= Solution
The <backward option pricing> recursion is
$$
V(n-1,s)
=\frac{qV(n,s(1+b))+(1-q)V(n,s(1+a))}{1+r},
$$
with terminal value $V(N,s)=g(s)$. At a node with $S_{n-1}=s$, define the <replicating portfolio in a binomial market>[replicating] stock holding by
$$
\boxed{
\theta_n
=\frac{V(n,s(1+b))-V(n,s(1+a))}
{s(b-a)}}.
$$
This is a <predictable process> because it depends only on information available at time $n-1$.
Indeed, if $X_{n-1}=V(n-1,s)$, substituting the risk-neutral formula for $q$ into the self-financing update shows separately in the up and down states that
$$
X_n=V(n,S_n).
$$
An induction over $n$ therefore gives $X_N=V(N,S_N)=g(S_N)$. The required initial capital is
$$
\boxed{x=V(0,S_0)=(1+r)^{-N}\mathbb E_Q[g(S_N)]}.
$$
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