Tuckman, equilibrium vs arbitrage-free

[EK]

Dear All,
I've done all questions and read the chapters assigned, yet there's one thing I cannot get enough clarity on. Tuckman describes 3 models (m1, m1+drift, Ho-Lee), then he makes some interesting points regarding model types (equilibrium and arbitrage-free), and then again gets back to models (Vasicek, lognormal, etc.). However, I cannot find anywhere the cross-reference between model types and described models! For instance, would "Ho-Lee is an arbitrage-free model" be an accurate statement? Vasiscek? m1/m2? etc.

 

[EK]

Another "general" question on Tuckman is - which models do support recombining trees and which do not?

 

[Bryon]

It is mentioned in the text, p.64 that M1 and M2 is equilibrium model. Ho-Lee is a class of arbitrage-free model. All 3 are recombining tree model.
The mean reverting model of Vasiscek is probably a mix of 2 depends on how the parameters are chosen/calibrated but not discussed anywhere in the text. This model can result in both recombining or non-recombining trees ( example in the text).

 

[perviz]

How can I differ arbitrage free model from equilibrium model (or vice versa) ?

 

[David Harper CFA FRM]

Hi @perviz In the revised Tuckman notes, I introduce the relevant LO with the following: "Equilibrium models (e.g., Model 1, Model 2, Vasicek, CIR) return a term structure as output and therefore do not match the current, observed term structure. Arbitrage models, on the other hand, take the observed term structure as an input and therefore do match the current, observed term structure."

Since you asked a good question, I also revised the summary table in order to add a "type" (ie, Equilib vs No-arb) column to the summary, see here https://forum.bionicturtle.com/threads/week-in-risk-ending-dec-11th.10048/#post-46989

@brian.field Thank you! FYI, since I had to fix the CIR function I revised the summary table as follows:

Finally, please note arkabose's comment at https://forum.bionicturtle.com/thre...ge-free-interest-rate-models.6701/#post-44758 which is pretty consistent and its an easy way to look at it, I think (ie., is the drift time dependent?)

Hi to the guys above, I know I may be late and you guys have possibly passed FRM, but here is how to distinguish between the equilibrium and arbitrage free models - Any model that has time dependent parameter (say lambda for example) is an arbitrage free model. This is because at that point of time, it tries to match with the market prices and hence its parameters are different, hence time dependent.
Thus, if we look at the models, Model 1 and Model 2 do not have time dependent parameters, so does CIR model. Thus, they are equilibrium models. On the other hand, Ho-lee model and model 3 have time dependent parameters, (lambda in case of the former and both lambda and volatility in case of the latter) they are arbitrage free models.

I hope that helps! Thanks,

 

[perviz]

Thanks, for reply.. It was helped..

 

[Martin B]

Thank you very much @David Harper CFA FRM . I was asking myself the same question regarding the Vasicek model. What confuses me a bit now is the following: Does arbitrage-free and no-arbitrage mean the same or is arbitrage-free equal to Equilibrium? I am asking regarding the Practice Question P2.T5.305.3 and Answer A.

 

[David Harper CFA FRM]

Hi @Martin B Right, but my question T5.305.3 has a mistake (apologies) per my comment at the source https://forum.bionicturtle.com/thre...interest-rates-tuckman.6710/page-2#post-47829 . In case it is convenient, my note there is copied below. My current, best understanding is reflected in the Study notes and below (per https://forum.bionicturtle.com/threads/week-in-risk-ending-dec-11th.10048/#post-46989)

So I view Vasicek as an equilibrium model as opposed to an arbitrage-free model.

Re: Does arbitrage-free and no-arbitrage mean the same or is arbitrage-free equal to Equilibrium?
In this context (at least, just to be safe) arbitrage-free = no-arbitrage. Tuckman's discussion of term structures concerns only the distinction between arbitrate-free versus equilibrium models. I tend to use "no-arbitrage" (Hull's adjective) as a synonym for "arbitrage-free," (Tuckman's adjective) probably because I am so familiar with Hull who uses the former; e.g.,

"The essential difference between an equilibrium and a noarbitrage model is therefore as follows. In an equilibrium model, today’s term structure of interest rates is an output. In a no-arbitrage model, today’s term structure of interest rates is an input. In an equilibrium model, the drift of the short rate is not usually a function of time. In a no-arbitrage model, the drift is, in general, dependent on time. This is because the shape of the initial zero curve governs the average path taken by the short rate in the future in a no-arbitrage model. If the zero curve is steeply upward-sloping for maturities between t1 and t2, then r has a positive drift between these times; if it is steeply downward-sloping for these maturities, then r has a negative drift between these times." -- Hull, John C.. Options, Futures, and Other Derivatives (Page 715). Pearson Education. Kindle Edition.

