R16.P1.T2.HULL_CH11_TOPIC:UPDATED_CORRELATION_using_GARCH

[gargi.adhikari]

In Reference to R16.P1.T2.HULL_CH11_TOPIC:UPDATED_CORRELATION_using_GARCH :-
Hi,
I have a couple of questions on this problem statement illustrated below:-
1) Omega (Correlation ) = .000001 This Omega is for the Correlation.Why are we using the same coefficient for the Covariance as well..?

2) The Volatilities for X & Y are given to be 1% and 2% - these are yesterday's or (n-1) day's Volatilities. But the problem statement says these are the 'Current' Volatilities. By 'Current' , i tend to think it's for today for for the 'n'-th day....

3) I have a similar issue with the Correlation Coefficient. The Problem statement says .5 is the " Correlation Coefficient " "at this time" - it is unclear whether this is for n or for (n-1)....

Would we have similar verbiage in the exam..if so, then I will train my brain to interpret that way....

 

[David Harper CFA FRM]

Hi @gargi.adhikari It just so happens that I just addressed this two days ago here at https://forum.bionicturtle.com/threads/p1-t2-703-ewma-versus-garch-volatility-hull.10151/#post-52546 See copy below, especially my spreadsheet snapshot please (you need to click on the XLS image in the quote below, and I think you will agree that my labels match your intuition about the timing). Specifically,

• This is Hull's end of chapter question 11, as I labeled in the note. I do agree with your implication that the phrasing with respect to timing could be (slightly) improved for clarity. In many cases, Hull's own questions are more precise re: the timing (especially if written in his OFOD which seems to occasionally be more precise) and he himself might instead say here as he often does in OFOD "Suppose that the daily volatilities of asset X and asset X, calculated at the close of trading yesterday, were 1.0% and 1.2%, respectively." But I think we need to be a little flexible and keep in mind: GARCH is just updating the most recent estimate to a "more current" estimate of volatility, which itself is a statistical artifact of a series of prices. Prices are instantaneously observable, but volatility and correlation are not (they are metrics that summarize a series of prices!) so that these semantic issues, for them, are more easily engendered. Sometimes "current" means "beginning of today" or really means "end of close yesterday" because that's the end of the most recent series.
• Re your (1), the ω(ρ) = 0.0000010 is called the "correlation omega" but, you are correct, is chiefly used to update the covariance; notice in my spreadsheet, this omega is a GARCH param that updates cov(X_n-1, Y_n-1) to cov(X_n, Y_n). I think my XLS is accurate with respect to timing. I do tend to think that calling this "covariance ω" would be more accurate; but it's understandable that he's using as synonyms, yes?. But there is only one covariance/correlation ω, okay? The other volatility omega, ω(σ), is using GARCH to update the variances of the individual (X) and (Y). Notice, here, too, we should probably call this a "variance omega" but we understand what he means by "volatility omega."
• Re (2) Yes, agreed, per my XLS and per the table in the note, I labeled these as σ(x-1) and σ(y-1) or better, σ(x,n-1) and σ(y,n-1)
• Re (3), same thing, it's really yesterday's correlation and I would prefer that the note matched my XLS with ρ[ X(n-1), Y(n-1)] = 0.50. After all, the entire point of the question is to update yesterday's correlation of 0.50 to today's, ρ[ X(n), Y(n)] = 0.5689
• Re: Would we have similar verbiage in the exam? I hope the exam is more precise. Behind the scenes, I have been working hard giving feedback to GARP in order to support a higher standard with respect to questions. I am working on a glossary that I plan to "open source" to GARP and the other EPPs in order to promote consistency. To be candid, GARP's own questions are often not sufficiently precise ... Thanks!

Hi @[email protected] That's really just an elaboration on Hull's End of Chapter Question 11.6, and I can't see any problem with it, although it's highly tedious!

I entered into into the more generic version (in our learning XLS), see here (and below) at https://www.dropbox.com/s/hyadoaavumn7a7g/1001-hull-11-06-correlation-update.xlsx?dl=0

It's true that there are two omega (ω) assumptions given but that looks okay (consistent) with Hull. Please note that GARCH is used twice here: once to update correlation, and once (for each of X and Y) to update individual asset volatility.

• Correlation ω = 0.0000010 is used in a GARCH model to update the covariance (n-1) from 0.000060 to covariance (n) = 0.000001 + 0.040*3.33%*2.00% + 0.940*0.000060 = 0.0000841
• Correlation ω = 0.0000030 is used in a GARCH model to update the variance/volatility of each of the (X) and (Y) variance/volatilities; e.g., for asset X, the variance (n-1) of 1.0%^2 is updated to 0.000003 + 0.040*3.33%^2 + 0.940*1.00%^2 = 0.0001414; volatility of (X) is 1.19%.

Here is snapshot of above XLS. Please let me know if you still think I missed something (and frankly, while I care about these details, I have a hard time imaging GARP asking a full version of this question on the exam, too tedious ....) Thanks!

 

[gargi.adhikari]

Thanks so much @David Harper CFA FRM for your inputs on this...always helpful....yes that's a bit of a fear I have about getting confused or misinterpreting the verbiage in the exam..and I need to train my brain accordingly ...hence your clarifications helped a lot as usual - again - my heartfelt gratitude