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.
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".