I don’t believe any of the comments below address the meat of your question - what value are you getting from the equation ?

I would answer- not much.

You can think of BS as a curried function. Since all the other params are fixed, you can curry and get a reduced equation that only depends on IV and underlying. If you do that, then its just - you give me iv and underlying, i give you spot. So, for a given strike(fixed), with the prevailing time left(fixed theta) under current interest rate(fixed), given the underlying, the historical vol gives you the wrong spot. You fudge it until you get the right spot. Call the fudged quantity the IV. Now plot that fudged quantity for a few other strikes and you get a smile. Then you can mess with that smile, plot the vol surface etc but end of the day, does the BS equation matter if the price of spot is going to be off and you have to fudge it with IV ? Its a good question. From an operational standpoint, the equation doesn’t matter. You can use bopm and get a more intuitive price anyways. Traders can trade the iv without knowing what effect BS has on the system.

When I was in 5th grade, they took us to the top of a tall building. We dropped a ball and measured the time it took to hit the ground. So if you square that time and multiply by 5, that’s how tall that building is. At that age I thought wow this is such magic! Then I grew up and reached 8th grade and worked out equations of motion with some basic differential calc, and derived the canonical equation s equals ut plus half at square. So since u is zero and a on planet earth happens to be g which is 9.8, half of which is about 5, that’s why 5t^2.

ok but does this equation matter ? I could have gone my whole life measuring height of buildings without knowing what is gravity.

> If you do that, then its just - you give me iv and underlying, i give you spot.

But... what really happens (in my opinion) is... options makers or writers or whatever might set a price based on what they feel is fair/good for them/whatever

Then a bunch of people on Robinhood make memes over it, hammer the bid, IV goes to 160%, voila...

Why does "spot" price matter in that equation? Robinhood buyers + supply/demand are what drives IV in reality I feel.

> options makers or writers or whatever might set a price based on what they feel is fair/good for them/whatever

Former options market maker. We basically made money because of (a) people setting prices based on gut feel and (b) retail investors buying options for leverage and then forgetting to exercise barely in-the-money contracts. The first has largely left the market; fortunately, the second came in with gale force.

> Robinhood buyers + supply/demand are what drives IV in reality

Of course. Supply and demand drive price. Volatility is a measure on price. Options are principally an instrument for trading volatility.

Former option trader here. The free parameter is actually the "thing" that you're actually trading when you trade an option. All the other parameters are just environmental, you look them up.

The short story is that the implied vol is a sort of balancing price between how much the option loses in value over time vs how much you can make performing the hedge.

Why do any of the other variables matter if the market is collectively fighting between "overpriced and underpriced" on premium/implied volatility?

The Black-Scholes equation describes the unconscious biases that influence the price of options. It is able to beautifully separate the one quantity that every trader prices differently from the rest. That’s volatility.

All the other factors, time including, are the same for everyone.

You can find an IV that makes sense for a single option with invalid other parameters, but things will break down when you go to price other expirations / strikes.

When trading, you don't want to wait to see an "updated" IV, you would want to respond directly to changes in important and well understood parameters like underlying price.

They don't really matter. They are just things you look up in order to a get a number out for what the option costs in dollars.

Was BS actually of any practical use? It just seems to be a fantasy like most maths in economics.

Yes, it has some use. First of all, if you don't have a common model, it becomes impossible to talk about vol. So even if everyone uses their own model, they convert back into BS vol to talk about vol. Second, all models are wrong, but some models are useful. BS is a good starting point because it captures something that is relevant in option pricing, namely that uncertainty matters.

It's not really implied volatility, it's the actual volatility between now and expiration. Everyone can only estimate at what that will be.

IV is essentially using prevailing prices to understand what everyone else has estimated that forward volatility to be.

Beyond that, you will also find that IV differs across strikes [1]. Still, being able to fit a vol smile from incomplete market data (and some other adjustments if you are very sophisticated) and then price an arbitrary option is pretty useful.

[1] https://en.wikipedia.org/wiki/Volatility_smile

The parameter (sigma) is the historical volatility (stdev of annualized returns). Implied volatility is what you get if you run the formula backwards and input the observed price to solve for volatility. In practice though many people use implied volatility as the input making the whole thing circular.

It's no really circular, just think of IV as the price/what is traded.

Implied volatility is really the standard deviation of the price over time. You can calculate it by look at prices in the market. Then interpolate values. Where banks get funky is that the market for options go out about 3 years, but a banks will write options going out much much further. For those options, they are really just guessing, no matter how much fancy math they do, it's all to dress up a guess. And the traders don't care since they won't be around when the option expires

> Implied volatility is really the standard deviation of the price over time.

Maybe you mean that “implied volatility is really the implied standard deviation of the price over time”.

Sorry, yes, that is the better definition.

This equation is an idealized option, a spherical cow. If one would actually use it to price options one would lose money.

There are many empirical option pricing features that this equation can't explain - the "smile", the "skew", ...

> If one would actually use it to price options one would lose money

If you try to fly a rocket across the solar system using only Newton's equations, it will crash. That doesn't make Newtonian mechanics useless. Almost every option-pricing engine in the market starts with Black-Scholes-Merton. Smiles and skews are all dealt with on the vol surface--it's an expandable variable.

One thing that seems not widely understood is that the assumption of normal or log-normal distributions in stochastic calculus is like the assumption of linearity in most engineering fields. It's known to be incorrect, but you can do a heck of a lot with piecewise-linear model. Similarly, using a normal or log-normal assumption within a range of parameters (such as the volatility smile) is really useful.

Part of the reason a Gaussian distribution is used so much is that you need a stable distribution if you want to be able to perform algebra on your random variables. The variance of the Cauchy distribution is undefined and the variance of the Levy distribution is infinite, so Gaussian is really the go-to distribution.