I was a bit hesitant to post this question because it seems so basic...but I wasn't able to figure it out on my own.
Say we setup a one-step binomial tree with $S_0=100$, $S_u=110$ and $S_d=90$, where $S_u$ and $S_d$ are the up and down possibilities for the stock price at time $T=1$. Let $K=100$ be the strike price of a call, and $r=10%$ be the continuously compounded risk-free interest rate.
Using a replicating portfolio (with some quantity $\Delta$ of the stock and borrowed money), I find the price of the call to be $c_0 = 9.28\$$ ($\Delta=0.5$).
Now I understand that I don't need to know what the real-world probabilities are (of $S_u$ and $S_d$), since the replicating portfolio...replicates the option payoff no matter the outcome.
But just for fun, let's say $Pr(S_1=S_u)=1\%$ and $Pr(S_1=S_d)=99\%$, in which case, on average, the call at time 1 would be worth 0.01*10 = 0.1$.
How would anyone be willing to pay 9.28$ for that ?
I'm pretty sure I'm missing something very basic, I hope someone can explain what it is.