Some of you might have know about the paper entitled "Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple S p e c i f i c a t i o n T e s t"By Lo and MacKinlay from U. of Pennsylvania.But on the other hand, many mathematical models runs under this assumption. How doespeople who work in the financial industry using stochastic calculus, brownian montions and the like see this issue?I'm a master candidate on applied math and I'm studying a lot of stochastic calculus, I would like to knowif these is somewhat in vain when you consider applying these know/ledge into the financial industry.Thanks for any help.

Haven't looked at this paper for a while, but just browsed it again, briefly.A Random Walk can mean a lot of different things. Thereare simple walks akin to Brownian motion, iid walks akin to Levy processes, Markov-modulatedwalks akin to stochastic volatility models, etc.The simple walk is easy to reject and Lo and Mackinlay reject it. Fine, nobody would argue and everybody would agree.Next, they reject a more complicated walk through a variance ratio test. Again, fine. This likely means the variance processis more complicated than their model. No issues so far, as far I can see. Broadly, most models in finance assume that security returns have very low serial correlation but arenot independent. Finally, Lo & Mackinlay try to argue that some CRSP index has significant (weekly & monthly) serial auto-correlation. This is more serious. But, this is only really a challenge to the dogma if the correlation is significant enoughto be (i) not the result of data mining; (ii) exploitable in some trading strategy, ideally in the future, aftertransaction costs. Since it is a very large index, the (unmet) burden is on the authors here. My advice; keep studying. Models are getting better and the broad claim that returns are largely unpredictableis not going to be refuted. The strongest indirect evidence for this will always be the performance recordsof the money management industry. Just keep asking yourself: in many industries (say software), a dominantsupplier often arises. Why does this happen?. Why has there -never- been a dominant supplier in investment management??

They do and they don't. Edit: Your paper was written in 1988 - check out the more recent additions to the literature for the low down ... Also, discussions on these forums as to why the BSM eqn is considered mostly crap .

QuoteOriginally posted by: AlanMy advice; keep studying. Models are getting better and the broad claim that returns are largely unpredictableis not going to be refuted. The strongest indirect evidence for this will always be the performance recordsof the money management industry. Just keep asking yourself: in many industries (say software), a dominantsupplier often arises. Why does this happen?. Why has there -never- been a dominant supplier in investment management??Are there not dominant suppliers in the money management industry (bulge bracket IB's and biggest HF's) just as there are dominant suppliers in the software industry (APPL, MSFT, IBM, ORCL, SAP etc.)?

Alan, thank you for your reply.Nowadays I'm just studying the underlying math of all of these. I still don't have any experience with actualmodeling. I'm the kind of guy who can't use a stochastic integral until I learn it with a somewhat deep mathematical understanding. So I'm sorry if my question seemed fool.

pishposh

QuoteOriginally posted by: TraderJoeQuoteOriginally posted by: AlanMy advice; keep studying. Models are getting better and the broad claim that returns are largely unpredictableis not going to be refuted. The strongest indirect evidence for this will always be the performance recordsof the money management industry. Just keep asking yourself: in many industries (say software), a dominantsupplier often arises. Why does this happen?. Why has there -never- been a dominant supplier in investment management??Are there not dominant suppliers in the money management industry (bulge bracket IB's and biggest HF's) just as there are dominant suppliers in the software industry (APPL, MSFT, IBM, ORCL, SAP etc.)?The Economist puts themoney management industry at $64 trillion (assets under management). MSFT has, what, 80% of the operating system business?Let's be generous and say a dominant supplier need only have 20% of the market, which meansabout $13 trillion. Now let's take the hedge fund darlings Renaissance Technology -- surely those guys have enough special expertiseto capture 20% of the market, right? But, they only have $27 billion, or so says wikipedia, which is a miserable 0.04% ofthe market. Why aren't they like MSFT? or at least as good as AAPL? I'll let you research the size of the asset management units of IBs. I'm pretty confident they will each manage well under 1% of the market. p.s. Just to be clear, I am just talking about active management here. For example, Vanguard group-could- over time become a dominant supplier by offering largely passive management at very low cost. This would simply reinforce the argument.

