I want to get some comments on the following: http://sambian.wordpress.com/2010/02/07/29/ To me, it is the same as saying: "Buy and hold; the stock market always goes up in the long run."

It's bollox... I didn't get any further than this glaring fallacy right here: "Letâs assume that we donât know anything about the future price movements. This means that for us an increase of 100% in eur/usd (A) is as likely as an increase of 100% in usd/eur (B). But B is the same as a 50% decrease in eur/usd. So an increase of 100% in eur/usd is as likely as a decrease of 50% in eur/usd. And in the same logic a 100% increase in usd/eur is as likely as a 50% decrease in usd/eur." If only we could all define outcomes for our binomial trees at will, with as free a hand as this guy has, we'd all be squillionaires in no time...

Hello, MathAndLogic, I am the author of the article to which you posted a link. Thank you for the interest. That's not what I say. My experience until now is that very very few people understood what I have written. For some of them the reason might be unwillingness to read the whole article and spend some time to understand it. But I suppose you've read it and thought about it. So I assume that I didn't write it clear enough, but unfortunately I have no idea how to make it more understandable. This is why here I give you a link to a spreadsheet which might make things more clear - http://rapidshare.com/files/372675452/eur-usd_example.xlsx.html In the example the starting price of eur/usd is 1. The price movements are random - either eur/usd goes up 100%, or usd/eur goes up 100%, with equal probability. We have two portfolios. In portfolio 1 our goal is to maximize our dollar returns, while in portfolio 2 our goal is to maximize our euro returns. Both portfolios start from 1000. In portfolio 1 our strategy is to buy euros with half of our dollars and make readjustments after every movement. In portfolio 2 our strategy is to buy dollars with half of our euros and make readjustments after every movement. So for example Portfolio 1 has $1000, we buy 500 euro and keep $500. Eur/usd goes to 2, and now we have $1500. We readjust our portfolio by again buying euros with half of our money - now we buy 375 euros and keep $750. Afterwards the price goes to 1 and we have $1125 dollars. In the same time the euro portfolio has 1125 euros.

You're taking two independent outcomes A and B that don't "span" the probability space and pretend they do... That's why, I think, you're getting positive expected value no matter what happens. If you were to try outcomes A) EURUSD rises 100%; and B) EURUSD falls 100%, and assume those are equally probable, you will get something more sensible. EDIT: Actually, if you want a technical summary, I think your problem is that your choice of numeraire is inconsistent.

Now that's another thing which is very important. If eur/usd falls 100%, which means falling to 0, this is the same as usd/eur going to infinity. Such things are impossible for people who believe in random walks. You are right, if eur/usd rising by 100% is equally probable as eur/usd falling by 100%, there will be no positive expected value. Let's assume that this is the case. Then A) 200% increase in eur/usd should be equally probable as B) 200% decrease. This means that if eur/usd is now trading at 1, it is equally probable A)eur/usd = 3 or B)eur/usd = -1. Do you think this is true? If not, then what is the equal probability of eur/usd increasing 200%?

Aha, now we're getting into the issues with the zero bound, but you should realize that it's a red herring. After all, the real argument you're making is not for +/- 100% or 200% moves, but rather for +/- epsilon, where epsilon tends to 0. You shouldn't confound the issues of the zero bound and the constraints it imposes on the resulting distribution with your basic binomial tree argument.

they are right. its because futures are based on real ticks. but random walk is based on multiples and fractions. futures rise of 6E to from $1.5 to $3 is 15000 pips drop from $1.5 to $0 is 15000 pips. On the other hand random walk implies rise to $3/1 is same as drop to 1/$3 which is $0.33. but the rise is then 15000 pips but the drop 11700 pips. so the money you can make is because of how a future operates.