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Option pricing impact of alternative continuous-time dynamics for discretely-observed stock prices

Author

Listed:
  • Damiano Brigo

    (Product and Business Development Group, Banca IMI, San Paolo-IMI Group, Corso Matteotti 6, 20121 Milano, Italy Manuscript)

  • Fabio Mercurio

    (Product and Business Development Group, Banca IMI, San Paolo-IMI Group, Corso Matteotti 6, 20121 Milano, Italy Manuscript)

Abstract

In the present paper we construct stock-price processes with the same marginal lognormal 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 assume any value in-between the no-arbitrage lower and upper bounds. We also explain that this is due to the particular way one models the stock-price process in between the grid time instants that are relevant for trading. The findings of the paper are inspired by a theoretical result, linking density-evolution of diffusion processes to exponential families. Such result is briefly reviewed in an appendix.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:finsto:v:4:y:2000:i:2:p:147-159
    Note: received: March 1998; final version received: March 1999
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    Citations

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    Cited by:

    1. Brigo, Damiano & Jeanblanc, Monique & Vrins, Frédéric, 2020. "SDEs with uniform distributions: Peacocks, conic martingales and mean reverting uniform diffusions," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 3895-3919.
    2. Damiano Brigo & Fabio Mercurio, 2008. "Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing," Papers 0812.4010, arXiv.org.
    3. 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..
    4. John Armstrong & Claudio Bellani & Damiano Brigo & Thomas Cass, 2021. "Option pricing models without probability: a rough paths approach," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1494-1521, October.
    5. 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.
    6. Luciano Campi, 2004. "Arbitrage and completeness in financial markets with given N-dimensional distributions," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 27(1), pages 57-80, August.

    More about this item

    Keywords

    Stock-price dynamics; Black and Scholes model; option pricing; discrete;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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