We created the SABR model because we realized that (a) option values were nonlinear in the volatility, and (b) volatilities are stochastic. This means that if one had an option (or portfolio of options) which have positive gamma in the volatility dimension, on average we'd make money from fluctuations in the volatility, and we'd lose money with negative vol-gamma. To be fair, these gains or losses should be compensated for in the daily carry ... it's just Black Scholes in the volatility dimension. We created the model so that we wouldn't leak money due to the volatility of the volatility. It turns out that at-the-money options are nearly linear in the volatility, so there is little vol-gamma for ATM options, and much higher levels of vol-gamma for options away from the money ... so the SABR price correction is much stronger away from the money, resulting in a volatility smile. Pat

The are implied parameters. You basically do a parametric dimension reduction by implying them across a range of observed prices, checking the errors, and then you might interpolate or even cautiously extrapolate. In reality, desks will have their own spreads but you can generate a volatility surface as a baseline from these parameters.

If I remember correctly, there might be some issues with this parametrization in terms of stability depending on what you are doing.

Let's relabel this as What (TF) is SABR?

Alpha, Beta and Rho are the point of the model. So explaining them is explaining the model.

A model of two processes

Unlike earlier models in which the volatility was modelled as a constant (Vasicek, Hull-White, etc), SABR assumes that as well as the price of the thing being stochastic, so is its volatility. That is, the volatility will also follow some stochastic path.

Thus we have two clearly related processes; the price, lets say of a forward rate (following the Wikipedia notation):

$$dF_t = sigma_t F_t^beta dW_t$$

Which just means the changes in the price are proportional to the price itself raised to power $beta$, and a Wiener process $W_t$, scaled by the now time-dependent volatility $sigma_t$.

We also have a process for the volatility $sigma_t$:

$$dsigma_t = alpha sigma_t dZ_t$$

Again, the changes in volatility are proportional to the volatility itself (so the behaviour is scale invariant) and to a second Wiener process $Z_t$, all scaled this time by $alpha$.

$alpha$ is then the (constant) volatility of the volatility. I mean, we could model that as stochastic too but that seems like hard work.

So where is $rho$?

$alpha$ was the volvol, $beta$ was the power in the price relation, we are missing $rho$.

Since the two processes (the price and its volatility) are very much related, the SABR model connects the two Wiener processes driving their movement by making them correlated with parameter $rho$:

$$dW_t dZ_t = rho dt$$

So changes in the two Wiener processes are correlated with $rho$ in time. Again, $rho$ is a constant.

So none of $alpha$, $beta$ or $rho$ are stochastic; perhaps the name should have been Stochastic Volatility, Alpha Beta Rho. But SVABR is much less catchy.

How am I going to price anything with SABR when no one quotes $alpha/ beta/ rho$?

Ah yes. While the market does quote volatilities, it doesn't quote these parameters, so it's hard to just knit a model in Excel and wear it.

The equations we have so far model the dynamics given the parameters, so in order to get the parameters we will have to essentially solve for the parameters given some other stuff, like market prices for options which are sensitive to those parameters.

Calibrating a set of parameter values to market quotes is the subject of much effort, e.g. this blog post.

All models are finite

No model is able to magically capture all the information available, and there would be no point; a model's power is in deriving simpler truths than the information you start with. With SABR the model better recreates the dynamics of the evolution of an interest rate, but note that there are just a small, fixed number of parameters. So it cannot calibrate perfectly to a market with tens or hundreds of inputs.