Let $S_n = \xi_1 + \dots + \xi_n$ be asimetric random walk such that $P(\xi_i = 1) = p > \frac{1}{2}$ and $P(\xi_i = -1) = q $. Let $\sigma^2 =1-(p-q)^2$ and let $X_n=(S_n-n-(p-q)n)^2 - \sigma^2n $
($X_n$ is martingale). Prove that $\text{Var} \tau = \frac{1 − (p − q)^2}{(p-q)^3} $
where $\tau = \min\{n:S_n=1\}$
I would use fact that $\text{Var}S_\tau = \sigma^2 E \tau + \mu^2 \text{Var} \tau$ where $\mu = E \xi_i$ and $\sigma^2 = \text{Var} \xi_i$. But I don't have $\text{Var}S_\tau$ and $E\tau$ to find $\text{Var} \tau$. Maybe I should use fact that $X_n$ is martingale but how?
I will grateful for yours help.