Consider the following binomial model which gives the price of a stock at different points in time:

The return in the up state (u) is 10%, the return in the down state (d) is -9.09%, and the risk-free interest rate (r) is 4%. We are interested in pricing American options that expire at time 2.

a) Find the risk-neutral probability (p) of an up state. Briefly interpret this measure.

b) At each node of the binomial tree compute the no-arbitrage price of a call option with an exercise price (X) of 95.

c) Derive the hedge ratio (h) between the stock and the call option at the nodes at time 0 and time 1. Briefly explain why h increases when the stock price goes up.

d) Repeat parts (b) and (c) for a put option with an exercise price of 105.

121 100 82.64+ Time 2 110 100 90.91 Time 0 Time 1