Question is: If B-S and Monte Carlo yielded two different prices for an option, which would you use? It seems as if the risk of mis-pricing an option is so bad you wouldn't want to deviate more than a penny or two from B-S. Otherwise, you are either never able to make a trade or you get taken advantage of by an arbitrageur using B-S.
I can't claim to know the subtle differences between BS and MC outcomes, beyond the likely possibility that MC probably takes longer (compute time) to resolve than BS does. I would expect the differences (if any) in the computed results to be tiny fractions of a penny if the dimensions of the problem are the same, although, I have no empirical evidence to offer in support of that assumption.
If there are more problem dimensions than BS can handle, then I would expect MC to converge on a more accurate solution. But that additional accuracy wouldn't, in and of itself, make MC a more viable solution.
To my way of looking at this, the most significant difference between applying a BS versus an MC solution would be defined by whatever the state of the art capabilities are in high performance computing hardware, as compared with the past. E.g., if MC computes a more accurate solution, but takes 5 seconds longer to do that than BS (or some combined heuristic), then could MC really be regarded as a practical (superior) solution in the context of an exchange trading tool? I don't really know.
With the unrelenting advances in computing horsepower, I would expect, at some point, that MC will become more viable to apply in a wider area of problem spaces. But is the computational horsepower yet available to offer a practical solution for this problem by applying MC now? I don't know.
If there is no arbitrage opportunity, that would be interesting too because then a pair of credit spreads like I'm trying here might offer a way to borrow large sums of money at close to the risk free rate - or at least the borrowing rates large funds have available.
I don't know that there would never be an arbitrage opportunity even if the options were accurately priced in real-time, regardless of the method used to price the options. To suggest otherwise would be to suppose that the main goal of pricing options is to eliminate the possibility of arbitrage.
I tend to think that the primary goal would be to match supply with demand for a particular issue, but I'd have to work in CBOE to know for sure. And I don't. ;)
I do, however, tend to think that any arbitrage opportunities should rapidly correct in response to supply and demand imbalances W.R.T. any correlated issues. Opportunities created by supply and demand imbalances wouldn't be systematic; you'd have to be Jonny-on-the-spot with timing in order to exploit those kinds of imbalances.
I am definitely not a Jonny. But I am certain that there are quite a lot of Jonnys out there. ;)
Update to add: Some of the bigger "Jonnys" have supercomputers co-located and connected by fiber optics, or use microwave network technologies (which is faster than fiber optics), to feed market data to their trading systems much more rapidly than a typical armchair trader could ever hope to do over the internet.
No matter how superior the armchair trader's system might seem, in theory, it suffers a distinct disadvantage in that the data it is using to base decisions on is already old by the time s/he gets it, and has likely already been acted on by someone having faster access to the information, faster computing hardware, and the ability to enter trades, bypassing a broker's system and any human interaction, directly into the exchange.
The race to exploit the information advantage using technology has been on ever since the invention of the telegraph.
The section on low-latency strategies in this article touches on that a little bit:
https://en.wikipedia.org/wiki/High-frequency_trading