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### Brownian motion, does this phrase make sense?

Posted: **May 8th, 2007, 10:12 pm**

by **Rez**

QuoteOriginally posted by: gardener3Cuchulainn,I am looking for a proof of the following assertions made by N:GBM is mathematically inconsistentStochastic calculus likewise rests on inconsistent assumptions and is therefore wrongStock price 'processes' (what ever that means) are continuousYou won't get any.Many have tried before you... That's his raison d'être 8->K

### Brownian motion, does this phrase make sense?

Posted: **May 8th, 2007, 10:24 pm**

by **N**

QuoteOriginally posted by: RezQuoteOriginally posted by: gardener3Cuchulainn,I am looking for a proof of the following assertions made by N:GBM is mathematically inconsistentStochastic calculus likewise rests on inconsistent assumptions and is therefore wrongStock price 'processes' (what ever that means) are continuousYou won't get any.Many have tried before you... That's his raison d'être 8->KI thought the introduction was pretty good. gardener3's interruptions and total ignorance of math make it impossible to convey even simple concepts. Somehow I get the feeling you're in the same league.

### Brownian motion, does this phrase make sense?

Posted: **May 8th, 2007, 10:54 pm**

by **Rez**

QuoteOriginally posted by: NQuoteOriginally posted by: RezQuoteOriginally posted by: gardener3Cuchulainn,I am looking for a proof of the following assertions made by N:GBM is mathematically inconsistentStochastic calculus likewise rests on inconsistent assumptions and is therefore wrongStock price 'processes' (what ever that means) are continuousYou won't get any.Many have tried before you... That's his raison d'être 8->KI thought the introduction was pretty good. gardener3's interruptions and total ignorance of math make it impossible to convey even simple concepts. Somehow I get the feeling you're in the same league.Cool your boots mate, no need to get uptight!I was just making an observationK.__.......................__ (....\..................../....). \....\................ /..../...\....\.............../..../....\..../´¯.I.¯`\./.../..../... I....I..(¯¯¯`\...I.....I....I...¯¯.\...\...I.....I´¯.I´¯.I..\...)...\.....` ¯..¯ ´.......'....\_________.·´.....l-_-_-_-_-_-|.....l-_-_-_-_-_-|

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 12:04 pm**

by **MCarreira**

N,QuoteOriginally posted by: NI thought the introduction was pretty good.I can't imagine you winning the hearts and minds of the traders in 1973 if your start talking about manifolds and do not give them a formula for the price and a related formula for the hedge ... quoting NNT (Dynamic Hedging, page 5):"... Traders, by contrast, are impatient and need brief simplistic descriptions, ...""...1. In a single sentence, explain the conclusion, before discussing the subject matter.2. In a single sentence, explain the subject matter.3. If unable to perform 1 or 2, then abandon the entire project...."Also worth bringing to the discussion (?!) are Derman's comments on models:Good-ModelsTheories-Part-IIof which I quote the most relevant part:"In finance, a good theory is something different. A good theory is something that gives you insight and a way of thinking about phenomena, even if it's not a wonderful predictor. A good theory starts from something intuitive about markets, or something qualitative, and then quantifies it, which leads to new concepts, which one then develops intuition about through the theory, which then becomes the subject of newer theories."I still have not found the references to any debunking of a well accepted book like Avellaneda's.

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 1:38 pm**

by **N**

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### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 1:45 pm**

by **N**

MCarreira, Yes back to the important question... Avellaneda's book is 100% correct if the returns are random. Very incorrect if returns are deterministic. Things not random are deterministic.Returns are only random if the distribution is Gaussian (think of sums of returns - central limit theorm). Is there anyone around who believes returns have a Gaussian distribution? Therefore Avellaneda's book is worthless.

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 2:12 pm**

by **N**

QuoteOriginally posted by: outrunQuoteReturns are only random if the distribution is Gaussian (think of sums of returns - central limit theorem). Is there anyone around who believes returns have a Gaussian distribution? Therefore Avellaneda's book is worthless.Your probably talking about independence (instead of random) with your central limit theorem. Returns can be dependent & still be random. Mean reverting processes are random but have autocorrelation. You can also have non gaussian distribution (e.g. the stable distributions) that keep the same (scaled) non gaussian distribution under summation.what I;ve seen: returns are non-Gaussian, have dependency in time & are not deterministic.If there is autocorrelation, one can simply use a linear predicitive filter to get the next return. A slam dunk for making money. Unfortunately there is absolutely no autocorrelation in returns of any financial instruments.

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 2:45 pm**

by **N**

QuoteOriginally posted by: MCarreiraN,QuoteOriginally posted by: NI thought the introduction was pretty good.I can't imagine you winning the hearts and minds of the traders in 1973 if your start talking about manifolds and do not give them a formula for the price and a related formula for the hedge .No problem if you could have enlisted Wigner, Lax or Simons as your trader.

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 2:48 pm**

by **MCarreira**

N,Now we have a chain of thought.Let's go though this in order:1. "Avellaneda's book is 100% correct if the returns are random."Why can't we say also "Avellaneda's book is a 100% correct model if the returns are modeled as random." ? In this way we are not making any assertion about returns that we cannot prove.2. "Very incorrect if returns are deterministic."I guess that if returns on financial markets were deterministic in the Laplace's Demon/Computer way, markets wouldn't be much fun.3. "Things not random are deterministic."Are your thoughts random ?4. "Returns are only random if the distribution is Gaussian (think of sums of returns - central limit theorm)."outrun has already addressed that; others can discuss it better than myself.5. "Is there anyone around who believes returns have a Gaussian distribution?"No, although it was a very convenient way of modeling returns for its tractability.6. "Therefore Avellaneda's book is worthless."Well, in 1. you assign to a book that tries to present a "toy model" the ambition of modeling reality, can't say I agree with you on that. I guess that the discussion now becomes a debate on the meaning of random and deterministic ... again. So I can't say that I agree with you on 1 (the way you phrased it), 2, 3 and 4. And therefore I cannot agree with you on 6.

