Why Short-Rate Models Use Rate Changes and How They Differ from Yield Models
Summary
The document asks why interest-rate models often describe changes in the short rate rather than percentage changes in bond yields, as commonly done for stock prices. The answers clarify that many classic models concern the instantaneous short rate, which is not directly observable, rather than a traded Treasury yield. Their aim is to represent interest-rate risk while excluding changes caused by maturity, credit quality, or liquidity.
Some models instead apply a logarithmic transformation to rates, which keeps modeled rates positive under a lognormal distribution and can matter for interest-rate option values. The discussion also distinguishes short-rate models from term-structure models such as HJM, which describe the yield curve and use no-arbitrage relationships among maturities. The document offers conceptual distinctions rather than a derivation or empirical comparison; model choice and the treatment of negative rates depend on the intended application.
Key ideas
- Many short-rate models represent changes in an instantaneous rate rather than changes in an observed Treasury yield.
- The instantaneous short rate is not directly observable in markets.
- Logarithmic rate models can constrain rates to remain positive under a lognormal specification.
- Term-structure models describe curve dynamics and connect maturities through no-arbitrage relationships.
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# Intuition behind interest rate models
# 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.
## Answer by not.so.quanty (score 2)
https://quant.stackexchange.com/a/12909
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.
## Answer by Kumar (score 1)
https://quant.stackexchange.com/a/12877
These are not yield. They are instantaneous short rates which are not directly observable in the market.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.