Hi.

If one posits the usual two-factor model as either:
df(t, T) = (...)dt + exp(-mrs1*(T-t))*sigma1*dW1 + exp(-mrs2*(T-t))*sigma2*dW2
or
dr(t) = (theta + u - lambda1*r(t))*dt + sigma1*dW1
u is another correlated OU process
where we assume <dW1, dW2> = rho.dt

then both representations are of course equivalent, but the parameters are not really intuitive, or orthogonal. 
For instance, rho is not the only parameter that controls the correlation between short rates and long rates, since the difference in MRS matter too. 

Does anyone know of a better more "orthogonal" (i.e. non-overlapping) where one parameter would clearly control the correlation between short rates and long rates ? 

Thanks!

The question is - what do you mean by "better"? What's the purpose of your model?

Thanks for your reply. 

I'm basically after a parametrisation where for instance the instantaneous correlation is driven by one parameter, e.g. <dr(t), df(t,+inf)> = rho.dt, and the mean reversion speed help to control only the mean reversion speed of the short rate and the long-short spread separately. 
But at the moment in the classical way to describe a 2F model, both effects are inter-twinned and massively overlapping.

Still - what's the purpose of your model? Econometric modelling, derivatives pricing or something else?

Derivatives pricing please.

Can you prove your model is arbitrage-free? I don't see it.

I don't know of such a parametrisation. But if you can compute the L-S correlation you are interested in as a function of the MRs and the correlation parameter, then you can presumably hold this quantity fixed (by adjusting rho) as you vary the MRs? 

Derivatives models must be arbitrage-free, otherwise your portfolio bleeds. This is the worst thing that may happen, you loose money without knowing it. The absence of arbitrage must be proven as the first thing.

The model as expressed in the forward rate notation can definitely be made arbitrage free by specifying the (…)dt bit in the drift correctly. I think the form of that is in fact in the original HJM paper. But it’s basically a special case of the standard family of Gaussian rate models or, if you are so inclined, lognormal zero coupon bond price models. The heart of the problem is that the covariance structure of a collection of rates, while deterministic, is a non-trivial function of five parameters: the two volatilities, the two mean reversion parameters, and the correlation coefficient. Note that it collapses to a one-factor model if one of the vols goes to zero, if the correlation coefficient goes to +/- 1, or if the two mean reversion coefficients coincide. I think your best bet is to generate the two non-trivial principal components from the covariance matrix given by the model and explore how they change as you change your parameters. That is not particularly hard, and should help build some intuition.

Derivatives models must be arbitrage-free, otherwise your portfolio bleeds. This is the worst thing that may happen, you loose money without knowing it. The absence of arbitrage must be proven as the first thing.

A) If you know your model is not arbitrage free you might lose money, and you might know it
B) Or you might make money with or without knowing it
C) If it is arbitrage free then you will still make or lose money since your model is unlikely to be perfect
D) But your brain will explode when you discover the profit or loss
E) And most importantly you cannot expect any salary or bonus because where would it come from?

Derivatives models must be arbitrage-free, otherwise your portfolio bleeds. This is the worst thing that may happen, you loose money without knowing it. The absence of arbitrage must be proven as the first thing.

A) If you know your model is not arbitrage free you might lose money, and you might know it
B) Or you might make money with or without knowing it

In theory. In practice all flaws of your models are commonly known and there are enough smart guys who know how to monetize it. Stupid guys lose and smart guys win but it is all right, isn't it?

I understand the practice. You are the TV evangelist who preaches temperance and collects donations while in private snorting cocaine and sleeping with prostitutes.

Would that be a good analogy for when I lecture about vol models? Would I get cancelled? I would have a trigger warning at the start of the lecture for the sensitive young people.

I think it would be OK. It somehow reminded me of a Jimmy Carr quip about Diana’s not wearing a seatbelt. That one would probably not be OK.

Just to be clear, arbitrage freeness is not in question here. The OP is referring to the standard formulation of gaussian IRMs, in HJM or shortrate form, they are definitely arb free (with the correct HJM drift as bearish notes, but that is not in question here).

On the other hand I am enjoying the hooker and blow analogies, reminds me of the bad old days. So fire away.