Intuition behind interest rate models

I am modelling the 3M yield of US Treasuries using an ARMA/ GARCH approach. Most interest rate models (e.g. Vasicek) describe the process as follows:

$r_{t}-r_{t-1} = some ARMA+ \epsilon_t$

Where $r_t$ is the yield of a bond. Why do you use the yield instead of the return of the yield (relative price change to last period)? For example if you model stock returns you use the return as well. I am very well aware of the relation between price and yield of a bond.

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See Section 3.1.2 symmys.com/sites/default/files/… –  John Jun 12 at 14:16

These are not yield. They are instantaneous short rates which are not directly observable in the market.

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Some models do use ln(r_t), like Black–Derman–Toy and the Black–Karasinski models. Mainly to avoid negative interest rates in low rates / high volatility environments through the use of the log-normal distribution. Negative rates can wreak havoc in option premiums for example.

They are interest rates indeed, that we call short rates, not yield on treasuries. The idea is to find a stochastic process that models only the risk of changes in instantaneous interest rate and not those of the change in maturity, creditworthiness, liquidity, etc.

Interest rate curve models that tackle dynamics of the interest rate curve like HJM take their root in non-arbitrage arguments by stating that equivalent securities must have the same price and that longer maturity treasuries are effectively composed of strips of shorter maturity bonds, which links the treasury curve components.

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