Here is the comment about my mistake in 305.3:.A:

Hi @RobKing Great point: my 305.3.A looks like a mistake. Sorry. I wrote it in 2013 (that's the meaning of 3xx) and I mis-interpreted the transition in Tuckman's text to suggest Vasicek was non-arbitrage. The notes reflect my subsequent research and understanding: although Tuckman (to my knowledge) does not say whether Vasicek is equilibrium/no-arbitrage, it's pretty clear to me now (per the notes) that Vasicek is an equilibrium model (for example, most of my authoritative references dub it "equilibrium" although I will note that I did find two exceptions but I took a pretty deep dive into the literature before settling on the Vasicek = Equilibrium designation that you shared above ...). My understanding is reflected in my study note:
"Equilibrium models (e.g., Model 1, Model 2, Vasicek, CIR) return a term structure as output and therefore do not match the current, observed term structure. No-arbitrage models, on the other hand, take the observed term structure as an input and therefore do match the current, observed term structure. The key issue in choosing between an arbitrage-free versus an equilibrium model is the desirability of fitting the model to match market prices. This choice depends on the purpose of the model. According to Tuckman:"
... I think @arkabose has does a good job of operationalizing this here (i.e., if you have time-dependent parameters, it's easy to imagine how you could fit the model to match the market prices/rates but if you do not have such "variables" in the model, as is the case with the Vasicek, it's sort of hard to see how you could do that because, once you have the initial parameters, the tree builds out effortlessly ) https://forum.bionicturtle.com/...ge-free-interest-rate-models.6701/#post-44758
Hi to the guys above, I know I may be late and you guys have possibly passed FRM, but here is how to distinguish between the equilibrium and arbitrage free models - Any model that has time dependent parameter (say lambda for example) is an arbitrage free model. This is because at that point of time, it tries to match with the market prices and hence its parameters are different, hence time dependent.
Thus, if we look at the models, Model 1 and Model 2 do not have time dependent parameters, so does CIR model. Thus, they are equilibrium models. On the other hand, Ho-lee model and model 3 have time dependent parameters, (lambda in case of the former and both lambda and volatility in case of the latter) they are arbitrage free models. I hope that helps! Thanks,
Thanks for catching my mistake, will fix!

 

[Martin B]

Thank you very much @David Harper CFA FRM .

 

[perviz]

Hi, David. There is a problem in question 305.3 (Tuckman chapters 6,7,8). The a variant true answer, because Vasicek model is an equilibium model. But in explanations, correct answer is d (which is also correct)

Okay I got it Rob King has written about it

 

[emilioalzamora1]

Hi @David Harper CFA FRM,

just wondering about the following: the CAIA material for level II assumes that for both (Vasicek and Ho-Lee) the short-term interest (r) is additionnaly added to the drift and diffusion term of the equation. See my attachment of the book.

Where does this come from? I looked it up at several different sources but no one actually adds the short-term interest rate.

For exam purposes it's clear what to do but it would be good to know any reason for this. Guess it could be helpful to a wider audience (FRM candidates) as well.

Critical comments are very much appreciated

 

[David Harper CFA FRM]

Hi @emilioalzamora1 Your version just parses out dr = Δr = r(t+1) - r(t). Tuckman's formulas gives the expected change in the short-term rate, which in constructing the tree, does need to get added to the current short term rate. So these look identical to me. Your attached could be re-arranged to: r(t+1) - r(t) = τ*[µ-r(t)] + σ*ε(t+1) --> Δr = τ*[µ-r(t)] + σ*ε(t+1). I hope that helps?! (I'm thinking you just over-analyzed it! ). Thanks!

 

[jf_actu2]

Hi,
Question 305.3 is still not corrected in R40.P2.T5.
Thanks,

 

[Nicole Seaman]

Hi,
Question 305.3 is still not corrected in R40.P2.T5.
Thanks,

Hello @jf_actu2

Thank you. This has already been noted in the source forum thread here: https://forum.bionicturtle.com/thre...el-for-interest-rates-tuckman.6710/post-57897. It is tagged for revision, but since it is an old question from 2013, we need to prioritize other (current) materials before revising that question.

Thank you,

Nicole