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If this preprint (which is discussed here on QSE) is correct in showing that there are mathematical mistakes in the Black-Scholes-Merton option pricing framework, are there strands of the economic literature that are likely to be affected, or would this mainly be an issue for the finance literature?

For example, we show in the preprint that the continuous-time budget equation of Merton (1971), which Black and Scholes (1973) use implicitly to derive their option pricing formula, is mathematically misspecified. In the finance literature, this budget equation is used to model the return on an investment portfolio when the portfolio weights are continuously adjusted and when asset prices follow stochastic processes such as the geometric Brownian motion. To better understand the implications of these findings, we wonder which are the main (if any) economic theories that use Merton's (1971) budget equation or the Black and Scholes (1973) formula?

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  • $\begingroup$ It would be more transparent if you would point out that you are one of the authors of the preprint. $\endgroup$
    – Michael Greinecker
    Jul 19 at 21:20
  • $\begingroup$ @MichaelGreinecker, the OP writes We show in the preprint..., so technically they do reveal their authorship, but yes, they could be more explicit. $\endgroup$
    – Richard Hardy
    Jul 20 at 5:30

3 Answers 3

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The Black-Scholes-Merton (BSM) option pricing model is one of the most important and widely used models in finance. It has been used to price a wide variety of options, including stock options, bond options, and foreign exchange options. The BSM model is based on a number of assumptions, including the assumption that asset prices follow a geometric Brownian motion.

The preprint you linked to argues that there are mathematical mistakes in the BSM model. If these mistakes are correct, it could have implications for a number of strands of the economic literature. For example, the BSM model has been used to study the relationship between stock prices and volatility, the impact of dividends on option prices, and the pricing of options on assets that do not follow a geometric Brownian motion.

If the findings are correct, it could lead to a rethinking of a number of economic theories that rely on the BSM model.

Here are some of the main economic theories that use the BSM model or Merton's budget equation:

  • Portfolio theory: Portfolio theory is the study of how to construct and manage a portfolio of assets. The BSM model is often used to calculate the value of a portfolio of options.

  • Risk management: Risk management is the process of identifying, measuring, and managing risks. The BSM model is often used to calculate the risk of an option position.

  • Corporate finance: Corporate finance is the study of how to raise and manage capital for a business. The BSM model is often used to calculate the value of a corporate option, such as a call option on a stock or a put option on a bond. It is important to note that the BSM model is not without its limitations. For example, the model assumes that asset prices follow a geometric Brownian motion, which is not always the case. Additionally, the model does not take into account factors such as dividends, interest rates, and transaction costs.

Despite its limitations, the BSM model is a powerful tool that can be used to price a wide variety of options. If the findings of the preprint are correct, it could lead to a rethinking of the BSM model and its use in economics

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    $\begingroup$ This answer is so generic and superficial, that it makes me wonder if it was written with ChatGPT? $\endgroup$
    – AKdemy
    Jun 18 at 7:18
  • $\begingroup$ from whatever I know... Don't know a whole lot $\endgroup$
    – user1874594
    Jun 18 at 8:56
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The Black-Scholes-Merton model makes a number of assumptions on the theoretical behavior of the involved quantities, for example that the asset prices behave like a geometric Brownian motion as written in user1874594 answer. It is known that these assumptions are not satisfied for real life financial assets. Sometimes real life patterns are well approximated by these assumptions, sometimes not even that.

Nevertheless the BSM model has proven to be useful in applications to real life finance. It is widely applied and it works (most of the time). This is a classic example of the physics saying: all models are wrong but some models are useful.

So the original BSM paper says if you make these theoretical assumptions then you can draw these theoretical conclusions. The preprint now claims you can't quite draw these conclusions. For practical purposes this preprint makes no difference whatsoever.

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  • $\begingroup$ This does not answer the question, which does not ask about practical purposes. $\endgroup$
    – Giskard
    Jun 19 at 13:48
  • $\begingroup$ The "physics" quote is usually attributed to the late statistician George E. P. Box. $\endgroup$
    – Richard Hardy
    Jun 20 at 8:27
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You can derive the Black-Scholes formula just by assuming (a) stocks obey a lognormal distribution and (b) this distribution has a risk neutral expectation. I don’t see how that can be ‘wrong’ due to some technicalities about self financing.

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  • $\begingroup$ Can you please back up your claim "You can derive the Black-Scholes formula just by..." with a reference? $\endgroup$
    – Giskard
    Jun 19 at 13:49
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    $\begingroup$ Yes, method 5 frouah.com/finance%20notes/Black%20Scholes%20Formula.pdf $\endgroup$
    – dm63
    Jun 19 at 17:42
  • $\begingroup$ Hi dm63, thank you for sharing your thoughts. We agree that one can derive the BS formula (a little trivially) if one assumes a risk-neutral world. While we are not in a risk-neutral world, the argument of Black and Scholes is that we can assume a risk-neutral world nonetheless because the option can be replicated. Their replication argument hinges on the self-financing condition, however, so without that condition one can no longer assume risk-neutrality. $\endgroup$
    – MMFdW
    Jun 19 at 18:01
  • $\begingroup$ Hi, I think that the proof I referred to utilizes a stock price distribution whose expected value is equal to the forward price. This surely must be correct because otherwise you violate put-call parity. Do you claim that the forward price of the stock is not equal to its risk neutral expectation ? $\endgroup$
    – dm63
    Jun 20 at 22:35
  • $\begingroup$ Hi dm63, we agree with you that the forward price is equal to the risk-neutral expectation of the stock price. However, for the claim of Black and Scholes that option prices can also be calculated by assuming risk-neutrality, one needs the self-financing condition. See for example Bjork’s (2009) textbook (and the preprint discusses this too). $\endgroup$
    – MMFdW
    Jun 21 at 10:40

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