Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 3 / 27J / ii

The European put has terminal values $0,0,0,5$. Discounted backward expectation gives value zero at stock 30 and $(4/5)(5/6\cdot 0+1/6\cdot 5)=2/3$ at stock 12. ThereforeFor replication, denote by $B$ the money currently in the bank, which grows to $RB$ next period. At time zero hold $\Delta =-1/27$ shares and $B=8/9$, costing $15\Delta +B=1/3$. At stock 30 rebalance to zero shares and zero bank balance. At stock 12 rebalance to $\Delta =-5/6$, $B=32/3$. This costs $12(-5/6)+32/3=2/3$ and gives terminal wealth zero at 16 and five at 10. The initial portfolio produces exactly the two node values, so these rebalancings are self-financing.