Exercise: Variance and Drift of a Biased Random Walk

Prerequisites: Random Walk, Expectation and Variance

Problem

A trader models intraday tick moves of a futures contract as an asymmetric random walk. Each tick the price moves up by $1 with probability $p = 0.51$ or down by $1 with probability $q = 0.49$. Let $S_n$ be the net price change after $n$ ticks, with $S_0 = 0$.

1. Compute $\mathbb{E}[X_i]$ and $\operatorname{Var}(X_i)$ for a single tick.
2. Give closed-form expressions for $\mathbb{E}[S_n]$ and $\operatorname{Var}(S_n)$.
3. Evaluate both at $n = 10{,}000$ (roughly one trading day of tick data).
4. Compute the signal-to-noise ratio $\mathbb{E}[S_n] / \sqrt{\operatorname{Var}(S_n)}$ at $n = 10{,}000$. How many ticks would the trader need to observe before the expected move is one standard deviation above zero?

Hint

Use linearity of expectation and independence-of-increments for variance. For part 4, set $\mathbb{E}[S_n] = \sqrt{\operatorname{Var}(S_n)}$ and solve for $n$.