Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 29L (Stochastic Financial Models)

Under a risk-neutral measure,Writing $q=Q({\xi }_n=1+b)$ givesEquivalence requires $0<q<1$, hence $a<r<b$. At every node the risk-neutral condition determines this same conditional probability $q$, so the multipliers remain independent and identically distributed under $Q$.

LetRisk-neutral valuation or direct backward induction givesThusSubstitution into the recursion verifies the terminal condition and the induction step.

At a node with time $n-1$ price $s$, choose the stock holdingThe two possible values of the stock position differ by exactly the difference between the two continuation claims. Choose the bank holding so that total wealth is $V(n-1,s)$. The pricing recursion makes its next value equal to the appropriate continuation value in both states. Backward induction from $V(N,s)=g(s)$ therefore replicates $g(S_N)$ from initial capital $V(0,S_0)$.

Before the barrier has been hit, the two next-step continuation values areThe replicating stock holding on $(n-1,n]$ is thereforeThe bank holding is chosen from current wealth $U(n-1,s)$ as before. If the barrier has already been hit, the claim and both holdings are zero. Backward induction proves replication of the up-and-out payout.