Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 3 / 27J (Stochastic Financial Models)

At each nondegenerate binary node, no arbitrage requires the riskless gross return $R=1+r$ to lie strictly between the two stock returns. The three intervals are $(4/5,2)$ at time zero, $(6/5,3/2)$ at the upper node, and $(5/6,4/3)$ at the lower node. Their intersection givesAt $R=5/4$, the upper-branch risk-neutral probabilities are $(RS-S_d)/(S_u-S_d)$. They are $3/8$ initially, $1/6$ after the up move, and $5/6$ after the down move. All are strictly between zero and one, so the martingale measure is equivalent and unique. The four terminal probabilities in the PDF's order $45,36,16,10$ are $1/16,5/16,25/48,5/48$.

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.

At the upper time-one node, both exercise and continuation values are zero. At the lower node, immediate exercise gives three while continuation gives $2/3$, so exercise is strictly better. At time zero immediate exercise gives zero, whereas continuing givesThus continue at time zero and exercise at time one after a down move; at the upper node, exercise or continued holding is immaterial. One initial replicating portfolio for these optimal stopping values is $\Delta =-1/6$, $B=4$, costing $3/2$ and producing zero at stock 30 and three at stock 12.

European put-call parity givesFor the American call, there are no dividends and the riskless rate is positive, so early exercise sacrifices the benefit of deferring the strike. Directly, at stock 30 its continuation value is $(4/5)[(1/6)30+(5/6)21]=18$, exceeding its exercise value 15; at stock 12 it is $(4/5)(5/6)=2/3$, exceeding exercise value zero. At time zero continuation is $86/15>0$. Thus $C_A(0)=C_E(0)=86/15$, with no optimal early exercise. European parity must not be applied using the American put price $3/2$.