To answer
@morganpbrown 's original question I try to make a very dumbed down explanation. Many options veterans may jump on me for this but bare with me, ok?
So the most important thing for understanding options is the fact that you can REPLICATE the options payoff through trades in the underlying. Let's make a simple example:
Let's say a stock is at 100$, you are bullish and want to go long.
You buy 50 shares and install the rule that every time the stock moves 1$ you buy or sell shares in a decreasing manner as the stock moves away from 100$. For every 1$ move away from 100$ stock price you trade 1 less share. You start with buying 10 shares at 101$ and selling 10 shares at 99$, you buy 9 shares at 102$ and sell 9 shares at 98$ and so on.
You will do this over 30 days and then stop, no matter where the stock price is then.
Let's say you are right and the stock goes up to 102$, so your original position of 50 shares is now 69 and stock continues to rise. at 107$ you have 99 shares.
The number of shares you have is your delta and the difference between the current add and the next one is your gamma, which is one share (delta) per 1$ move. Voilà, you replicated a 100$ at the money call option with 30 days to expiration.
Stocks rarely go up in a straight line, and the problem that results from this is that your position is always the highest the more the stock moves in your favour, so you lose more, when the stock retraces.
Let's mix this up a bit.
You start at 100$ with 50 shares. Stock goes to 101, you add 10, it goes to 102$, you add 9 shares, it goes to 103$, you add 8 shares and now it retraces to 102$, where you will sell 8 shares.
The stock goes up to 103$, you add 8 shares again. Your position is 77 shares just like before the retracement, but you have lost 8sharesx1$ = 8$ on the way. The stock keeps on trucking to 107$ without any further retracements after that.
It will be intuitive that the more violently the stock moves back and forth during those 30days the more you will lose. When it goes from 100$ to 105$, back to 102$ before it goes to 106$, the losses you make due to the fact that you buy high and sell low will be higher comparet to a stock that goes from 100$ to 100.20$ to 100.10$ and back to 100.50$.
Still with me? Fine.
The losses you make during the stocks path to target are comparable to the premium you pay for an option.
In the first example we replicated a 30 day 100$ call option that had 8$ premium, as we lost 8$ on the way.
When the stock moves more violently aka is more volatile (or it's "average daily returns are higher"), you lose more, so you pay more premium and vice versa.
With that info fed into our brain, we can go ahead and talk more about volatility:
The examples I gave above generated returns through realized volatility. Thats the volatility the stock actually had during those 30 days. You can calculate it yourself in excel.
Back to options. Let's say the 30day100$ option for our stock in the first example costs 10$. Without crunching the numbers we know that the option is more expensive than our replication by trading shares.
So the option IMPLIES a higher volatility and if we replicated it via trading the stock and sold the option short, we would have made a 2$ profit.
Now to a more sophisticated explanation:
In order to compare nonlinear instruments, you need a common denominator. And some dude figured out, that you can take the options price, add the stock price, time to expiration, option strike to a formula that converts the options $ premium into a volatility figure...because that is the driver of all options premiums.
You cannot trade annualized interest rate of a 3m Short Sterling contract, but you use annualized interest rates to compare that 3m contract to a 10y treasury contract. It's just a math concept. So is implied volatility.
The BS formula is NOT a model that tells you how options should be priced according to the stocks historical volatility. That's how LTCM blew up.
Instead it is a mathematical framework that allows you to compare apples to apples aka compare a 60 day 20 delta option to a 10 day 50 delta option.
Options are driven by supply and demand and that ALWAYS comes first, NOT the volatility figure. You do NOT ask yourself why a 10d 5 delta option trades at 80 vols although realized 10d vol is at 20.
However you ask yourself how you can capture the 60 vols difference as a profit and at this point you speculate.
On the one hand you have a known return distribution which is implied by the options price, on the other hand you have an unknown return distribution which will be realized by how the stock moves during the next ten days.
