Guest Post: Product Over Process: Balance Complexity With Liquidity

Tyler Durden's picture

Submitted by JM

Product Over Process: Balance Complexity With Liquidity (pdf)


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Biggus Dickus Jr.'s picture

Wow it's been a while since I have plowed through some statistics.  However I noticed toward the end you said that it is not possible to time the markets, but later you said that nonstationary creates opportunity for investors who maintain liquidity.  Has anyone ever studied the opportunity cost of maintaining liquidity for these "non stationary events" and might it historically be worth foregoing  some return in order to take advantage of these conditions?

Bob's picture

Has somebody discovered the meaning of life, you mean?  LOL.  If only.  You definitely nail the question. 

Who would have thought that the Weiner was such a pivotal element in the whole statistical artifice? 

That was a great read, I thought.  Ya still risks y'own money in the end, but at least it puts alot of the pretenses of modeling in proper perspective. 


Biggus Dickus Jr.'s picture

My weiner is bigger and more pivotal than yours.

jm's picture

Risk premium is hwere the money is.  After a meltdown, you realy get compensated for the risk.  That, and luck, you can count on.

Biggus Dickus Jr.'s picture

But does it beat buy and hold of the same risk adjusted asset class. That is the opportunity cost of keeping some liquidity in reserve for non stationary events. You can't have it both ways. Either you can time the market or you can't. Which is it?

jm's picture

There is more to it than opportunity cost.  Some have to worry about getting margined.  Some people also have complex books that they don't understand until the SHTF.

Because of this, some try to get flat to the market instead of taking the hit.  Taking the hit puts them out of business.  Real hard to get flat with minimal scroungable capital, because hedging becomes expensive.

Biggus Dickus Jr.'s picture

Ok it looks like I am just wanking with myself here.  I'm used to that.  Maybe one of the cerebral Tyler Durdens can answer my question, but in the meantime if anyone who doesn't have a statistical background wants a great read about the history of statistics and insurance, a really fascinating book, it is Against the Gods.  It will show in an easy to follow way how Pascal developed probability theory in order to address an interrupted game of dice.  How would you split the pot?  Once you truly understand how statistics and probability theory developed then you can go as far as you want.  It's an easy read trust me.  And very interesting!

HarrisonBergeron's picture

Thanks JM, love your stuff.

delacroix's picture

page 2 paragraph 5 last sentence?

jm's picture

Sorry.  Wish I had a budget and a secretary.  The sentence preceeding it captures the point.  The future depends on the present because any information content available re: the future is embedded in the present state.

The filtration concept would help here, but it takes a lot of machinery to develop.

Biggus Dickus Jr.'s picture

Ahhh..the theory of

scratch_and_sniff's picture

Cant really get much out of the guy's argument for some reason(should i read it again?).

"Stop trying to time markets, or trying to outthink the markets (you can’t),

or relying on luck (while calling it prediction). Take risk to get reward..."

So, if i am not timing the market, with no thoughtful edge, and i cant rely on luck("while calling it prediction" therefore we assume that its neither luck or prediction in JM's mind), then what exactly is it that i am doing?

Well its fairly straightforward to me, i choose my times to trade, i think carefully about each position, and i usually get lucky (correctly) scores of times a day(while calling it prediction)...ahhh, thats umm, err, well sorry you wont get me admitting that its luck. And, just because you wont win anything if you dont put your balls on the table, doesn’t mean that every outcome is down to a chance that excludes all manner of prediction.

If you know your market inside out, you can predict quite a lot actually, quite often, random or not. I know what the average Euro trader is going to do if we fall 40 pips in 2 ticks, my only talent is to be there when it happens.

jm's picture

I understand your points, although I disagree. I could be as wrong as the next guy.

The big picture...

I don't think people can time the "big moves" I refer to--meaning moves where relative pricing models collapse.  I could be wrong.  But if the big Weiner processes on Wall street were taken to the brink, I'm not sure how anyone can.  These big moves are clustered together via some dependence relationship a la black noise.  This dependence at best corrodes the value of current probability modelling including everyone favorite "new thing that will save us," Extreme Value Theory.

Addressing your points...

"What am I doing?" 

