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Risk-Free Internal Gains – Black And Scholes Re-Examined


  • Gergei Bana

    (University of Pennsylvania)


In this paper we first show that if a not-necessarily-self-financing portfolio has instantaneously riskless internal gains, then on an infinitesimal time-interval, the increase in the internal gains on the portfolio is the same as the change in the price of that amount of bonds which has the same wealth as the portfolio has. Then, using this result, we re-examine the original derivation of the Black-Scholes formula, and conclude that contrary to common belief, the argument of Black and Scholes can be made completely rigorous, employing the same ?-hedge portfolio that they used and keeping all their mathematical formulas; but the explanations they gave to support their formulas must be replaced by others.

Suggested Citation

  • Gergei Bana, 2005. "Risk-Free Internal Gains – Black And Scholes Re-Examined," Finance 0509015, University Library of Munich, Germany.
  • Handle: RePEc:wpa:wuwpfi:0509015
    Note: Type of Document - pdf; pages: 11

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    References listed on IDEAS

    1. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    2. 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.
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    More about this item


    mathematical finance; Black-Scholes formula; internal gains; Wiener process; self-financing portfolio;

    JEL classification:

    • G - Financial Economics


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