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Fermion
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Brownian motion, does this phrase make sense?

May 7th, 2007, 9:53 pm

QuoteOriginally posted by: crowlogicTrue, it is inherently discrete, but could it not be said that the stochastic process is that which is between the discrete observations? A hybrid continuous-discrete setup.Yes, but what does that give us other than an untestable theory of non-observables? It's like trying to model a system with a finite number of degrees of freedom by a function that has an infinite number of parameters. There are plenty of solutions but which one is correct?I'm not saying (like N apparently does) that stochastic calculus can't give us any insights, but ultimately we have to have to apply them to a discrete model if those insights are to be useful.
 
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crowlogic
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Brownian motion, does this phrase make sense?

May 7th, 2007, 10:03 pm

QuoteOriginally posted by: FermionQuoteOriginally posted by: crowlogicTrue, it is inherently discrete, but could it not be said that the stochastic process is that which is between the discrete observations? A hybrid continuous-discrete setup.Yes, but what does that give us other than an untestable theory of non-observables? It's like trying to model a system with a finite number of degrees of freedom by a function that has an infinite number of parameters. There are plenty of solutions but which one is correct?I'm not saying (like N apparently does) that stochastic calculus can't give us any insights, but ultimately we have to have to apply them to a discrete model if those insights are to be useful.Two points of each interval are observable, and one end point is the next begin point, it's not really unobservable...
 
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N
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Brownian motion, does this phrase make sense?

May 7th, 2007, 10:15 pm

QuoteOriginally posted by: FermionQuoteOriginally posted by: MCarreiraIs there a way to express your assertions in a way I could follow or at least a book/paper I could read ?I'm not a particularly strong mathematician, but from what I do understand from the multitude of N's posts, his abhorrence of stochastic calculus seems to stem from the fact that it uses a continuous process to represent a system that is inherently discrete. (Successive observations necessarily have a finite time difference.)If this is, as I suspect, the core of his case, then I wish he would state it as simply as this and stop the endless distractions with hyperbolae, obscurantism and tirades against other mathematicians and physicists. By using simple English like this, instead of trying to quell rational thought by bombing us with mathematical jargon, he might give myself (and, I suspect, others) a better chance of both understanding and joining in a reasoned discussion.No, the process is continuous. The particular solution to the heat equation is e^-at. Adding boundary conditions, we get the expected Gaussian temperature/time curve. [The equivalent circuit is a network of resistors with capacitors at each node, in case you're an EE] Notice that this particular solution also applies to compound interest. In one case the 'rate' is heat conduction and with the other it's the compound interest rate. All Lie groups have a 'rate' (or infinitesimal automorphism ) including this most simple example. Please note that not only are these processes continous, but there is no noise (BM). Aslo note that only rational numbers are needed for the math.Agreed?
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Fermion
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Brownian motion, does this phrase make sense?

May 8th, 2007, 5:32 am

QuoteOriginally posted by: NNo, the process is continuousWhat??? You've told me before in this thread and in another months ago that in time series analysis the only application of probability is in frequency counting and you've equated determinism with discrete states and told us all that everything is deterministic. Now you tell me the opposite!QuoteAgreed?Are you kidding? It's as clear as mud. Is this some sort of competition over who can throw out the most impressive math buzz words with the most riddles, twists and turns? If so, you win. I hereby give up all attempts to understand. Please collect your prize.
 
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N
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Brownian motion, does this phrase make sense?

May 8th, 2007, 8:20 am

QuoteOriginally posted by: FermionQuoteOriginally posted by: NNo, the process is continuousWhat??? You've told me before in this thread and in another months ago that in time series analysis the only application of probability is in frequency counting and you've equated determinism with discrete states and told us all that everything is deterministic. Now you tell me the opposite!QuoteAgreed?Are you kidding? It's as clear as mud. Is this some sort of competition over who can throw out the most impressive math buzz words with the most riddles, twists and turns? If so, you win. I hereby give up all attempts to understand. Please collect your prize.Look, the manifold I just described is clearly continuous, has a rational number domain, is deterministic, is fully integrable and has single discrete state (the "rate"). Since probability cannot be complex, this smooth manifold is the only manifold where probability can be defined.Fermion, it may not be obvious, but if probability is a rational number then probability *is* frequency counting, right? Black-Scholes is defined on this manifold. (although the model itself may not be robust enough).So far, there's no Brownian motion.
 
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MCarreira
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Brownian motion, does this phrase make sense?

May 8th, 2007, 12:00 pm

N,Although your quotes:QuoteOriginally posted by: NNo, the process is continuous. The particular solution to the heat equation is e^-at. Adding boundary conditions, we get the expected Gaussian temperature/time curve. [The equivalent circuit is a network of resistors with capacitors at each node, in case you're an EE] Notice that this particular solution also applies to compound interest. In one case the 'rate' is heat conduction and with the other it's the compound interest rate. All Lie groups have a 'rate' (or infinitesimal automorphism ) including this most simple example. Please note that not only are these processes continous, but there is no noise (BM). Aslo note that only rational numbers are needed for the math.And:Look, the manifold I just described is clearly continuous, has a rational number domain, is deterministic, is fully integrable and has single discrete state (the "rate"). Since probability cannot be complex, this smooth manifold is the only manifold where probability can be defined.Fermion, it may not be obvious, but if probability is a rational number then probability *is* frequency counting, right? Black-Scholes is defined on this manifold. (although the model itself may not be robust enough).So far, there's no Brownian motion.are a bit more elaborated than your previous quotes, I'd still like you to give me some reference that takes a book like, let's say, Avellaneda's "Quantitative Modeling of Derivative Securities" ,and shows where things are wrong, and, most importantly, what would be the correct way of price derivatives instead.My EE course is 15 years behind me and I've almost succeeded in forgetting everything but the software part, so mentioning resistors triggers bad memory circuits for me.
 
