Binomial call with moves to 90 or 110

With zero interest rates, a stock now at $100$ will be worth either $110$ (real-world probability $p$) or $90$ after one period. What is a call struck at $100$ worth?

$5$, whatever $p$ is. The risk-neutral probability of the up move satisfies $110q+90(1-q)=100$, so $q=\tfrac12$, and the call pays $10$ in the up state: $\tfrac12\times10=5$. Equivalently, half a share minus $45$ in cash replicates it. The real probability $p$ never enters.