Yep. You can make money off of using options as a way of betting on what the volatility measure itself will be. If you think the historical standard deviation is lower than what it will be because of some new change to the company or the world environment, and your view is different from the market's view. It's why sometimes very out of the money call options will paradoxically go up in price after really bad news - you're so far away from the price of the share that the increase in volatility from the price drop increases the option's value even though it's moved even more out of the money

Is that a bug in the equation that one could take advantage of?

In a way, yes. A lot of money follows these standard formulas for pricing which do not necessarily reflect accurate probabilities of the underlier price movement. After an idiosyncratic price shock (disappointing earnings, geopolitical news etc), people blindly following a trailing 1 month volatility or something will misprice the option as volatility reverts back to the mean. This probably has been arbed away to a large extent by trading algorithms.

The underlying assumption Black-Scholes makes, that stock price movements can be modeled by a log-normal distribution, is known to be false. However not since the 1980s has this lead to the ability to make money of the model itself being imperfect.

The true distribution of the market beliefs in future stock prices can be understood by empirically studying the volatility smile [0]. That is, because investors know Black-Scholes is not a perfect mathematical model of real world stock behavior, every strike price has a different implied volatility. By looking at these different IVs you can get a sense of what the market believes are the true probabilities of "long tail" events.

In theory, the opportunities you have to make money should be cases where you believe the market has mispriced risk. In my amateur experience, I have found that virtually every time you think the market has mispriced some extreme event, when you look at the volatility smile, you realize you are mistaken.

0. https://en.wikipedia.org/wiki/Volatility_smile

Not really. The equation is just saying "based off these assumptions here is the best price" and you would make money if your assumptions differ from market assumptions in a favorable direction. Arbitrage is the closest to exploiting "bugs" in finance to get risk free returns but in a liquid enough market all these obvious opportunities quickly close (if there's free money on the ground, someone will pick it up, and then there's no more free money ond the ground. Some hedge funds build ultra fast private internet networks just to be able to pick up that free money nanoseconds faster than someone else). It's more that the equation is telling you if you think you have a better estimate for some of these values, what you should be willing to pay.

Need to separate two different situations here:

1) where there are pretty complete markets for implied volatility, looking at the past matters less to little, because there is a market for the "future volatility" you can hedge and interact with

2) when there isn't a good volatility market and hedging future volatility exposure is difficult, looking towards the past for some guidance increases in importance

Both things can get complicated at times and in both cases it isn't strictly speaking the stddev you care about, but the quadratic variation (which can be the same under some assumptions).

This is why options traders (prototypical ones at least, I suspect most trading is still directional) trade on the implied volatility, as a projection of future volatility.

You take a view on volatility by buying or selling an option, if you are right then you will make money proportional to the options gamma (i.e. the convexity of the option is where the money comes from)

In practice the standard deviation used is implied from option prices, making the whole thing completely circular. To match the market you need to use different standard deviations (volatilities) for different options!

It’s about using the best information you have to quantify and systematize risk.

Volatility is a bit more predictable than price. And there are more complex formulae that also model volatility rather than treat it as a constant.

If volatility is predictable, it would quickly be traded until it became unpredictable and unprofitable.

dr1ver is correct here and you are not.

Actual volatility (not implied!) is much easier to predict than price.

It’s also much more difficult to trade than price changes. So your intuition about this is correct though.

It is not super difficult to predict tomorrows volatility sign (up/down compared to today) with +60% success. Even textbook GARCH models do well here.

If you could do that with the price, you’d quickly become filthy rich.

It is a well known empirical fact that volatility is mean reverting.

the hardest thing obviously is to time the moment when it will start mean reverting.

there is possibility you can take trading position with expectation of vol reverting to the mean, and vol will keep increasing (what happened to tesla and gamestop short sellers)

and vice versa

People have been trading options and futures on the volatility of the S&P500 for years:

https://www.cboe.com/tradable_products/vix/vix_options/

This is also (slightly) incorrect.

When trading the VIX, you are trading the implied volatility not the actual (realized) volatility.

VIX represents the implied vol of options on S&P500 expiring 30 days into the future.

Trading the realized volatility is not easy :)

You can make money just indiscriminately selling option premium, many do. You just have to do it in a way that you're sure you won't blow yourself up. You make money because you are paid for taking on that volatility risk. Just like an insurance company.

The variance component, implied volatility, is more often than not treated as the _output_ of the equation. By looking at the prices of options you can determine what the market currently, implicitly, estimates the future variance of underlying to be.

Lots of options trading involves taking a position on whether you think that implicit estimate is too high or too low. Generally, a long options position encodes belief that volatility is cheap and visa versa. Options are also a very specific kind of instrument and can be used to craft very specific bets on volatility. For instance, you might feel that the options at a $200 strike are pricing too high of an implied volatility compared to those at the $195 and $205 strikes.

Traders build an intuition around the model instead of treating it as in and of itself predictive. They instead try to price or take bets on certain derived quantities from it (the "greeks").

The saying goes that implied volatility is "the wrong quantity put into the wrong model in order to make the right decision".

Adding to this, it's very worthwhile exercise for any curious programmer to work out the IV of a stock based on options pricing and compare it to other measure of volatility (for example historic volatility, or even your own beliefs about volatility based on what you think future returns might be).

Black-Scholes/Merton makes a lot more sense once you work it all out yourself in code.

I'd actually suggest doing this through modeling the underlying geometric Brownian motion and ensuring that your simulated results match up to the analytic formula.

everybody has different model of pricing options, but because everyone knows BS model then the IV (implied vol) becomes as a quoting instrument.

Option Traders can quote each other in IV without disclosing their asset pricing models and assumptions (trade secret tech).