Monte Carlo Scaling and Black–Scholes Limits for Complex Options

Summary

The discussion examines a paper’s claim that Black–Scholes has familiar limitations for liquid option types such as American and barrier options, and that simulation becomes costly when payoffs depend on multiple securities. The response questions whether these points imply a general dimensionality problem. It notes that crude Monte Carlo’s convergence rate scales with the square root of sample count and does not itself worsen with the number of dimensions, which is a central advantage of simulation for high-dimensional problems.

It also separates difficulties caused by the underlying price model from those caused by payoff structure. Barrier payoffs can be discontinuous, while American options involve an early-exercise decision based on the path. These features complicate valuation even when the underlying follows a familiar model. The response is explicitly tentative: its author had only read the paper’s abstract and introduction, and it does not provide a full comparison of numerical methods, variance reduction, or computational costs. Its challenge is therefore a qualified critique, not a complete treatment of option pricing.

Key ideas

• Crude Monte Carlo has a convergence rate that depends on sample count rather than problem dimension.
• Simulation can be useful for high-dimensional payoffs because it avoids the usual grid-based curse of dimensionality.
• Barrier discontinuities and American early exercise create payoff-related valuation challenges.
• The response distinguishes payoff complexity from the choice of underlying price model.
• The critique is preliminary and does not compare detailed pricing algorithms.

Tags

# BS-Model not suitable for $n$-dimensional options


# BS-Model not suitable for $n$-dimensional options












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.

## Answer by g g (score 1)

https://quant.stackexchange.com/a/58621

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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