IDEAS home Printed from https://ideas.repec.org/p/arx/papers/0812.4010.html
   My bibliography  Save this paper

Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing

Author

Listed:
  • Damiano Brigo
  • Fabio Mercurio

Abstract

In the present paper we construct stock price processes with the same marginal log-normal law as that of a geometric Brownian motion and also with the same transition density (and returns' distributions) between any two instants in a given discrete-time grid. We then illustrate how option prices based on such processes differ from Black and Scholes', in that option prices can be either arbitrarily close to the option intrinsic value or arbitrarily close to the underlying stock price. We also explain that this is due to the particular way one models the stock-price process in between the grid time instants which are relevant for trading. The theoretical result concerning scalar stochastic differential equations with prescribed diffusion coefficient whose densities evolve in a prescribed exponential family, on which part of the paper is based, is presented in detail.

Suggested Citation

  • Damiano Brigo & Fabio Mercurio, 2008. "Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing," Papers 0812.4010, arXiv.org.
    • Handle: RePEc:arx:papers:0812.4010
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/0812.4010
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Martin Schweizer, 1995. "Variance-Optimal Hedging in Discrete Time," Mathematics of Operations Research, INFORMS, vol. 20(1), pages 1-32, February.
    2. Manfred Schäl, 1994. "On Quadratic Cost Criteria for Option Hedging," Mathematics of Operations Research, INFORMS, vol. 19(1), pages 121-131, February.
    3. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    4. 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.
    5. Harrison, J. Michael & Pliska, Stanley R., 1983. "A stochastic calculus model of continuous trading: Complete markets," Stochastic Processes and their Applications, Elsevier, vol. 15(3), pages 313-316, August.
    6. L. C. G. Rogers & S. E. Satchell, 2000. "Does the behaviour of the asset tell us anything about the option price formula? A cautionary tale," Applied Financial Economics, Taylor & Francis Journals, vol. 10(1), pages 37-39.
    7. Brigo, Damiano, 2000. "On SDEs with marginal laws evolving in finite-dimensional exponential families," Statistics & Probability Letters, Elsevier, vol. 49(2), pages 127-134, August.
    8. Martin Schweizer, 1994. "Risk‐Minimizing Hedging Strategies Under Restricted Information," Mathematical Finance, Wiley Blackwell, vol. 4(4), pages 327-342, October.
    9. Damiano Brigo & Fabio Mercurio, 2000. "Option pricing impact of alternative continuous-time dynamics for discretely-observed stock prices," Finance and Stochastics, Springer, vol. 4(2), pages 147-159.
    10. Bouleau, Nicolas & Lamberton, Damien, 1989. "Residual risks and hedging strategies in Markovian markets," Stochastic Processes and their Applications, Elsevier, vol. 33(1), pages 131-150, October.
    11. Schweizer, Martin, 1991. "Option hedging for semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 37(2), pages 339-363, April.
    12. Tina Hviid Rydberg, 1997. "A note on the existence of unique equivalent martingale measures in a Markovian setting," Finance and Stochastics, Springer, vol. 1(3), pages 251-257.
    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. Damiano Brigo, 2008. "The general mixture-diffusion SDE and its relationship with an uncertain-volatility option model with volatility-asset decorrelation," Papers 0812.4052, arXiv.org.
    2. Brigo, Damiano, 2000. "On SDEs with marginal laws evolving in finite-dimensional exponential families," Statistics & Probability Letters, Elsevier, vol. 49(2), pages 127-134, August.

    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. Mercurio, Fabio, 2001. "Claim pricing and hedging under market incompleteness and "mean-variance" preferences," European Journal of Operational Research, Elsevier, vol. 133(3), pages 635-652, September.
    2. repec:dau:papers:123456789/5374 is not listed on IDEAS
    3. Ke Du, 2013. "Commodity Derivative Pricing Under the Benchmark Approach," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2013.
    4. Møller, T., 2002. "On Valuation and Risk Management at the Interface of Insurance and Finance," British Actuarial Journal, Cambridge University Press, vol. 8(4), pages 787-827, October.
    5. D. Brigo, 2023. "Probability-Free Models in Option Pricing: Statistically Indistinguishable Dynamics and Historical vs Implied Volatility," World Scientific Book Chapters, in: David Gershon & Alexander Lipton & Mathieu Rosenbaum & Zvi Wiener (ed.), Options — 45 years since the Publication of the Black–Scholes–Merton Model The Gershon Fintech Center Conference, chapter 4, pages 47-61, World Scientific Publishing Co. Pte. Ltd..
    6. Prigent, Jean-Luc & Renault, Olivier & Scaillet, Olivier, 2004. "Option pricing with discrete rebalancing," Journal of Empirical Finance, Elsevier, vol. 11(1), pages 133-161, January.
    7. Ke Du, 2013. "Commodity Derivative Pricing Under the Benchmark Approach," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2, July-Dece.
    8. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    9. Timothy Johnson, 2015. "Reciprocity as a Foundation of Financial Economics," Journal of Business Ethics, Springer, vol. 131(1), pages 43-67, September.
    10. Alexandre Carbonneau & Fr'ed'eric Godin, 2021. "Deep equal risk pricing of financial derivatives with non-translation invariant risk measures," Papers 2107.11340, arXiv.org.
    11. Hosam Ki & Byungwook Choi & Kook‐Hyun Chang & Miyoung Lee, 2005. "Option pricing under extended normal distribution," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 25(9), pages 845-871, September.
    12. 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.
    13. Gondzio, Jacek & Kouwenberg, Roy & Vorst, Ton, 2003. "Hedging options under transaction costs and stochastic volatility," Journal of Economic Dynamics and Control, Elsevier, vol. 27(6), pages 1045-1068, April.
    14. Jos'e Manuel Corcuera, 2021. "The Golden Age of the Mathematical Finance," Papers 2102.06693, arXiv.org, revised Mar 2021.
    15. Carbonneau, Alexandre, 2021. "Deep hedging of long-term financial derivatives," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 327-340.
    16. Dilip Madan, 2011. "Joint risk-neutral laws and hedging," IISE Transactions, Taylor & Francis Journals, vol. 43(12), pages 840-850.
    17. Chong, Wing Fung, 2019. "Pricing and hedging equity-linked life insurance contracts beyond the classical paradigm: The principle of equivalent forward preferences," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 93-107.
    18. Zhang, Xiaolan, 1995. "Formules quasi-explicites pour les options américaines dans un modèle de diffusion avec sauts," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 38(1), pages 151-161.
    19. Ludovic Gouden`ege & Andrea Molent & Antonino Zanette, 2023. "Backward Hedging for American Options with Transaction Costs," Papers 2305.06805, arXiv.org, revised Jun 2023.
    20. Wayne King Ming Chan, 2015. "RAROC-Based Contingent Claim Valuation," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 3-2015.
    21. Wayne King Ming Chan, 2015. "RAROC-Based Contingent Claim Valuation," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 21, July-Dece.

    More about this item

    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:arx:papers:0812.4010. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.