Math

A simplistic approach to stock movements

Suppose a stock is currently worth $1, has a probability P of going up 1% and a probability (1-P) of going down 1% (for the day).

The expectation of the stock after two days is:
  P^2(1.01)^2 + 2P(1 - P)(1.01)(.99) + (1 - P)^2(.99)^2
= (P(1.01) + (1 - P)(.99))^2
= (1.01P + .99 - .99P)^2
= (.02P + .99)^2

If P = 1/2, the expectation is 1 (no change in money). If P = 0.51 = 51%, the expectation is $1.00040004.

Repeat the math with up 3% and down 3% and you get
= (P(1.03) + (1 - P)(.97))^2
= (.06P + .97)^2

If P = 1/2, the expectation is 1 (no change in money). If P = 0.51 = 51%, the expectation is $1.00120036. Note that this is MORE than 3 times the above.