Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 27J (Stochastic Financial Models)

At the node with stock price thirty, both possible next prices exceed thirty-three, the value of thirty invested at gross rate $11/10$. Take no position at time zero. If the first move is good, borrow thirty at time one and buy one share; if it is bad, do nothing. At time two sell the share for forty-five or thirty-six and repay thirty-three, leaving twelve or three. The strategy has zero initial cost, no possible loss, and positive profit on the good first move, so it is an arbitrage.

At gross rate $R=5/4$, the conditional up probabilities of the equivalent martingale measure areAll lie strictly between zero and one. The four terminal path probabilities are $1/16,5/16,25/48,5/48$ in the order good-good, good-bad, bad-good, bad-bad.

The terminal European put option payoffs are zero at forty-five, thirty-six and sixteen, and five at ten. At the good first node its value is zero. At the bad node its continuation value is $(4/5)[(5/6)0+(1/6)5]=2/3$, below the exercise payoff three, so optimal exercise occurs there. At time zero the continuation value is $(4/5)[(3/8)0+(5/8)3]=\boxed{3/2}$, exceeding the immediate payoff zero.

To hedge one sold American put option, short $1/6$ of a share and put four in the bank at time zero. The portfolio costs $-15/6+4=3/2$, exactly the premium. At time one the bank holds five. After a good move the short position costs five to close, exhausting the portfolio and liability. After a bad move it costs two to close, leaving three for the exercised put. Thus after two bad periods the rational buyer already exercised at time one, and the hedge pays three and closes then; there is no remaining time-two obligation. Physical delivery gives the same accounting by immediately selling the stock received on exercise.

If the first move is bad but the buyer misses exercise, the remaining European claim is worth $2/3$, whereas the existing hedge is worth three. At that node rehedge the continuation claim: its stock holding is $(0-5)/(16-10)=-5/6$ and its bank holding is $2/3+(5/6)12=32/3$.

Starting from the original short $1/6$ and bank balance five, short another $2/3$ share for eight, giving bank cash thirteen. Retain $32/3$ for the new hedge and withdraw $\boxed{7/3}$ as guaranteed time-one profit. At time two the retained bank grows to $40/3$. If price is sixteen, covering the short costs $40/3$ and the put expires worthless. If price is ten, covering costs $25/3$ and the remaining five pays the put. Thus the profit does not depend on the second move. Investing the withdrawn profit at the riskless rate makes it $35/12$ at time two.