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Black-Scholes-Merton Option Pricing Revisited: Did we Find a Fatal Flaw?

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  • Mark Mink
  • Frans J. de Weert

Abstract

The option pricing formula of Black and Scholes (1973) hinges on the continuous-time self-financing condition, which is a special case of the continuous-time budget equation of Merton (1971). The self-financing condition is believed to formalize the economic concept of portfolio rebalancing without inflows or outflows of external funds, but was never formally derived in continuous time. Moreover, and even more problematically, we discover a timing mistake in the model of Merton (1971) and show that his self-financing condition is misspecified both in discrete and continuous time. Our results invalidate seminal contributions to the literature, including the budget equation of Merton (1971), the option pricing formula of Black and Scholes (1973), the continuous trading model of Harrison and Pliska (1981), and the binomial option pricing model of Cox, Ross and Rubinstein (1979). We also show that Black and Scholes (1973) and alternative derivations of their formula implicitly assumed the replication result.

Suggested Citation

  • Mark Mink & Frans J. de Weert, 2022. "Black-Scholes-Merton Option Pricing Revisited: Did we Find a Fatal Flaw?," Papers 2202.05671, arXiv.org, revised Oct 2024.
    • Handle: RePEc:arx:papers:2202.05671
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    References listed on IDEAS

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    1. Merton, Robert C., 1977. "On the pricing of contingent claims and the Modigliani-Miller theorem," Journal of Financial Economics, Elsevier, vol. 5(2), pages 241-249, November.
    2. Merton, Robert C, 1973. "An Intertemporal Capital Asset Pricing Model," Econometrica, Econometric Society, vol. 41(5), pages 867-887, September.
    3. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
    4. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    5. 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.
    6. 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..
    7. 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.
    8. Yaacov Z. Bergman., 1981. "A Characterization of Self-Financing Portfolio Strategies," Research Program in Finance Working Papers 113, University of California at Berkeley.
    9. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
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