IMHO, you are successfully trading only in the interval between liquidity crises, when correlations are weakly stable and stationary conditions hold.  Here you can understand upper and lower bounds in some process control sense. Not taking anything away from your skill.  But when liquidity conditions change, the reference points you had all reset and predictions fail.  I'm coming at this from a probabilistic standpoint, because it is so beautiful and consistent.  Nothing else comes even close to it from a prediction standpoint.

Michael Steinhardt and Soros may be the exceptions that prove the rule?  There is something intuitive that lies beyond deduction... I call it pattern recognition.  It just doesn't work when the context changes.  Better is to recognize the fallibility of any prediction framework and prepare for accordingly.

On taking risk:

Given what I've said, which you may disagree with...  When crisis happen, that is the best time to take risk because you are compensated for it.  I'm not saying don't take risk.  One has to in order to make money.  BUT because the floor could flat out of your model always have powder dry to buy the dips or cover if margined or blow up in some way.  Not a huge fan of shorting, although I've done my share of it.


Bob's picture

Funny, I was just thinking that your explanation was really good . . . and then realized that you're the frickin' author of the paper.  Duh. 

Thanks.  It is beautiful to see probability put into perspective this way.  That was stripped to its bare elements like lines of a poem.

jm's picture

That last part was very kind.  It took some effort.

Biggus Dickus Jr.'s picture

That of a quote from against the gods....the simplest way to frame is that nature is predictable but only for the most part.

scratch_and_sniff's picture


The big picture...

“These big moves are clustered together via some dependence relationship a la black noise”

On the one hand you are accepting randomness, and on the other you are obviously holding out for some device to dismiss it. I sympathize, and urge you to give up the ghost before it drives you nuts.

Can big moves be predicted? It depends on how you frame prediction; crystal ball prediction or a careful quantification of the likelihood of a set of future decision outcomes based on what is known about some causality chain. In the first sense, no, in the second sense, the market is the sum total of all prediction/ analysis/speculation, so yes most of the time they are in a sense predicted, if they weren’t then theoretically they wouldn’t have happened. Sorry but that’s the kind of nonsense we are dealing with here.

Can they be modelled mathematically with any accuracy? Can you model when trader x is going to pull liquidity out of the market, or similarly when it will be there, or when some cunt will "hijack a plane"? Well I wouldn’t think so, but that’s not to say experienced traders and investors cant know what’s coming down the pipe in a shitstorm.


IMHO, you are successfully trading only in the interval between liquidity crises, when correlations are weakly stable and stationary conditions hold. Here you can understand upper and lower bounds in some process control sense“

I largely agree(you make it sound so dirty), but here you don’t necessarily need to understand upper or lower bounds in a process, of course its only human to identify them as real and stable at some points(and admittedly a lot of my trading revolves around identifying the ebb and flow of daily order flows, I like to think I can feel it, and usually I can, but of course I am caught out frequently by a change in rhythm or the odd piece of news, it doesn’t necessarily have to be a liquidity “crisis” that gets me). But you can use a system to negate the potential for being wrong-footed or fooled, if you are that way inclined, by any kind of crisis, such as not bailing out until you get some technical indication that a prescribed tolerance has been met, etc.

Therefore accepting that you cant completely predict becomes a powerful prediction tool in itself, and ironically its the one that is the most profitable depending on who’s trading. For example, if I buy when some sequence of moving averages cross in some time frame, I don’t need to understand anything, I don’t even need to look at the market after I am positioned(depending on the trader, not looking at the market is probably the best option), all that’s required is to accept that i cant predict and let the market run its course. But here is the rub, I make more money from prediction than I do with medium to long term trend following. I don’t predict where or how far the price will go, I just make a prediction about how I should react when something happens. The market is free flowing randomness, Brownian motion, so the risk( i.e. we want to be talking about controllable risk) stems from the traders reactions, or the inherent risk in the design of the system or model. Personally I don’t see the point in talking about “out there” risk, in fat tail land... more or less because of my finite intelligence. 

I am not blowing my own trumpet here, but your language is so rigid, you don’t leave me much room for manoeuvre, so for illustration purposes, here are some average trading days in which I am taking advantage of my understanding of what is happening in a daily market, all very much statistically impossible in a random market lol.

There is a point where adaptability and instinct will trump, and of course you need to have the rigidity in the system and the neutrality to recognize and accept the change when it comes. I predict how I should react, and there is no mathematical answer to the chemical processes involved in that…don’t get me started on that shit.

