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

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  • 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.

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  • 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
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    References listed on IDEAS

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    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.
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    Cited by:

    1. 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.
    2. 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.

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