Infinite Variance

Fermion · August 8th, 2005, 3:54 pm

QuoteOriginally posted by: erstwhile... it can be shown through simple arguments that if the gamma is large relative to the liquidity of the underlying, you can get a runaway short-gamma process that is effectively an option-induced crash. sellers sell, and get so long they have to sell an untenable amount of stock, and down it goes. .......and you find that the market should "jump from one stability point to the other" in much the same way that a material jumps from solid to liquid.....Given the above model, I still can't make the leap to an infinite variance of something...Me neither. A jump or crash doesn't mean infinite volatility. Is there any evidence to suggest that actual price volatility even increases in a crash?Certainly |Delta S|^2 gets very large, but so does |Delta S|. Assume an Ito-like process. Is the jump contributing to drift or volatility or both? It clearly contributes significantly to drift: mu = <Delta S> per time step. For finite time steps, you must subtract the drift term out to compute volatility: sigma = root mean square |Delta S - <Delta S>|. What do you have left?

erstwhile · August 8th, 2005, 4:20 pm

I would think that when analyzing data like a crash you would put the jump purely into volatility process bucket and leave the drift as something near zero (as in a risk neutral analysis).I suppose the thing that would be quasi-infinite in this model would be something like the specific heat. Here you could define a "liquidity delta" that means "how much the market goes down if i sell 0.01% of the company's free float". This value would increase and increase as you approach the "instability point" of the negative gamma, and finally the value would jump to a large value as you hit the critical point. I suppose it would be undefined in between the stability points, and would take on a fixed value at the other stability point. Given the discontinuity in behaviour, i guess the "liquidity gamma" which might mean "how much the liquidty delta changes if i sell 0.01% of free float", would become infinite.This liquidity gamma would be sort of like a specific heat of the stock market, no? Still not an infinite variance, but at least maybe an analogy with specific heat...

hammerbacher · August 8th, 2005, 9:26 pm

Erstwhile,While I can't answer the question of how to test statistically for infinite variance, the standard example of a distribution with infinite variance, the Cauchy distribution, can be generalized to a stochastic process with densityf_t(x) = t/(pi*(x^2+t^2)).This process is called a Cauchy process, and is just one example of a stable Levy process which has infinite variance (to contradict the claim of N below). Other stable Levy processes which have infinite variance can be obtained by considering the process with Levy characteristics (b,0,v), where b is in R^d (d a positive integer) and v is a Levy measure given byv(dx) = (C/abs(x)^(alpha+d))*dx,with 0<alpha<2 and C>0. Setting d=alpha=1 gives the Cauchy process. Also, when alpha<=1, these processes also have infinite mean.Hope that was interesting,Jeff

hammerbacher · August 8th, 2005, 9:40 pm

Oh, and for another (intimately related) example of a process which exhibits infinite variance: consider the process constructed when each web page is a random variable and the number of links pointing to the web page gives the value of the random variable. The intuition here is that there are a TON of little web pages floating around which have very little connection to the rest of the web. So the mean is around 1 or 2. But there are a few sites, like Google.com and Wilmott.com, which have tons of links to them. If you were to let the number of web pages approach infinity, the variance of the process will increase without bound.Bear in mind I'm trying to be as non-technical as possible, so the above statements shouldn't be taken at face value. But I find that example easy to visualize.Also, the reason the above distribution has infinite variance is because it is a self-similar process. Rotationally invariant stable processes are a subset of this class of processes.And, lastly, this is some of the mathematics behind for the phenomena chronicled in the popular books "The Tipping Point" by Malcom Gladwell and "Small Worlds" by Duncan Watts that came out a few years ago.

N · August 8th, 2005, 9:45 pm

Hey hammerbacher,The variance, second order moment, for the Cauchy distributioin doesn't exist. How can it be infinite?BTW, when alpha <= 1 the mean doesn't exist either. It isn't infinite.The theory is very very mature in electrical engineering, over 40 years old. You should get current before you confuse folks.N

erstwhile · August 8th, 2005, 9:47 pm

Jeff - that is interesting, as it gives me something I think I can set up in a spreadsheet.Let's see if I understand it correctly: if I generate a random number with uniform density on the interval (0,1), I can convert it to a random number from this distribution by using the inverse of the integral of the distribution. If I'm lucky, that might even be analytic, though it doesn't matter, as I can use a table.The numbers i generate will represent returns, so I basically integrate to get my series.I can then run little intuitive tests on finite amounts of data, such as measuring the variance of larger and larger samples to see what happens to the variance and higher moments. Maybe I could Fourier transform the series and conclude something from the power spectrum?I know nearly zero about Levy processes - once I get some intuition for an infinite variance process, I will search the forum and the web to learn the basics, and then come back and try and understand the second half of your message.