For the record, i do think you could make more money in a crisis because of the volatility, but as you know it works both ways dude, there are no set rules that say you must get paid in a volatile market. I have kept buying all the way down some days, took huge heart attack risk, and got little reward or a big slap on the mouth!


jm's picture

It's all good.  I'll say some stuff about the randomness, because its on the tip of my tongue. I'll come back to the trading because I need to think about it.

White noise--the good randomness that obeys conventional probability models only happens in intervals of the market where quants can do their thing.

There is black noise which is random too, bu doesn't obey probability laws that make conventional models work... this is bad randomness. 

Bad randomness isn't really bad, it just doesn't fit models.  It is nonstationary, meaning there is no finite population mean and variance.  People estimate sample properties of the mean and variance, but they in the don't work.  It's like riding a bull when you thought you were on a hobby horse.

So what I'm saying is not binary... randomness or not randomness.  What I'm saying is that there are many differetn kinds of randomness, but assuming the one you work with is tractable using probability is a dangerous assumption. 

jm's picture

Hello.  Sorry this took a while.  I had some puzzles to work through about the implications.

Assume the future is unpredictable... if dW(t) (meaning the differential part of Ito's equation not it's derivative, thank you very much) is white noise then with enough capital one can profit simply from random perturbations moving a trade into the money, not because of cointegration.  The driver is randomness, not prediction.  This is being in the right place at the right time... luck.  if you are really lucky, then you can get upside for some non-stationary moves in your favor.

You can also trade in a way that exploits non-stationarity by being risk minimizing.  Take a bond and buy CDS protection on it.  THEN swap the cashflow from fixed coupon to floating rate.  Again, this is risk minimization and waiting for a big move to present itself.  When it does close the position and take a fixed payment that is worth it to you.  This seems to be Warren Buffett's way as I understand it, although I could be wrong.

This is in itself worthy of more detail.

bingocat's picture

I am perplexed. Your piece reads nicely but I remain perplexed as to what it is supposed to bring to the table.

First order of business...

When crisis [sic] happen, that is the best time to take risk because you are compensated for it.  I'm not saying don't take risk.  One has to in order to make money.

How does this jive with the concept that, paraphrased, "one cannot know the future; only the present matters"? How does one predict the best time? How does one predict how one will be compensated for it?


Anything which allows you to say that Michael Steinhardt and George Soros are "exceptions to the rule" means that your fundamental premise is flawed. They are either LUCKY or markets are not random, even when one is in a black noise regime.

Third... The conclusion of your piece appears to say that one will make more money if one is less levered in the down markets and can decide to get fully invested at the "right time". How does that jive with the idea that one cannot know the future? Or is this simply some kind of it-is-better-to-buy-low-and-sell-high-than-buy-high-

and-sell-high-if-one-looks-at-it-over-a-long-period argument?

Either scratch n sniff's DECISION to trade a successful 'model' is proof that predictions can be made (thereby rendering incorrect, or perhaps just incomplete, the argument that the future cannot be 'known' to some degree), OR he is just lucky. Philosophical dogma which says that a random process is random by definition and therefore results cannot be predictable does not allow for pattern recognition.

As to the idea of keeping powder dry, there are any number of treatises which address the calculation of optimal bet sizing. In essence, this is the same thing. In either case, you have to be able to decide a) what is your optimum return distribution profile, and how do you measure your expected return. The stochastic process of addressing the probability of return profile on your 'next bet' takes care of the "returns are better if you take risk after the crisis hits" part of your argument, but predicting that is a bear.

jm's picture

Saying it reads nicely is high praise from Bingocat.  Thanks.

Regarding the first order of business...

"How does one predict when compensation is right for risk?"

I would say this isn't an issue of future prediction as much as it is a present situation punching you in the face.  When relative pricing models break, and everyone is running in the opposite direction, compensation is good.  This is one benefit of having some cash-like assets... when your model breaks, you can take advantage. 

Regarding Soros and Steinhardt:

I'm humble enough to admit I could be wrong, and these guys could be the couterexamples that disprove what I'm saying.  I don't think so, but I leave it for the reader to consider.  Clearly they have an "intuitive feel" for markets.  Clearly they know what they are doing.  They may have the benefit of herd behavior following them.  But on a long enough timeline, they will encounter things that trump their intuition.

