[mykey1961]

[ QUOTE ]
But this doesn't mean it has no positive expectation. Let's take a simple example: We play the exchange rate game for 4 periods. The initial exchange rate is 1 and every period it halves or doubles with probability 1/2. We buy 100€ before the first period for $100 and whenever the exchange rate hits 2 we sell and quit the game. If it hasn't reached 2 in the 4th period we sell for whatever the exchange rate is.

The probability of the exchange rate hitting 2 before the 4th period is (1/2+1/8). With prob. 1/2 it hits it in the first period (up), with prob. 1/8 in the third (down, up, up). The remaining cases are 1 (p=1/8), 1/4 (p=3/16), 1/16 p=(1/16) in the 4th period.

So the EV for this game is

5/8 * 200$ + 1/8 * 100$ + 3/16 * 25$ + 1/16 * 6$ = 142$.

Weird ... [img]/images/graemlins/smile.gif[/img]

[/ QUOTE ]

Why do you find this weird?

"The initial exchange rate is 1 and every period it halves or doubles with probability 1/2."

If you start with equivilant of $100, and the exchange rate halves, you have the equivilent of $50, if it doubles, you have the equivilant of $200.

You in effect are risking $50 to win $100. If you win with probability 1/2 then you should have a +EV.

If however the probability of the exchange rate doubling was only 1/3, then your EV drops to 0.