How HJM and Markovian Short-Rate Models Represent Yield-Curve Dynamics
Summary
The document distinguishes what HJM and commonly used Markovian short-rate models say about interest-rate curves. HJM specifies the current forward curve and models its evolution across maturities. A short-rate model specifies the evolution of the instantaneous rate; in a one-factor Markovian model, bond prices and yields at each maturity are functions of that single state variable.
The answer uses principal component analysis as intuition: yield curves can exhibit parallel shifts, level changes that invert the curve, and twists. A model with multiple Brownian drivers can represent separate curve movements, while a one-factor Markovian short-rate model cannot produce independent movements of that kind. The distinction is not that every HJM model captures every possible curve dynamic; its number and structure of factors still constrain its behavior. The discussion concerns the commonly used Markovian short-rate models and does not claim that every model labeled a short-rate model has the same limitations.
Key ideas
- HJM models the forward-rate curve across maturities, including its current shape.
- A one-factor Markovian short-rate model makes bond prices at each maturity functions of a single short-rate state.
- A single state variable constrains the yield curve to move along linked shapes rather than independently twisting or shifting.
- HJM models also have limits: their ability to represent curve movements depends on their factors and specification.
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# Difference HJM Framework versus Short rate model
# Difference HJM Framework versus Short rate model
Recently I study some interest rate models.
When I moved on to forward rate models, I see this documents
https://en.wikipedia.org/wiki/Heath-Jarrow-Morton-_framework
It said "HJM-type models capture the full dynamics of the entire forward rate curve, while the short-rate models only capture the dynamics of a point on the curve"
What I don't understand is that why short rate model can't not capture the full dynamics of curve? It seems that the only difference in the model is the short rate 'r(t,T)' is substituted for instantaneous forward rate 'f(t,s)'
I mean, the short rate model : $dr(t,s) = μ(t,s)dt + σ(t,s)dW_t$
and the HJM Framework : $df(t,s) = μ(t,s)dt + σ(t,s)dW_t$
So I think the short rate model can capture the dynamics of full term structure. Also any no arbitrage short rate model could make current term structure.
I tried to understand meaning of the above bold&italic sentences with the fixed income securities textbook such as Tuckman, Veronesi.. But it fails.
What I'm misunderstanding now?
## Answer by Kurt G. (score 3, accepted)
https://quant.stackexchange.com/a/70581
Most principal component analyses (PCAs) on historical data of yield curves find that typically a yield curve
- moves parallel
- flips from normal to inverse (or vice versa)
- twists (changes its curvature)
As in every PCA the drivers of these movements are uncorrelated standard normals of one dimension each.
Mathematically you could model this by a HJM model that is driven by three Brownian motions.
- I do not believe that every HJM model captures the full dynamics of the entire forward curve. If that were true even the poorest HJM model (namely the one you get when you start with the Ho-Lee model) could capture the full dynamics.
- What is true is that a HJM model by definition captures the shape of the current yield curve because that is just the curve $T\mapsto f(0,T)$ which is part of the model.
- Short rate models are typically Markovian (at least those that are mostly used such as Vasicek, CIR, Black-Karasinski, etc.). This Markov property means that every conditional zero bond price $$ P(t,T)=\textstyle\mathbb E\Big[\exp\Big(-\int_t^Tr(s)\,ds\Big)\Big|{\cal F}_t\Big] $$ is a deterministic function of the single variable $r(t)$: $$ P(t,T,r(t))\,. $$ In turn, this means that the yield curve $$ Y(t,T)=-\log P(t,T)/(T-t) $$ or its continuously compounded sister $$ f(t,T)=-\frac{\partial}{\partial T}\log P(t,T) $$ must also be a deterministic function of a single variable $r(t)$.
This precludes that the dynamics of the yield curve in a Markovian short rate model can perform uncorrelated parallel shifts, flips or twists.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.