In other words, these intervals of liquidity can be somewhat complex.  An interval from 1955 to 2010 (what I would call the "Age of Liquidity" either for deomgraphic or other factors) can have subintervals like from 1955 to 1973 (Bretton Woods), and 1999 to 2010 (a secular bear market in equities).  They may be able to handle some intervals, but ultimately the world will shift in such a way they can't. 

Third, as I thought more about this thanks to prompting from "The answer is 42", liquidity is really just a device that may or may not work in an unpredictable world.  I talked about the Kolmogorov 0-1 law and its import below.  Liquidity is powerful as a device to steady your nerve as a trader.  I'm not naturally an "all-in" trader, and this may be just my perversities justified.  But at least I'm kicking the tires.   


Bob's picture

I'm sure he didn't mean it personally, but I think he said that you're probably a fairly lucky guy! 

It doesn't really violate the model.  Enjoy. 

He didn't claim that all is truly random by any means . . . merely that we still lack robust models for the complex dynamic system we actually operate in. 

The takeway, imo, is to never lose your respect for risk.  Being ready to fire on the tick is a very real concrete advantage--with the current generation of HFT algorithms, at least, even co-located.  If it works, it works, i.e., until it doesn't. 

scratch_and_sniff's picture

I'm not too sure if he saying that all is truely random, but i am sure that i am saying it. But i am also saying that just because its random, don't mean you cant predict.

Bob's picture

I see what you'd like to do, but if things truly were even predominantly random within normal bounds you'd be looking at a hopeless situation predicting across a serious/realistic number of parameters. 

Atomizer's picture

Please put on your rose colored eye wear.

This Labor Day, Secretary Solis wanted to talk directly with you -- the American worker. Tune in as she shares what she's seen and heard around the country... and what your Department of Labor is doing to get America back to work.

Biggus Dickus Jr.'s picture

You still didnt answer my question. The opportunity cost of maintaining liquidity for non stationary events is the return you would get for using that same liquidity to buy and hold of an equivalent risk and return adjusted asset class through more benign conditions. Either you can time the market or you can't. Which is it. Author speak with forked tongue in article.

jm's picture

See above.  Maybe I'm missing your point.

revenue_anticipation_believer's picture

"Either you can time THE market or you can't. Which is it"

"The Law of the Excluded Middle" goes back a while, since at least Aristotle 2300 years ago...

Is "THE Market"  susceptible to this Logical Analysis?  No, of course not, 

"THE MARKET" is an example of 'misplaced concreteness'  Whiteheads favorite Phrase...

Oh, He and Russell co-wrote Principia Mathematica - in which 'mathematics' was  presented as a subset of LOGIC..

.I mean, Whitehead KNEW what he MEANT when he said 'misplaced concreteness' is one of the great logical fallacies..

.Used in courtrooms daily to mislead, obscure, and make knowingly, misusing words = Rhetoric to falsifying REALITY

words and things

the model and the actuality

Yes, the words/model are great for 'compressing reality', but such is NOT lossless compression....


revenue_anticipation_believer's picture

Regards the statistical commentary:

i believe what was said, is that the gaussian distribution is convenient, so much so that the Black-Scholes options model is based on THAT particular distribution....white noise, which has about 99% of the spectral power/peaks within 13db ..... mostly pretty well bounded analogous to the bollinger bands at the typical 2x standard deviations...

AND THIS MODEL WORKS PRETTY WELL, IN STATIONARY DISTRIBUTIONS...long term market movements for instance...where there is noise = predictable envelope, and SIGNAL a definite trend your you can statistically plan your hedging...

BUT at times of extreme market conditions, the DISTRIBUTION IS NO LONGER STATIONARY, but is in dynamic/chaotic transistion....Black Swan conditions...AND YOU WHO 'WON' OVER AND OVER SUDDENLY START LOSING OVER AND OVER...

the 'distribution' changed from white noise, thru a black noise non-'normal' distribution... that is indeed, the REAL NORMAL, the normal you use for how to engineer the capacity of a dam, for the 100 year or the 1000year flood...the fractal for 'normal' = 2.0 dimensions, the fractal for the dam filling was (i forget..) Something like 1.6 dimensions = a FRACTAL dimension...

as applied to investing, this means that the long term maximum return investing....and factory building, etc is unsure....could become 'allentown 1982' worthless assets...

In cases. like NOW of non-stationary non-gaussian distributions....SHORT TERM ASSETS = highly liquid investing = day trading, so to speak, makes sense....

thats what i THINK is being said...





