(3), pages 205-231, September.
    9. Nan Chen & Zhengyu Huang, 2013. "Localization and Exact Simulation of Brownian Motion-Driven Stochastic Differential Equations," Mathematics of Operations Research, INFORMS, vol. 38(3), pages 591-616, August.
    10. Mariano González-Sánchez & Eva M. Ibáñez Jiménez & Ana I. Segovia San Juan, 2022. "Market and model risks: a feasible joint estimate methodology," Risk Management, Palgrave Macmillan, vol. 24(3), pages 187-213, September.
    11. Roger Lord & Remmert Koekkoek & Dick Van Dijk, 2010. "A comparison of biased simulation schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 177-194.
    12. Christian Bayer & Chiheb Ben Hammouda & Raul Tempone, 2020. "Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities," Papers 2003.05708, arXiv.org, revised Oct 2023.
    13. Andrei Cozma & Matthieu Mariapragassam & Christoph Reisinger, 2015. "Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets," Papers 1501.06084, arXiv.org, revised Oct 2016.
    14. Benjamin Jourdain & Mohamed Sbai, 2013. "High order discretization schemes for stochastic volatility models," Post-Print hal-00409861, HAL.
    15. Andrei Cozma & Christoph Reisinger, 2017. "Strong order 1/2 convergence of full truncation Euler approximations to the Cox-Ingersoll-Ross process," Papers 1704.07321, arXiv.org, revised Oct 2018.
    16. Michael A. Kouritzin, 2016. "Explicit Heston Solutions and Stochastic Approximation for Path-dependent Option Pricing," Papers 1608.02028, arXiv.org, revised Apr 2018.
    17. Michael A. Kouritzin, 2018. "Explicit Heston Solutions And Stochastic Approximation For Path-Dependent Option Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(01), pages 1-45, February.
    18. Jaehyuk Choi & Yue Kuen Kwok, 2023. "Simulation schemes for the Heston model with Poisson conditioning," Papers 2301.02800, arXiv.org, revised Nov 2023.
    19. Maya Briani & Lucia Caramellino & Giulia Terenzi & Antonino Zanette, 2019. "Numerical Stability Of A Hybrid Method For Pricing Options," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(07), pages 1-46, November.
    20. C'onall Kelly & Gabriel J. Lord, 2021. "An adaptive splitting method for the Cox-Ingersoll-Ross process," Papers 2112.09465, arXiv.org, revised Feb 2023.

    More about this item

    Keywords

    Quantitative methods in Finance;

    JEL classification:

    • D53 - Microeconomics - - General Equilibrium and Disequilibrium - - - Financial Markets
    • D46 - Microeconomics - - Market Structure, Pricing, and Design - - - Value Theory
    • C0 - Mathematical and Quantitative Methods - - General
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

    Statistics

    Access and download statistics

    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:pra:mprapa:21523. 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: Joachim Winter (email available below). General contact details of provider: https://edirc.repec.org/data/vfmunde.html .

    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.