IDEAS home Printed from https://ideas.repec.org/a/taf/quantf/v13y2013i1p111-123.html
   My bibliography  Save this article

Derivatives pricing with marked point processes using tick-by-tick data

Author

Listed:
  • Álvaro Cartea

Abstract

I propose to model stock price tick-by-tick data via a non-explosive marked point process. The arrival of trades is driven by a counting process in which the waiting time between trades possesses a Mittag--Leffler survival function and price revisions have an infinitely divisible distribution. I show that the partial-integro-differential equation satisfied by the value of European-style derivatives contains a non-local operator in time-to-maturity known as the Caputo fractional derivative. Numerical examples are provided for a marked point process with conditionally Gaussian and with conditionally CGMY price innovations. Furthermore, the infinitesimal generator of the marked point process derived to price derivatives coincides with that of a L�vy process of either finite or infinite activity.

Suggested Citation

  • Álvaro Cartea, 2013. "Derivatives pricing with marked point processes using tick-by-tick data," Quantitative Finance, Taylor & Francis Journals, vol. 13(1), pages 111-123, January.
    • Handle: RePEc:taf:quantf:v:13:y:2013:i:1:p:111-123
    DOI: 10.1080/14697688.2012.661447
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1080/14697688.2012.661447
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1080/14697688.2012.661447?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Alfonso Dufour & Robert F. Engle, 2000. "Time and the Price Impact of a Trade," Journal of Finance, American Finance Association, vol. 55(6), pages 2467-2498, December.
    2. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    3. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2003. "Stochastic Volatility for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 345-382, July.
    4. Robert F. Engle & Jeffrey R. Russell, 1998. "Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data," Econometrica, Econometric Society, vol. 66(5), pages 1127-1162, September.
    5. Cartea, Álvaro & del-Castillo-Negrete, Diego, 2007. "Fractional diffusion models of option prices in markets with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(2), pages 749-763.
    6. M. Montero, 2008. "Renewal equations for option pricing," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 65(2), pages 295-306, September.
    7. BAUWENS, Luc & HAUTSCH, Nikolaus, 2006. "Modelling financial high frequency data using point processes," LIDAM Discussion Papers CORE 2006080, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    8. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    9. Clark, Peter K, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," Econometrica, Econometric Society, vol. 41(1), pages 135-155, January.
    10. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    11. Álvaro Cartea & Thilo Meyer-Brandis, 2010. "How Duration Between Trades of Underlying Securities Affects Option Prices," Review of Finance, European Finance Association, vol. 14(4), pages 749-785.
    12. Enrico Scalas & Rudolf Gorenflo & Hugh Luckock & Francesco Mainardi & Maurizio Mantelli & Marco Raberto, 2004. "Anomalous waiting times in high-frequency financial data," Quantitative Finance, Taylor & Francis Journals, vol. 4(6), pages 695-702.
    13. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    14. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    15. 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.
    16. Alvaro Cartea & Diego del-Castillo-Negrete, 2007. "On the Fluid Limit of the Continuous-Time Random Walk with General Lévy Jump Distribution Functions," Birkbeck Working Papers in Economics and Finance 0708, Birkbeck, Department of Economics, Mathematics & Statistics.
    17. Robert F. Engle, 2000. "The Econometrics of Ultra-High Frequency Data," Econometrica, Econometric Society, vol. 68(1), pages 1-22, January.
    18. Guido Germano & Mauro Politi & Enrico Scalas & Ren'e L. Schilling, 2008. "Stochastic calculus for uncoupled continuous-time random walks," Papers 0802.3769, arXiv.org, revised Jan 2009.
    19. Alvaro Cartea & Sam Howison, 2009. "Option pricing with Levy-Stable processes generated by Levy-Stable integrated variance," Quantitative Finance, Taylor & Francis Journals, vol. 9(4), pages 397-409.
    20. repec:dau:papers:123456789/1392 is not listed on IDEAS
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Scalas, Enrico & Politi, Mauro, 2012. "A parsimonious model for intraday European option pricing," Economics Discussion Papers 2012-14, Kiel Institute for the World Economy (IfW Kiel).
    2. S. Banihashemi & A. Ghasemifard & A. Babaei, 2024. "On the Numerical Option Pricing Methods: Fractional Black-Scholes Equations with CEV Assets," Computational Economics, Springer;Society for Computational Economics, vol. 64(3), pages 1463-1488, September.
    3. Zhang, Meihui & Jia, Jinhong & Zheng, Xiangcheng, 2023. "Numerical approximation and fast implementation to a generalized distributed-order time-fractional option pricing model," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    4. H. Mesgarani & M. Bakhshandeh & Y. Esmaeelzade Aghdam & J. F. Gómez-Aguilar, 2023. "The Convergence Analysis of the Numerical Calculation to Price the Time-Fractional Black–Scholes Model," Computational Economics, Springer;Society for Computational Economics, vol. 62(4), pages 1845-1856, December.
    5. Ahmad Golbabai & Omid Nikan, 2020. "A Computational Method Based on the Moving Least-Squares Approach for Pricing Double Barrier Options in a Time-Fractional Black–Scholes Model," Computational Economics, Springer;Society for Computational Economics, vol. 55(1), pages 119-141, January.
    6. Yuanda Chen & Zailei Cheng & Haixu Wang, 2023. "Option Pricing for the Variance Gamma Model: A New Perspective," Papers 2306.10659, arXiv.org.
    7. Vasily E. Tarasov, 2019. "On History of Mathematical Economics: Application of Fractional Calculus," Mathematics, MDPI, vol. 7(6), pages 1-28, June.
    8. Fadugba, Sunday Emmanuel, 2020. "Homotopy analysis method and its applications in the valuation of European call options with time-fractional Black-Scholes equation," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).