The Answer Is 42's picture

This is all very well, except that I don't think it answers the question of opportunity cost of holding a liquidity crisis reserve.

Here's what I think: the very notion of "expected value" is undefined, and undefinable, because there's no convergent, stationary distribution. The best you can do is to speak of "expected value given this set of conditions", which is meaningless for the investor/trader since we all know "this set of conditions" will be broken some day -- and we all know how bad the consequence can be when we forget this.

Terms like "expected return" or "opportunity cost" are misleading in that they convey a false sense of rigor and meaning. Truth is nobody knows what the fuck the "expected return" is or what the "opportunity cost" is if you don't do this and this.

Fact is, many in the credit derivatives business had known for years that shit would hit the fan some day, correlation would go to 1, and super-senior tranche would be as good as equity. But at least at the individual level, it didn't make any financial sense for them to hedge it, nor any political sense for them to raise the issue -- bause everybody knew that the higher-up guys (at least some of them) also knew and it's an unsolvable problem but if it's put on the table then somebody would have to pay for it. It's a prisoner's dilema in real life. It's a rational thing to do, just like Californians living in California despite the inescapable future of devastating earthquake.

My suggestion to the author is to expand deeper on some of the math, but cut off the conclusions -- they're weak and at best loosely related to the arguments.


jm's picture

I tried to make the conclusions a little less mathy and practical so one could "see" the effect of losses on different types of portfolios.  To me, it is all very fuzzy in conclusion.  For example, research desk output is complicated things getting priced.  Trading desks take the output and apply their subjective +/- haircuts to it a lot of times.   

As far as math goes, I agree with your statements, but would add something about the Kolmogorov or Levy 0-1 therems... there are extreme events which will either converge to 1 or 0.  They either happen for sure or they won't happen ever.  The problem is determining which they converge to is extremely hard to do theoretically.  Practically, there is no chance of determining rate of convergence (timing).  The sun goes supernova in my lifetime... or it doesn't.

So what do you?  You can either live in California on a fault line and take out a small insurance policy that may never pay out for peace of mind.  Or you can move to Oklahoma and get killed by an F5 tornado. Horribly unsatisfying view on liquidity: it may save your life or it may be worthless.  I showed at least in some cases it keeps you alive.  Ambiguities.

You mention CDS.  Should the Fed have paid out these insignificant earthquake insurance policies to keep worse things from happening is the ultimate policy question of the age.


The Answer Is 42's picture

In other words, buy the fucking dip. ;)

Some excellent intro and very good points. They should lead to much more meaningful conclusion than "buy the fucking dip" or "never fire all your bullets", no?

Liquidity risk and its value/premium is an unsolved problem, but as we all know now extremely important. But I think there's a very simple reason why it's still unsovled -- nobody wants to pay it. When you sell insurance and calculate your reserve, you have to stop somewhere -- say, once a decade, once a century type event -- beyond which poin tyou just say "ugh fuck it." It's simply mathematically impossible and financially nonsensical to have enough reserve to cover everything. Mathematically, I suspect the integral is divergent; financially, it simply doesn't pay. So, all insurance has an embedded cap on claims, which is not specified anywhere in the policy but applies implicitly on the policy holder collective. Hedging liquidity risk is similar mathematically.

You raised some very good points. But some points can be taken further in that, the market, by nature, defies either structure or statistical description for the simple reason that, if it doesn't, then somebody will figure it out and consistently win and it will be traded away. It's not only unpredictable, it's even statistically unpredictable. This is not an accident, it's the due to the game nature of the market. But lots of empiricaly evidence (the existence of consistently successful traders) indicates that it is highly, consistently predictable in some time scales and areas. And this is not an accident, either.

jm's picture

I didn't read this comment until after I wrote the reply above it.  Remarkably similar things said in different ways.



mcguire's picture

speaking of liquidity, this is an issue all of my zerohedge friends should pay attention to:

TraderTimm's picture

The only thing that keeps me analyzing market data for underlying structural components is the Hurst Exponent. In the most simplified form, it demonstrated prior changes in equity prices indeed "feed-forward" into the present price. This persistence of structure is the only thread upon which I've devoted countless hours of study.

I don't have the "holy grail", or anything near that - but I feel that I have some useful tools that expose the 'guts' of underlying price structure. Unfortunately, I may be up against an inherent limit much like that of the Heisenberg Uncertainty Principle. In this context, what I've been working on suggests that I may deduce a system to calculate turns - but not their intensity.

