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Stuart_Armstrong comments on When the uncertainty about the model is higher than the uncertainty in the model - Less Wrong
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The model used is the Black-Scholes model with, as you point out, a normal distribution. It endures, despite being clearly wrong, because there doesn't seem to be any good alternatives.
I doesn't endure, not in Risk Management, anyways. Some alternatives for equities are e.g. the Heston Model or other stochastic volatilities approaches. Then, there is the whole filed of systemic risk which studies correlated crashes: events when a bunch of equities all crash at the same time are way more common than they should be and people are aware of this. See e.g. this anaysis that uses a Hawk model to capture the clustering of the crashes.
B-S endures, but is generally patched with insights like these.
I think I can see what you mean, and in fact I partially agree, so I'll try to restate the argument. Correct me if you think I got it wrong. In my experience it's true that B-S is still used for quick and dirty bulk calculations or by organizations that don't have the means to implement more complex models. But the model's shortcomings are very well understood by the industry, and risk managers absolutely don't rely on this model when e.g. calculating the capital requirement for Basel or Solvency purposes. If they did, the regulators will utterly demolish their internal risk model.
There is still a lot of work to be done, and there is what you call model uncertainty at least when dealing with short time scales, but (fortunately) there's been a lot of progress since B-S.
Why were you using an options-pricing model to predict stock returns? Black-Scholes is not used to model equity market returns.
How about the Black-Scholes model with a more realistic distribution?
Or does BS make annoying assumptions about its distribution, like that it has a well-defined variance and mean?
It assumes that the underlying model follows a Gaussian distribution but as Mandelbrot showed a Lévy distribution is a better model.
Black-Scholes is a name for a formula that was around before Black and Scholes published. Beforehand it was simply a heuristic used by traders. Those traders also did scale a few parameters around in a way that a normal distribution wouldn't allow. Black-Scholes then went and proved the formula correct for a Gaussian distribution based on advanced math.
After Black-Scholes got a "nobel prize" people stated to believe that the formula is actually measuring real risk and betting accordingly. Betting like that is benefitial for traders who make bonuses when they win but who don't suffer that much if they lose all the money they bet. Or a government bails you out when you lose all your money.
The problem with Levy distributions is that they have a parameter c that you can't simply estimate by having a random sample in the way you can estimate all the parameters of Gaussian distribution if you have a big enough sample.
*I'm no expert on the subject but the above is my understanding from reading Taleb and other reading.