The current level of the underlying is S = 100, and the size of up{ and down{moves are

u = 1:10 and d = 0:90 per period, respectively. The risk-free interest rate will be specied in

each of the sub{questions. The rates are given as continuously compounded and per period

in the tree (i.e., you can think of each period as 1 year).

1. When there are no dividends, the early exercise of an American put depends on a trade{

o between insurance value (which comes from volatility) and time value (a function

of interest rates). Thus, for example, for a given level of volatility, early exercise of the

put becomes more likely if interest rates are higher. This question provides a numerical

illustration.

Consider an American put option expiring after two periods with a strike price of 95.

(a) First, consider a "low"interest rate of r = 1:98%. Show that early exercise of the

American put is never optimal in this case. What is the price of the put today?

(b) Next, consider a "high" interest rate of r = 4:879%. Show that it now becomes

optimal to exercise the put early in some circumstances. What is the early exer-

cise premium in this case (the dierence between the prices of an American and

European put today)?