Such is the riddle. Would you like to know every turn, including the inconsequential ripples that would flatten any minor gains into a fine paste, or would you rather know the magnitude (but not the direction) of the changes? I'm still struggling to put some kind of framework around this, but it may all end in a featureless expanse of null-profitability.

The FED-jections and other financial jackassery certainly doesn't help. If anything, I think it adds additional bias and 'noise' to data certainly rife with it in the first place.

jm's picture

The Hurst exponent has strong ties to many distributions. 

Let H=Hurst exponent:

1/H = 2, then the times series is not only stable but normal.


TraderTimm's picture

Ah, makes sense. Where a completely random series would produce a Hurst exponent of 0.5, that indeed does equal 2 when divided into 1. For other readings such as S&P data, I've seen Hurst exponents of 0.78 which would equate to 1.28205 (repeating).

I swear, when someone finally cracks this nut it will end up being a simple solution that has evaded all prior effort of discovery. Much like the elegant double-helix of DNA. From simplicity arises complexity.

jm's picture

Watch out.  The sequential calculation of a Hurst exponent over a time series may be nonstationary.  You're back to square one.

Markets absorb everyone's good ideas until they aren't good ideas anymore.


Variance Doc's picture

The author of this is article is very confused about many topics in measure theory and probability.  Some highlights are:

"...the queen of measure theory: probability."

No dude, you have it backwards: probability is the result of restricting the Lebesque measure to the interval [0,1] and considering the sigma field of measureable sets of [0,1]; Thus, constructing a probability measure. This can be extended to [a,b] via the restriction to omega, the universal set. Measure theory is the queen of probability.

"The unifying method of distilling stochastic processes into a unified mathematical subject is informed by measure theory, which in turn rest upon rigorous notions of integration."

Again, you get this wrong: Integration is the extension of the basic concepts of measure theory to measurable functions.  That is, “integration” is just a fancy term (a bit of misnomer) for measureable functions.  Complete probability spaces are the foundations of stochastic processes.

And it gets worse:

“There is nothing to support the notion that the probability distribution is identical over time.”

And that is why the measure is adapted to a filtration (increasing sigma algebra), so we don’t have to change the measure.

“The differential dW(t)…” 

This is NOT a differential!  I do hope you know that it is shorthand for integration since these Weiner processes are NON DIFFERENTIALBE!!!!!

Really, you need to step back and learn the material before you “teach” it to others.  There are glaring holes in your command of the subject of probability and statistics.  Garbage In Garbage Out at its best.

jm's picture

This is the kind of nit-picky shit that verifies you are merely a pain in the ass. 

How you view measure theory and it's relation to probability is your business.For this context, measure theory and probability are pretty much identical

“There is nothing to support the notion that the probability distribution is identical over time.”

That you have problems with this statement is confusing to me, but I hane no intention of garnering you opinion on the issue.

As to the "differential" stuff, you are wasting poeple's time... like most academics.



scratch_and_sniff's picture

JM is right VD, you are just being a pain in the ass. The wiener process(Brownian motion) is non differentiable only if it is discrete, therefore we would use simple binomial functions to treat it(and all pretty useless to real life modelling), but since JM went on to talk about noise, i.e. I think he was referring to the noise function(sigma t) in a continuous Black-Scholes model with drift(not dWt which deals with incremental changes in the Brownian motion component), then calculus is the standard way of working. Sooo, atleast JM’s head is in the right place, in this case.

Variance Doc's picture

"The wiener process(Brownian motion) is non differentiable only if it is discrete...."

You really have no idea what you are talking about. BM/WP is a continuous process and it is not differentiable.  Newtonian calculus does not work in stochastic analysis, hence the need of stochastic calculus.  Learn the difference.

You really are a poster child for the Dunning–Kruger effect.


scratch_and_sniff's picture

My bad VD, was thinking of time discrete Brownian motion, haven’t went near any math in over 8 years, and still obviously don’t appreciate the virtues of a quick look at wikipedia. Yes, differentiable nowhere and continuous everywhere...okay, stick me up there as the Dunning–Kruger effect poster child, no doubt you will be joining me at some stage.

jm's picture

No worries.  He's just a bitch trying to protect what he thinks is his turf.