Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 29K / c / Solution

The backward option pricing recursion iswith terminal value $V(N,s)=g(s)$. At a node with $S_{n-1}=s$, define the replicating stock holding byThis 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 thatAn induction over $n$ therefore gives $X_N=V(N,S_N)=g(S_N)$. The required initial capital is