Re: GARCH and EVT

[eddfung]

Hi David

I am afraid I need some more guidance after reviewing the QA.

GRACH:
1) The equation fpr volatility at time n depends on previous period volatility and mean. What if it is at time=0, how do we get the initial volatility value and mean value? Does it mean that somehow we have to estimate the inital values for volatility and mean by using some ways? Or, simply take the long term volatility as the given value at time zero?

2) I saw you estimate alpha, beta and omega by using maximum likehood approach, but you have not explained how it was used. I guess it could be out of symballus. I remember my high school teacher once tell me there are two ways of estimating parameters: one is OLS and another is maximum likelihood, but he never taught me anything on maximum likelihood and used OLS for regression. That seems MLE is an exotic approach. Could you briefly explain what it is? How does it do the job of estimating these three parameters?Similarilties and Dissimilarities to OLS?

EVT:
1)The xi parameter- I don't really understand why both EVT models consist of two eqts based on the value of xi parameters. Could you tell me why we need two instead of one? What does it mean when the xi value is zero? Which portion of the graph is it refered to? How is it estimated?

2)Block Maxima- you said it depends on three parameters. Where is the "location" parameter in the equation?

3)POT- what does the beta parameter stand for? Why is it not found in Block maxima?

4) The concept of mixture of distribution (pg 117 in the QA notes)-
I don't really get it if two normal distribution have the same mean , the combine will have a fatter tails. I always have the percepton is if there are two normal distributions, the sum will also be in normal distribution, That means contradicting to the earlier statement, as distributions with fat tails are not normal. Putting one more step forward, that may also mean fatter tails can be decomposed into two normal distributions with the same mean. Pls correct me where I am wrong and how I should grasp this concept.

Many thanks!!

Ed

 

[David Harper CFA FRM]

Ed


1) The current GARCH estimate is a function of last variance and last return^2. If you are thinking mean because of the u, understandable but Hull here uses ui to refer to the daily return, so GARCH = omega + alpha * last return ^2 + beta * last variance

At t=0, with no history, i am not sure that GARCH has meaning any more that historical volatility has meaning. The 'C' in GARCH is conditional, as in, conditional on the information set and history informs the information set. Similarily, what is historical volatility at t=0? I don't know because we need a period (two points) to start measuring...but in any case, it's not particularly important for our purposes though maybe philosophically interesting. Wilmott is really good on this topic of the paradoxical nature of volatility, I think somewhere he says that it doesn't exist at point in time or to that effect

2) I don't have the time to here expound deeply on MLE (sorry, i am unable to be succinct, i am working on the notes, and it's not particularly important for the exam), it is the most popular way to generated estimates (to find the params that would be "most likely" explain the dataset that we observe). Please note that the THIRD SHEET of the learning XLS 2.b.5 Volatilities Compares contains an MLE exercise. The easiest way to understand MLE is to inspect an XLS like this, IMO.
Please take a look at this thread: http://forum.bionicturtle.com/viewthread/361/#837
and, operationally, it is not too bad: you merely solve to maximize the log likelihood value.

3) True EVT coverage is coming when we get to Dowd. Can you look at this sheet, I tried hard to put them side by side @
http://www.bionicturtle.com/premium/spreadsheet/2.c.3._extreme_value_theory_evt/

As the "deep coverage" of EVT is coming in L2, can you wait until then?
(the appearance of EVT in L1 FRM makes no sense as it it: it is very briefly mentioned in Rachev, and would not be comprehensible as an introduction. IMO, EVT should only be in L2)
However, if you want the best reference, it's the chapter in L2 that i recommended to GARP: Dowd Ch7

But, okay my advice with EVT is *before* trying to memorize pamars, see that we have two distributions that characterize two approaches to an extreme tail:

1. POTS: must select a threshold and losses above the threshold are extreme. The GPD distribution characterizes this distribution. Approached this way, one can start to see why, having selected a threshold, we don't need to "center" the distribution on a location parameter

2. Block maxima: instead of picking a threhold, chop up the history of losses into blocks. GEV characterized the distribution of each maximum (local) within each block. This distribution, not unlike a normal, needs a location and a dispersion (sigma). But being EVT, it also gets a fat tail param (xi).

Both of the distributions have a "heavy tail" param (xi)

4) Yes, this is a very *astute* observation!
Please do not doubt this: "I always have the percepton is if there are two normal distributions, the sum will also be in normal distribution" You are very correct, a linear combination of normals is itself normal. But this is not what it means to mix two normals into a third density function.

But please read the first two paragraphs of this helpful wikipedia article: http://en.wikipedia.org/wiki/Mixture_distribution
i.e., a mixture distribution is a different concept than summing two distributions!

we normally are not referring to the mixture, we are typically referring to convolution: the mathematical operation on 2 or more functions (e.g., sum of two functions)

but the mixture, rather than solve for the value as a convoluted sum, builds a density function by mixing two. The "output" in the mixture is a new density function.

Hope this helps, David