Suppose each day for $365$ days, we flip a coin. If it lands heads, I get \$10. If it lands tails, I lose \$10. What is the expected value of the absolute value of the amount of money I have (or owe) after the $365$ days?
For 1 and 2 days, the answer works out to be 10. Is this always true? We may assume the CLT, but I don't know how to proceed.
On
Hint: If you assume the distribution of results is normal, it has a mean of $0$ and standard deviation of $\sigma=\sqrt {\frac 14\cdot 365}$. Your result is then $2\int_0^\infty np(n) dn$ because the absolute value reflects the density below zero to above zero. I believe it will scale with the square root of the number of trades.
On
Let $X_{n}$ be the amount of money you win after $n$ days. We know that $X_{n}$ follows a shifted, scaled binomial distribution (since it's basically just an adjusted count of the number of heads flipped), so we can approximate your winnings after a year as $X_{365} \sim \mathcal{N}(0, 100 \times 365 / 4) = \mathcal{N}(0, 9125)$ based on the variance of a binomial.
\begin{eqnarray} E(|X_{365}|) & = & \int_x {|x| \cdot p(x) dx} \mbox{ where $x$ ranges over all possible amounts} \\ & = & \int_{x<0} {-x \cdot p(x) dx} + \int_{x \ge 0} {x \cdot p(x) dx} \\ & = & \int_{x \ge 0} x \cdot \left(p(x)+p(-x) \right) dx \\ & = & \int_{x \ge 0} 2x \cdot p(x) dx \mbox{ since p(x) is symmetrical} \\ & = & 2 \int_0^\infty x\cdot \frac{1}{\sqrt{9125}\sqrt{2\pi}}e^{-\frac{x^2}{2\times 9125}}dx \\ & = & \frac{2}{\sqrt{9125\times 2\pi}}\times\frac{1}{9125} \mbox{ left as an exercise}\\ & \approx & 9.15 \end{eqnarray}
In particular, noting that both of the 9125s come from the fact that after $n$ days, $\frac{X_n+n/2}{10} \sim Bin(n, 0.5)$ so that $E(X_n) = 0$ and $Var(X_n) = 100 n/4$, the expected value of the absolute value of your winnings/losses after $n$ days is given by $E(|X_n|) = \frac{2}{n^{3/2}\sqrt{2\pi}}$
What is the starting amount of cash you enter the game with? Can you run bankrupt? If 0 and no, then the final sum is also zero.
It is true for expectation of your cash position after any number of days (1 and 2 days including) as the expectation does not depend on number of iterations of this game.