IMO, you are trying to quantify a incredibly complex, random situation into mathematical terms - it is impossible with the market. It's not like flipping a coin and calculating odds of heads or tails or even predicting odds of poker hands... My arguement? Because the mindset of 5 Million + traders, reacting to endlessly different news/stories and chart patterns/prices, emotions, etc. can never be duplicated from one moment to the next. Any mathematical probability that could be derived through a formula is endlessly skewed into irrelevance on a second-by-second basis. You have 50 million + variables in the minds of traders and price is just a reflection of all that combined. And it changes moment by moment. You have a scenario that cannot be quantified mathematically because the formula variables are changing every second or less. Open-minded to other opinions but I'm pretty solid on this one - this is why we have to "trade what we see". Paul

What we should try to prove is that a free market has a tendency to become more and more "random" as it becomes more efficient. This should be similar to data becoming more and more like white noise the better you compress it.

First, I really like how you set the problem up -- nice bit'o algebra there. Second, your results just verify the conclusion that the expectation at the outset of the trade is 0 (as you put it, m = 0). That p = 1 just shows that IF one reaches the stage of having 1 point of open profit, one can't expect to gain any further profit -- a random market does not display any momentum. The expected profit equals the open profit at all times (including when the trade was initiated with 0 open profit). If I have misunderstood your math, let me know. Cheers, Traden4Alpha

Re slapshot said "IMO, you are trying to quantify a incredibly complex, random situation into mathematical terms - it is impossible with the market." I definitely agree that the market is NOT a coin flip process for the very reasons that you state. Nonetheless, I see two good reasons for studying simplified mathematical random processes to gain a better understanding of the market. First, if I don't know what a random market would look like, how can I tell that the actual market is nonrandom. Although I agree that the market is composed of a large number of participants all behaving in very nonrandom ways, there are powerful deterministic forces that drive the market to apparent randomness. The never-ending quest of the 5 million traders to profit from any residual pattern in the market has the effect of removing those patterns from the market and making it look more random. By comparing actual market data to synthetic random market data I can assess whether what I see in the market is real or just an artifact of randomness. Second, randomness is a useful first-approximation to some fraction of the market's movement for two reasons. First, although the pattern of previous prices may help us guess the likely future behavior of the price action, there are no guarantees that the prices will evolve exactly as we have predicted. For example, news is an effectively random element that affects price action. Second, you accurately stated that the shear complexity of the system implies that nobody can model it in full detail. All of the models of the market are approximations that involve nebulous aggregate quantities of bulls and bears, hope and greed, buyers and sellers, etc. The unmodelled details will appear as random perturbations of any aggregate model. Therefore, news events and unmodelled details conspire to make future prices appear random. Although the markets are not purely random, understanding randomness does help us understand the random and nonrandom elements of market behavior. Cheers, Traden4Alpha

While the results are the same, if I understand correctly, actually you are stating the solution to another problem, which is E = Sum{e(i) } where e(i) is independent from each other with outcome of either +1 or â1 of equal probability Kay's problem is a restricted version of above, with restriction being that the e(i) process ends when E either reaches -1 or +3. Now we have concluded that such restriction has no impact on the expectation.

This is decidely true. BTW, in your sentence you have hit on something that few people consider...Try using one of the compression algorithms (or just use winzip,etc programs) on data of past, say, the first five years of the SP500, the next five years, etc. Now compare how well the data from the different years compress relative to each other! Repeat with smaller time frames, and even try including data that overlays with the past data, the way that a walk-forward program would do it, but instead the algorithm would be to compress the different time windows... Interesting!! nitro

This a modified binomial distribution, essentially same principles are used for pricing options via binomial model. Simple way to approach it, is just to set-up a simple binomial distribution tree and see what happens. It is obvious that from purely mathematical perspective the expectation to get from 0 to 3 is DEFINITELY not "0". AND if someone really wants to get fancy with math this problem can be also solved with some matrix algebra.. But that is for those Ph.D. types... Hehe, we just put a trade on see what happens.

the expectation is 0. one way to demonstrate this is to use binomial tree, and analyse all cases like mkmps suggested. however, there's much easier (yet equally stringent) way to demonstrate this, using symmetry: 1. if the market is random, then the expectancy of your system won't change if you replace every up-move of the market with down-move and vice versa. ie, expectancyFlipped = expectancyOriginal 2. on the other hand, by flipping the market moves, you'll negate the expectancy of your system. ie expectancyFlipped = -expectancyOriginal 3. from (1) and (2) it follows that expectancyFlipped = expectancyOriginal = 0. - jaan

Have you ever heard this: "It's all about risk/reward! If you are only right on 50% of your trades (which is like flipping a coin), but maintain W/L ratio of 2:1 then you'll be doing real well." Well the above posts prove that the notion above is nonsense (assuming random market), don't they? Because in a random market, with random entries, you might get 2:1 W/L easily but you'll only get 33% wins and the expectancy will be 0. So what are the conclusions coming out of this? It could be either of this: 1) The market is not random, instead it's autocorrelated. I have a strong suspicion that price time series are autocorrelated and at times are highly correlated (trends?). if they weren't then "letting profits run" would be impossible without reducing the Win% to a number which will effectively bring the expectancy back down to 0, rendering the whole affair moot. This scenario is one where we get advantage thru pushing up W/L. 2) The market is random but somehow people are still able to achieve positive expectancy. Man, this really sounds crazy. Is that even possible in theory? 3) The market is not random. But it's based on historical probabilities. Meaning it's possible to look back and pick out high % setups from the past and use same setups in the future with same % success rate. This scenario is where people take advantage of high win%. 4) The market is both 1) and 3) and you can have high win % and W/L >1. Believe it or not I've seen systems like that. But the most common profile of a winning system I've seen is 50/50, W/L=2, VERY common. So ... I got "the plate" full here on THX giving. Good discussion so far, looking forward to more. P.S. To kick it up another notch Emeril style here's a link that looks at price from a purely statistical perspective (can read all of it online although cumbersome http://www.iuniverse.com/bookstore/book_detail.asp?isbn=0595012078