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Brownian motion, does this phrase make sense?

May 8th, 2007, 1:33 pm

Most books on these topics are unreadable, so let me wrap up smooth manifolds before I recommend something. The one-manifold has sin and cos solutions, e^iwt. The domain is the complex numbers and the 'rate' is frequency. These are obvious solutions to well known differential equations. It turns out that smooth manifolds are the solutions and the only solutions to differential/partial differential equations.Here's a question... Can you have a path integral not on a smooth manifold, for example one where motion is determined by dW?
 
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gardener3
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Brownian motion, does this phrase make sense?

May 8th, 2007, 3:17 pm

QuoteOriginally posted by: FermionIf this is, as I suspect, the core of his case, then I wish he would state it as simply as this and stop the endless distractions with hyperbolae, obscurantism and tirades against other mathematicians and physicists. By using simple English like this, instead of trying to quell rational thought by bombing us with mathematical jargon, he might give myself (and, I suspect, others) a better chance of both understanding and joining in a reasoned discussion.Taleb, (in his dynamic hedging book) referring to Feynman, says that those who really know their subject well can speak of it in plain language without the need to into technical jargon which obscures or confuses the point they are trying to make. Those who are uncomfortable with the fundamentals, or those who are insecure in their knowledge hide behind technical jargon, and tend to throw out technical terms and talk excessively of details. That, Taleb says, is good way to spot a bullshitter.
 
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Brownian motion, does this phrase make sense?

May 8th, 2007, 3:42 pm

QuoteOriginally posted by: gardener3QuoteOriginally posted by: FermionIf this is, as I suspect, the core of his case, then I wish he would state it as simply as this and stop the endless distractions with hyperbolae, obscurantism and tirades against other mathematicians and physicists. By using simple English like this, instead of trying to quell rational thought by bombing us with mathematical jargon, he might give myself (and, I suspect, others) a better chance of both understanding and joining in a reasoned discussion.Taleb, (in his dynamic hedging book) referring to Feynman, says that those who really know their subject well can speak of it in plain language without the need to into technical jargon which obscures or confuses the point they are trying to make. Those who are uncomfortable with the fundamentals, or those who are insecure in their knowledge hide behind technical jargon, and tend to throw out technical terms and talk excessively of details. That, Taleb says, is good way to spot a bullshitter.gardener3,I know manifold has 8 letters, but sometimes words are longer than four letters. Sorry.Also didn't realize 'differential equation' was way over your head. Next time I'll try to avoid technical jargon like 'compound interest' and 'frequency' as examples of rates.Hey, I guess rational and complex can be better explained in plain language. Why don't you help me there.n
 
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MCarreira
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Brownian motion, does this phrase make sense?

May 8th, 2007, 4:02 pm

N,Let me return to this question:It's 1973, options are starting to trade, and you step out of a time machine fully armed with all that you know up to 2007.You then decide to be the one who'll win the Nobel for developing the pricing model for options and other derivative products.What could you publish that would make everyone ignore B&S, Merton, Ed Thorp, Bachelier, Mandelbrot, and use your approach ? How would it differ from what people chose to use ?Please note that we're speaking about a model that people can use to price things quickly.
 
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gardener3
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Brownian motion, does this phrase make sense?

May 8th, 2007, 4:35 pm

N, dude, I have yet to see you put forth a coherent argument or write down a single mathematical proof. You want to use complicated math, by all means show me the maths, show me the proofs.
 
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Brownian motion, does this phrase make sense?

May 8th, 2007, 5:33 pm

QuoteOriginally posted by: gardener3N, dude, I have yet to see you put forth a coherent argument or write down a single mathematical proof. You want to use complicated math, by all means show me the maths, show me the proofs.I was simply showing some well known relationships between Lie groups, differential equations, and field extensions as an introduction to the basic concepts. If this looks like complicated math, then I'm afraid there is no hope.
 
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gardener3
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Brownian motion, does this phrase make sense?

May 8th, 2007, 6:19 pm

Perhaps complicated for me, but not for others on the forum. And who knows maybe I'll follow as well. So why don't you show us the maths, and the proofs. Write down equations not words.
 
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Cuchulainn
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Brownian motion, does this phrase make sense?

May 8th, 2007, 7:21 pm

QuoteOriginally posted by: gardener3Perhaps complicated for me, but not for others on the forum. And who knows maybe I'll follow as well. So why don't you show us the maths, and the proofs. Write down equations not words.This might help:Here is one example of using Lie groups to solve some fluid problems. I am not sure if they are a real improvement on 'traditional' methods. The authors need to do a lot of <quote>tedious algebra<unquote> to get the new equations.Olver's book on Lie PDE tends to give examples that are not so directly relevant to many of the PDE in QF.Lie
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gardener3
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Brownian motion, does this phrase make sense?

May 8th, 2007, 8:13 pm

Cuchulainn,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 continuousNow if his assertion is simply that GBM is a poor proxy for actual stock prices, then I don't think many people will disagree with him. Stock price increments are not continuous, are not independent and they are not normally distributed. With respect to the last point either Fama or Mandelbrot in their J of Bus article refer to a study that goes back as far as 1914 that commodity prices are not normally distributed. These are not new or insightful observations. You cannot make excess trading profits with GBM (afterall it essentially means that you have no view with respect to the future returns not already incorporated into the current price) that does not mean that GBM has useful applications in other parts of finance. Same goes for B/S.
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