Can a 20 percent up or down walk return to start

Each day an asset's price is multiplied by either $1.2$ or $0.8$, each with probability $\tfrac12$, independently of other days. What is the probability that after $10$ days the price equals its starting value?

0. With $k$ up-days the price ratio is $(\tfrac65)^k(\tfrac45)^{10-k}=\dfrac{6^k\,4^{10-k}}{5^{10}}$. The numerator has no factor of $5$, so the ratio can never equal $1$ for any $k$; the event is impossible.