$1.$ The price of a non$-$dividend$-$paying stock is $$19$ and the price of a $3-$month

European call option on the stock with a strike price of $$20$ is $$1.$ The

risk$-$free rate is $4\%$ per annum. What is the price of a $3-$month European

put option with a strike price of $$20?$ Describe an arbitrage strategy if

the price of the put option is $$2.$

$2.$ The price of an American call on a non$-$dividend$-$paying stock is $$4.$ The

stock price is $$31,$ the strike price is $$30,$ and the expiration date is in $3$

months. The risk$-$free interest rate is $8\%.$ Derive upper and lower bounds

for the price of an American put on the same stock with the same strike

price and expiration date.

$3.$ A stock price is currently $$40.$ It is known that at the end of $3$ months it

will be either $$45$ or $$35.$ The risk$-$free rate of interest with quarterly compounding is $8\%$ per annum. Calculate the value of a $3-$month European

call and put options on the stock with stike price of $$40.$

$4.$ A price of an asset is currently S$0.$ At the end of T months it will be either

S$0$u or S$0$d$.$ The risk$-$free rate of interest with continuous compounding

is r$\%$ per annum. Find the $$ used for risk$-$free portfolio. Calculate the

value of a T$-$month European call and put options on the stock with strike

price of $K