Alternating Up and Down Moves

Each fall multiplies the price by $0.99$ and each rise by $1.01$. A fall followed by a rise therefore multiplies it by

Let $P_k$ be the price after $k$ moves. Then

Both subsequences tend to $0$ geometrically, since $0<0.9999<1$, so the whole sequence does too. The price never actually reaches zero, but it falls below any positive level eventually. For example, it halves after about $\ln 2/10^{-4}\approx 6931$ pairs of moves.

Key idea. Equal percentage moves up and down do not cancel. Their product is $1-r^2<1$, so alternating gains and losses of the same size steadily erode the value. This effect is called volatility drag.

Generalisation. With alternating moves of $\pm r$ for $0<r<1$, the price after $2n$ moves is $P_0(1-r^2)^n\to 0$.