Why options need a price model: the one-period binomial setup

In short

• Forward commitments can be valued by replication alone, without assuming anything about the underlying's future price, because their payoff is symmetric.
• Options and other contingent claims have asymmetric payoffs, so their value depends on which future prices are possible: we need a model.
• In the one-period binomial model the underlying moves from $S_0$ either up to $S_1^u = R^u S_0$ or down to $S_1^d = R^d S_0$, with $R^u > 1 > R^d$.
• q is the actual probability of the up move and 1 − q of the down move. Surprisingly, q is not needed to value the option.
• The gap between $S_1^u$ and $S_1^d$ stands for the underlying's volatility, a key driver of option value.