Hi fnisky,I wanted to reply to your question, because I have had a somewhat similar experience. Since I am now finishing my MSc applied math, I have also had a number of courses that simply assume GBM and do not discuss the validity of this assumption. Feeling not satisfied, I also read Lo and MacKinlay (actually, they published a book called 'Non-random walk down wall-street' in which the combine several papers, I think it is available in pdf-format online). I can also recommend the paragraph from Lo and MacKinlay about the 'Efficient Market Hypotheses', which you have perhaps already, or may soon, stumble(d) upon in your program, and which will probably also come as questionnaible.In my experience, most practitioners accept the fact that GBM is not completely true, but on the other hand are very skeptical towards technical analysis. I think that for most practitioners the question how true GBM actually is, is not very relevant. However, it does not directly follow that finding some kind of mathematical rule that will give you a consistent above-average trading result (trading costs taken into account!) is an easy task.So, though academia does researches more complex mathematical market processes (local- and stochastic volatility, jump diffusions, Levy processes, etc.), there are still many books (and teachers for that matter), that ignore this issue and will simply assume GBM. The assumption of zero trading costs is another often encountered example of where theory does not match practice. A fixed interest rate (static over time, as well as a flat yield curve) is another. My advice would be: be aware of the differences between theory and practice and how these influence theoretic results, and accept that theory oversimplifies reality.Regards

fniski: But on the other hand, many mathematical models runs under this assumption. How does people who work in the financial industry using stochastic calculus, brownian montions and the like see this issue?The reason mathematical models assume GBM as an approximation is that it's mathematically easy to handle GBM, you then correct for the fact that the markets aren't GBM by adding lots of correction factors. It's a lot like assuming that planets are point size spherical objects that travel in elliptical orbits around the sun. They aren't. You make the assumption, get out some numbers, and then you think about what is going on to think of how good/bad your assumptions are. For some things, they are pretty good. For other things that they are so bad that you are better off tossing the model away completely.

Alan: But, this is only really a challenge to the dogma if the correlation is significant enoughto be (i) not the result of data mining; (ii) exploitable in some trading strategy, ideally in the future, aftertransaction costs.And then you end up with market feedback effects. If you find an exploitable trading strategy then one of two things is going to happen. Either everyone starts exploiting it and get a negative feedback cycle at which point you can't much money/any money exploiting it. This is happening with statistical arbitrage in which people are putting more and more effort and making smaller and smaller returns. Or you get a positive feedback cycle in which the market blows itself apart (people buy real estate, prices rise, people buy more real estate until things blow up).

QuoteOriginally posted by: RoderickSo, though academia does researches more complex mathematical market processes (local- and stochastic volatility, jump diffusions, Levy processes, etc.), there are still many books (and teachers for that matter), that ignore this issue and will simply assume GBM.This is more of a teaching issue more than anything else. No one in the industry that I know of uses pure GBM for anything, but you start teaching with GBM because once you understand how that works, then you can introduce the models that people *do* use as corrections of the basic GBM models. QuoteThe assumption of zero trading costs is another often encountered example of where theory does not match practice. A fixed interest rate (static over time, as well as a flat yield curve) is another. My advice would be: be aware of the differences between theory and practice and how these influence theoretic results, and accept that theory oversimplifies reality.Theory is always oversimplifying reality, but the name of the game is to figure out how reality is being oversimplified and what the consequences of those simplifications are. Zero trading costs is actually a reasonable assumption in most industrial situations involving pricing because if you have to consider trading costs, you probably aren't going to do the trade. However, it does figure in when you do hedging. Fixed interest rates work if you have an option that is dated short enough so that interest rates aren't likely to fluctuate, but if you have a long dated 20 year option, then things will break down completely.