Early Bird 2009 FRM Session#2 Questions

[sudeepdoon]

Hi David,
I was going through one of the reference books which is Introduction to Quantitave Finance by Paul Wilmott. The formula given in this book seems to be different from the one on slide 33. I was looking at the formula on Page 207 (Second Edition). Could you please clarify if one of them is wrong or have i interpretted things wrongly

Thanks,
Sudeep

 

[David Harper CFA FRM]

Hi Sudeep,

That's a nice observation, I haven't really seen Wilmott's expression elsewhere. They are the same GARCH(1,1) only Wilmott is approaching it from a different angle. There is no substantive difference here, rather just a difference in the way the *weights* are expressed. My slide 33 is Hull's (I don't want you to think i am original!) where the weights are:

alpha (weights the lag return^2) + beta (weights the lag variance) + gamma (weights the lag long-run variance).
Note: sometimes you see gamma * L.R. variance = omega, but I like to use gamma*L.R. variance to remind that this additional term is not the L.R. variance per se, it's the weight multiplied by the LR variance.
(it is an exam trap, if you see: GARCH(1,1) = x +beta(y) + alpha(z). You want to remember that x is not the LR variance. The LR variance is "inside" X!).

And Wilmott has combined the weights such that
Hull gamma = Wilmott alpha
Hull beta = Wilmott's lambda * (1-alpha)
Hull alpha = Wilmott's (1-alpha)(1-lambda)

for example, if Wilmott's lambda = 96% and alpha = 2%, then that roughly translates into Hull's weights:
gamma= 2%
beta = 96%*98% = about 94%
Hull's alpha = 98%*4% = about 4% (I'm not being exact)

Wilmott can do use only two params because, importantly, the weights must sum to one: Hull's alpha + beta + gamma = 1.0

IMO Wilmott's expression is instructive because is really makes explicit that, on one level, all GARCH(1,1) does is add the mean reversion to the EWMA. or, equivalently, if no mean reversion the GARCH(1,1) reduces to EWMA.

thanks, David