Problem $3$

For a two$-$period binomial model, you are given:

Each period is one year.

The current price for a non$-$dividend paying stock is $$20.$

$u=1\mathrm{.}3,$ where $u$ is one plus the rate of capital gain on the stock per period if the

stock price goes up$.$

$d=0\mathrm{.}7,$ where $d$ is one plus the rate of capital loss on the stock period if the

stock price goes down.

The risk$-$free interest rate is $5\%.$

$($a$)$ Calculate the price of an American call option on the stock with a strike price of

$$20$ which expires in $2$ years. How does it compare to the price of a European call

option?

$($b$)$ Calculate the price of an American put option on the stock with a strike price of

$$20$ which expires in $2$ years. How does it compare to the price of a European put

option?

$($c$)$ Check the put$-$call parity for the European options and American options $($sepa$-$

rately$).$ Comment on what you find.