Anyone could explain me what the authors of this paper mean when they say that "The Black–Scholes model, in spite of its popularity, has some well-known deficiencies. Firstly, a closed form formula is not known for many liquid option classes such as American and Barrier options. This forces one to use computationally expensive simulation based methods to price these options, which do not scale well when the payoff of the option depends on the dynamics of multiple securities, that is when the option is high dimensional"?

I know that the results of BS can be always extended to the $n$-dimensional case.

Thanks in advance for any clarification.

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    $\begingroup$ The point is BS does not explain the smile at all. That is its biggest deficiency. It's not about closed form formulas. $\endgroup$ Oct 9, 2020 at 13:13
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    $\begingroup$ Ah well, the context is clearer now, and Bertsimas et al clearly know their maths, so I retract my earlier statements. That said, I still don't understand what Black-Scholes has to do with high-dimensionality, all models have issues with that. Furthermore, I do wonder whether there aren't simpler methods than the one they propose. $\endgroup$ Oct 9, 2020 at 19:49
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    $\begingroup$ I think what the authors are saying is that monte-carlo simulation of options whose price depends on the paths of multiple underlyings is computationally expensive, because most naive monte-carlo methods scale poorly with dimension. Often times, the needed sample size or number of paths one needs to draw to obtain a reasonable estimate of the needed expected value scales exponentially with $d$ - where $d$ is the number of asset paths simulated. $\endgroup$ Oct 9, 2020 at 20:21
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    $\begingroup$ This is an issue with all naive monte-carlo however, not so much Black-Scholes itself. I'd think Black-Scholes would have been optimized for such things, perhaps through numerical PDE approaches or otherwise, but now that I think of it, that most likely scales very poorly as well as numerical PDE runs into the same issues with lots of dimensions. $\endgroup$ Oct 9, 2020 at 20:22
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    $\begingroup$ @rubikscube09 I disagree with your statement that montecarlo scales poorly with dimension, as the number of dimensions increases, I would say it becomes the most appropriate method. $\endgroup$
    – will
    Oct 9, 2020 at 21:17

1 Answer 1


I have not read the paper except for the Abstract and the Introduction but I completely agree with the OP: The statements by the authors are confusing.

The convergence rate of crude Monte-Carlo is $\mathscr{O}(\frac{1}{\sqrt{n}})$ which is independent of the dimension of the problem. This is arguably THE greatest strength of Monte-Carlo: It avoids the curse of dimension.

Furthermore, the issues with barrier and American options are not - in my opinion - related to the law of the underlying but to the pay-out, being either discontinuous (barrier) or relying on the full history (early exercise).

Finally, it is true that there are no Black-Scholes closed form solutions to barrier or American options. But are there any other models offering those?


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