erstwhile · August 8th, 2005, 10:03 pm

N: If you were to define a sort of quasi-variance as the variance of a finite amount of data, like a monte carlo integral of the distribution as a function of a number of steps, would you get a larger and larger variance for increasingly accurate integration, or increasingly large number of steps? Is the sense in which the variance doesn't exist?Also, as an ex-ham radio operator, I was always interested in, though not educated in electrical engineering. Is there an intuitive real-world example of non-existent variance you can give us? Like maybe from signal processing maybe?Thinking back to your comment that variance is energy, is there some connection with a power spectrum? Maybe a Cauchy distributed power spectrum represents an infinite energy signal?

Rez · August 9th, 2005, 2:38 am

erstwhile:I think that if you disentangle trading viz calendar time you can simulate Levy processes in an intuitive way.Loosely spreaking, you can start with a Brownian motion in trading time (think of it as log-returns which are i.i.d. normally distributed from one trade to the next, rather than from one minute to the next).Then use a subordinator, a process that gives you the trading intensity across calendar time. If this happens to increase rapidly, you travel faster through the trading process and in a discrete sample you would observe an outlier. By controlling this 'time-changing' process you can generate 'calendar processes' with infinite variance.In the attached file you can see how this looks like for a VG process. (This is just an illustration of the simulation, VG has finite variance. Also apologies for the stupid mdi format.) We start with a normal brownian motion in trading time (NW), and a process that gives the trading intensity (NE, with the 45o line). Then, in calendar time our process would look a bit jumpy (SE). If we sample daily, the returns can give all sorts of sample stats (SW, the x-axis is years, not days). Changing the trading intensity \nu can make our process jumpier.Cheers,K.

erstwhile · August 9th, 2005, 7:10 am

Rez: Is that an AutoCAD file or a Microsoft MDI file? Both have an mdi extension...

Rez · August 9th, 2005, 8:26 am

Sorry about that. It is an MS print image file. Here is the pdf version.And here is a copy of the text:QuoteIn the attached file you can see how this looks like for a VG process. (This is just an illustration of the simulation, VG has finite variance. Also apologies for the stupid mdi format.) We start with a normal brownian motion in trading time (NW), and a process that gives the trading intensity (NE, with the 45o line). Then, in calendar time our process would look a bit jumpy (SE). If we sample daily, the returns can give all sorts of sample stats (SW, the x-axis is years, not days). Changing the trading intensity \nu can make our process jumpier.Kyriakos

erstwhile · August 9th, 2005, 10:13 am

Interesting - thanks for that! Very thought provoking. I have often thought of market movement as if there was a "business clock" that ticks at a different and variable rate. For example, nowadays with low interest rates and low market volatility the market acts very much as if time itself has slowed down. If only investors could measure hedge fund returns according to ticks of the business clock instead of demanding returns according to the real clock!I think I will construct a series whose returns are distributed like the Cauchy distribution, just to give me something extremely simple to play with.

ClosetChartist · August 9th, 2005, 1:39 pm

Erstwhile,Trying to imagine something with infinite variance may be an unhelpful route to go. Any finite sample of real data will always have a finite variance regardless of the underlying process. The question at hand is, "Is the process so 'jumpy' that it reasonably appears to have come from an infinite variance generator?"Looking at a Cauchy process is a good place to start. There is nothing "infinite" in its appearance and you can begin to develop an intuition of how an infinite variance process actually manifests.-CC-

erstwhile · August 11th, 2005, 11:14 am

For anyone following this thread who would like to have a look at a process whose returns are Cauchy distributed(therfore "infinite" or more accurately, undefined variance), see the attached spreadsheet. It generates a one (business) year history when you calc the spreadsheet, and you can control the "full width at half maximum" of the distribution by varying the parameter "t" (FWHM = 2t).It is an outrageous process in that you get a run that looks nice and mild, and then you get one that explodes or implodes!Here's a thought: none of the moments of the cauchy distribution exist, as discussed earlier. What if you multiplied the probability density by something like exp(-abs(x)/b), to guarantee convergence? In that case, you could set up the (for lack of a better term) "damped Cauchy distribution" to match the volatility of the stockmarket. I wonder what you would then get for skew and kurtosis? Massively higher than the stock market?Or maybe the integrals would be easier if the damping factor was exp(-x^2/b^2)? It might be interesting to see how the moments vary with the damping distance "b". I must get Mathematica one day...Please improve the attached spreadsheet and republish it.Even better: redo this calculation for a damped Cauchy distribution!