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 2:52 pm**

by **MCarreira**

QuoteOriginally posted by: NQuoteOriginally posted by: MCarreiraN,QuoteOriginally posted by: NI thought the introduction was pretty good.I can't imagine you winning the hearts and minds of the traders in 1973 if your start talking about manifolds and do not give them a formula for the price and a related formula for the hedge .No problem if you could have enlisted Wigner, Lax or Simons as your trader.Would they have been convinced back in 1973 ? Would thay trade against each other ? Or would they find counterparties willing to trade "blindly" ?

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 3:15 pm**

by **N**

QuoteOriginally posted by: MCarreiraQuoteOriginally posted by: NQuoteOriginally posted by: MCarreiraN,QuoteOriginally posted by: NI thought the introduction was pretty good.I can't imagine you winning the hearts and minds of the traders in 1973 if your start talking about manifolds and do not give them a formula for the price and a related formula for the hedge .No problem if you could have enlisted Wigner, Lax or Simons as your trader.Would they have been convinced back in 1973 ? Would thay trade against each other ? Or would they find counterparties willing to trade "blindly" ?They'd go directly to widely-traded, highly-liquid instruments like RenTec does today. Would they trade against each other - no way.

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 3:23 pm**

by **gardener3**

QuoteOriginally posted by: NMCarreira, Returns are only random if the distribution is Gaussian (think of sums of returns - central limit theorm). Is there anyone around who believes returns have a Gaussian distribution? Therefore Avellaneda's book is worthless.So basically argument is this: the average of a series converges to a normal distn under CLT, and we don't observe a normal distn, therefore randomness does not exist. Since you have not specified which CLT you are referring to I'll assume you are refering to the standard definition which assumes stationarity and iid. Since returns are not iid we know that CLT will not hold, so there goes your argument. Second what you suggest is internally inconsistent. On the one hand you claim returns are predictable based on past information in which case CLT will not hold, and at the same time you want to use CLT to prove a point. You claim there's no randomness yet you need independent draws to apply CLT. Again inconsistent. Of course certain physical processes are not random, people use the word random to refer to things they cannot model or compute. As before this is not particularly an insigthful observation.

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 3:24 pm**

by **gardener3**

QuoteOriginally posted by: NQuoteOriginally posted by: outrunQuoteReturns are only random if the distribution is Gaussian (think of sums of returns - central limit theorem). Is there anyone around who believes returns have a Gaussian distribution? Therefore Avellaneda's book is worthless.Your probably talking about independence (instead of random) with your central limit theorem. Returns can be dependent & still be random. Mean reverting processes are random but have autocorrelation. You can also have non gaussian distribution (e.g. the stable distributions) that keep the same (scaled) non gaussian distribution under summation.what I;ve seen: returns are non-Gaussian, have dependency in time & are not deterministic.If there is autocorrelation, one can simply use a linear predicitive filter to get the next return. A slam dunk for making money. Unfortunately there is absolutely no autocorrelation in returns of any financial instruments.zero linear correlation does NOT equal independence

### Brownian motion, does this phrase make sense?

Posted: **May 9th, 2007, 3:40 pm**

by **N**

QuoteOriginally posted by: MCarreiraN,Now we have a chain of thought.Let's go though this in order:1. "Avellaneda's book is 100% correct if the returns are random."Why can't we say also "Avellaneda's book is a 100% correct model if the returns are modeled as random." ? In this way we are not making any assertion about returns that we cannot prove.2. "Very incorrect if returns are deterministic."I guess that if returns on financial markets were deterministic in the Laplace's Demon/Computer way, markets wouldn't be much fun.3. "Things not random are deterministic."Are your thoughts random ?4. "Returns are only random if the distribution is Gaussian (think of sums of returns - central limit theorm)."outrun has already addressed that; others can discuss it better than myself.5. "Is there anyone around who believes returns have a Gaussian distribution?"No, although it was a very convenient way of modeling returns for its tractability.6. "Therefore Avellaneda's book is worthless."Well, in 1. you assign to a book that tries to present a "toy model" the ambition of modeling reality, can't say I agree with you on that. I guess that the discussion now becomes a debate on the meaning of random and deterministic ... again. So I can't say that I agree with you on 1 (the way you phrased it), 2, 3 and 4. And therefore I cannot agree with you on 6.1. Fine.2. Deterministic doesn't mean easy or without technical limitations (bandwidth etc)3. A good example of random is the random walk. Convergence is very rapid to Gaussian distribution. Each step is independent of the previous one. A constant increase of entropy. You can't predict the direction of the next step.4. I claim that the limit of the sum of random variables is always a normal distribution. I know this concept nowdays is a bit controversial. It wasn't when I studied OR 30 years ago.5. Suppose returns were actually a superpostion of sine waves. Would you still use Gaussian assumption for tractability?6. Why would there be a debate on random vs deterministic? If you can predict the next value in a timeseries it's deterministic. If not, it random. (Of course a variable can have both random and deterministic components). The book assumes the return timeseries is random. A random timeseries has a normal distribution for elements. But in the real world, the return distrubution is far from normal. I guess you could claim that the text is not so bad approximation. I claim close is only good in horseshoes.