The Black–Scholes-Merton (BSM) model is a renowned option pricing model used widely in financial markets. It was published by Fischer Black, Myron Scholes [1], and then Robert Merton in the early 1970s. Scholes and Merton later received the Nobel Memorial Prize in Economic Sciences for their work (Black died before the prize announcement). The model was initially developed to determine the fair value of stock options. It has since then been extended to the pricing of other derivatives such as interest rate options, currency options, commodity options.

Recently, Reference [2] argued that there might have been an error in the derivation of the BSM model,

*The hedging argument of Black and Scholes (1973) hinges on the assumption that a continuously rebalanced asset portfolio satisfies the continuous-time self-financing condition. This condition, which is a special case of the continuous-time budget equation of Merton (1971), is believed to mathematically formalize the economic concept of an asset portfolio that is rebalanced continuously without requiring an inflow or outflow of external funds. Although we sometimes find it hard to believe our results, we believe that we show with three alternative mathematical proofs that the continuous-time self-financing condition does not hold for rebalanced portfolios. In addition, we pinpoint the mistake in the derivation that Merton (1971) uses to motivate the continuous-time budget equation. Specifically, by inadvertently equating a deterministic variable to a stochastic one, Merton (1971) implicitly assumes that the portfolio rebalancing does not depend on changes in asset prices. If correct, our results invalidate the continuous-time budget equation of Merton (1971) and the hedging argument and option pricing formula of Black and Scholes (1973).*

Our thoughts are the following,

- Regardless of whether the derivation was correct or not, there exist assumptions embedded in the BSM model that are not realistic.
- All models in financial markets are wrong. The BSM model is no exception. It’s just a wrong model that gives correct numbers.
- BSM model, despite the fact that some of its assumptions are unrealistic, has proved to be useful and robust in both theoretical and practical contexts.

Let us know what you think in the comments below.

**References**

[1] F. Black, and M. Scholes, *The pricing of options and corporate liabilities*, Journal of Political Economy 81, 639–654, 1973

[2] M. Mink, FJ. de Weert, *Black–Scholes Option Pricing Revisited?*, 2022, https://doi.org/10.48550/arXiv.2202.05671

Guys, hello! Very interesting! Based on my quick look at the reasoning in your arXiv post, I’d say you have found something “real” but you interpret it differently than I do. To compress to just a few sentences, the math of BSM is definitely quirky. It’s not a mathematician’s math. It’s better to think about it from the derivative dealer’s standpoint. There is no rebalancing DURING the time increment delta-t. BSM then use the expectation of changes in market values (first and second moment) – hence eliminating the stochastic aspect at the end of the time increment. The rebalance at the end of the time increment always chases what the market did during the time increment. Taking the limit as time increment goes to zero is, therefore, not really sound or defined in a math sense. Unlike you, apparently, I believe BSM is still “right” – but I’d add that the value of the enunciated trading strategy to generate an option payoff is a greater contribution than the option price formula.