Volatility in GBM

[Liming]

Dear David,

Please allow me to ask you a quite basic question about the “constant volatility” in the GBM and question 4e on pagehttp://forum.bionicturtle.com/viewthread/1146/.

“4e. Is it homoskedastic?Yes, the volatility (sigma) is constant; this is a distinct feature (drawback) of GBM/ABM. ”

My questions are:

1) When you mentioned randomizing volatility in GBM simulation by altering epsilon, I think this contradicts with the notion that the volatility is constant under GBM because strictly speaking, we can't randomize a constant, right? To me, it seems like that it is only randomizing (normal variable x the constant volatility sigma) in order to perform a simulation of stock path.

2) When I look at the formula, I see that there is delta(t)^1/2 at the last term. I think the reason for this term is that the variance for the stock return is addictive over time (scaling), therefore if we divide the whole horizon into a few interval of delta(t), the volatility can then be scaled by delta(t)^1/2. So here is my confusion: if I draw stock return distribution over time, shouldn’t I expect to see the distributions getting more and more dispersed as the t is increasing? for example:
1) 1 week - sigma * randomized variable * 1
2) 2 week - sigma * randomized variable * 2^(1/2)
3) 2 week - sigma * randomized variable * 3^(1/2)

Or if the answer to my question is that: we are only looking at the equal time step (delta(t)) and only considers the distribution for the tiny equal time unit/step, instead of scaling? It is in this case that the volatility for all intervals are equal ?

Hope my above thought hasn’t confused you.

Thanks for enlightenment!

Cheers!
Liming

 

[David Harper CFA FRM]

Hi Liming,

1) I think, in a sense, you are correct to say "it seems like that it is only randomizing (normal variable x the constant volatility sigma) in order to perform a simulation of stock path"
I quickly created this XLS to illustrate, please see:
http://sheet.zoho.com/public/btzoho/0930-sim-vol

inputs are: volatility per annum = 20%
Time step = 1 day/250 = 0.004; i.e., delta t = 0.004 years
Standard deviation (daily) = 20% * SQRT(1/250) = 1.26%

okay, then I ran 100 rows (simulations).
Sigma = 1.26% * NORMSINV(RAND()) = random return

And cell F4 computes the standard deviation of the series...if you re-compute, you will see that the standard deviation of the series fluctuates around 1.26% (variation is called sampling variation)

this illustrates that the sigma is a constant, and we are randomizing the normal deviate;
or, put another way, the random variables are standard normal random variables: by definition they have standard deviation of 1.0, so most will fall between -1.0 and +1.0, fewer between -2.0 and +2.0. But those are not scaled by units. So we multiply by the sigma to *give scale/units* to the random standard normal which, in this sense, is unitless.
So you might this of it this way: the sigma is giving units to the random standard normal

(another way to think about this: if we did not multiply the the random, it would be a straight line and the vol would = 0. Volatilty requires fluctuation)

2) I don't quite follow ... but i think you are basically right...as Hull shows, this is a so-called Weiner process and the time interval (T) can be divided into N intervals such that N = T/(delta_t) so delta_t is the small interval. And the variance scales directly; e.g., variance (N * delta_t) = N*variance(delta_t).
So, variance scales directly with time; standard devation scales with the square root of time (as you suggest)
...but note this is really just follows from an underlying assumption that the returns are independent and identically distributed (i.i.d)
...this is commonly misunderstood: the square root rule does not require normality, it only requires i.i.d. (and finite variance)
because variance (r + r) = variance (r) + variance (r) + 2*Covariance(r,r); i.e., Gujarati's variance property
...if we assume i.i.d., then
variance (2r) = 2* variance(r) and Standard Deviation (2r) = SQRT(2)*standard deviation(r)
...so this square-root-rule may be easier than you thought...but let me know because the GBM is challenging, indeed!

David