We have a stock that is currently trading for $100$, and we know that it randomly goes up by $1%$ or down by $1%$ in a given trading day with equal probability. What is the expected value of the stock after 1 year? Assume 1 year = $256$ trading days. I was able to solve the problem, but the answer seems either wrong or unintuitive to me.
This is how I solved it: We must find the expected value of the stock so we use the standard formula for expected value which is the probability of an event occurring multiplied by the event's value, summed over all events.
The probability that the stock experiences X UP days and 256-X DOWN days is ${256 \choose X}0.5^X0.5^{256-X}$. If the stock experiences X UP days and 256-X DOWN days, it's final price can be written as $100*1.01^X0.99^{256-X}$. When I multiply the two expressions and do a summation from $X=0$ to $X=256$ (encapsulating all states), it spits out that the expected value is exactly 100.
Why? If a stock goes up $1%$ and then down $1%$, it is $1.01 * 0.99$ its original value, which is $0.9999$ time its original value. Similarly, if the stock goes down first and then up, it is also the same. Shouldn't we then expect the value of the stock to be below $100$ assuming a symmetric distribution?