    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. Álvaro Cartea & Thilo Meyer-Brandis, 2010. "How Duration Between Trades of Underlying Securities Affects Option Prices," Review of Finance, European Finance Association, vol. 14(4), pages 749-785.
    2. Alvaro Cartea & Thilo Meyer-Brandis, 2007. "How Does Duration Between Trades of Underlying Securities Affect Option Prices," Birkbeck Working Papers in Economics and Finance 0721, Birkbeck, Department of Economics, Mathematics & Statistics.
    3. Mr. Noureddine Krichene, 2006. "Recent Dynamics of Crude Oil Prices," IMF Working Papers 2006/299, International Monetary Fund.
    4. Jiang, Zhi-Qiang & Chen, Wei & Zhou, Wei-Xing, 2008. "Scaling in the distribution of intertrade durations of Chinese stocks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(23), pages 5818-5825.
    5. Torricelli, Lorenzo, 2020. "Trade duration risk in subdiffusive financial models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).
    6. Ni, Xiao-Hui & Jiang, Zhi-Qiang & Gu, Gao-Feng & Ren, Fei & Chen, Wei & Zhou, Wei-Xing, 2010. "Scaling and memory in the non-Poisson process of limit order cancelation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(14), pages 2751-2761.
    7. David S. Bates, 2009. "U.S. Stock Market Crash Risk, 1926-2006," NBER Working Papers 14913, National Bureau of Economic Research, Inc.
    8. Gurgul Henryk & Machno Artur, 2017. "Trade Pattern on Warsaw Stock Exchange and Prediction of Number of Trades," Statistics in Transition New Series, Statistics Poland, vol. 18(1), pages 91-114, March.
    9. Jiang, Zhi-Qiang & Chen, Wei & Zhou, Wei-Xing, 2009. "Detrended fluctuation analysis of intertrade durations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(4), pages 433-440.
    10. Scalas, Enrico, 2006. "The application of continuous-time random walks in finance and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 362(2), pages 225-239.
    11. Jingzhi Huang & Liuren Wu, 2004. "Specification Analysis of Option Pricing Models Based on Time- Changed Levy Processes," Finance 0401002, University Library of Munich, Germany.
    12. Lorenzo Torricelli, 2016. "Valuation of asset and volatility derivatives using decoupled time-changed Lévy processes," Review of Derivatives Research, Springer, vol. 19(1), pages 1-39, April.
    13. Antoine Jacquier & Lorenzo Torricelli, 2019. "Anomalous diffusions in option prices: connecting trade duration and the volatility term structure," Papers 1908.03007, arXiv.org, revised Apr 2020.
    14. Claudia Yeap & Simon S Kwok & S T Boris Choy, 2018. "A Flexible Generalized Hyperbolic Option Pricing Model and Its Special Cases," Journal of Financial Econometrics, Oxford University Press, vol. 16(3), pages 425-460.
    15. Askari, Hossein & Krichene, Noureddine, 2008. "Oil price dynamics (2002-2006)," Energy Economics, Elsevier, vol. 30(5), pages 2134-2153, September.
    16. Enrico Scalas, 2006. "Five Years of Continuous-time Random Walks in Econophysics," Lecture Notes in Economics and Mathematical Systems, in: Akira Namatame & Taisei Kaizouji & Yuuji Aruka (ed.), The Complex Networks of Economic Interactions, pages 3-16, Springer.
    17. Enrico Scalas & Mauro Politi, 2012. "A parsimonious model for intraday European option pricing," Papers 1202.4332, arXiv.org.
    18. Li, Junye & Favero, Carlo & Ortu, Fulvio, 2012. "A spectral estimation of tempered stable stochastic volatility models and option pricing," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3645-3658.
    19. Ruan, Yong-Ping & Zhou, Wei-Xing, 2011. "Long-term correlations and multifractal nature in the intertrade durations of a liquid Chinese stock and its warrant," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(9), pages 1646-1654.
    20. Detlef Seese & Christof Weinhardt & Frank Schlottmann (ed.), 2008. "Handbook on Information Technology in Finance," International Handbooks on Information Systems, Springer, number 978-3-540-49487-4, November.

    More about this item

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C41 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Duration Analysis; Optimal Timing Strategies
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

    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:taf:quantf:v:13:y:2013:i:1:p:111-123. 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: Chris Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/RQUF20 .

    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.