Problem $13\mathrm{.}5.$A stock price is currently $$100.$ Over each of the next two six$-$month periods it is expected to goup by $10\%$ or down by $10\%.$ The risk$-$free interest rate is $8\%$ per annum with continuouscompounding. What is the value of a one$-$year European call option with a strike price of $$100?$

Problem $13\mathrm{.}13$A stock price is currently $$50.$ Over each of the next two three$-$month periods it is expected to goup by $6\%$ or down by $5\%.$ The risk$-$free interest rate is $5\%$ per annum with continuouscompounding. What is the value of a six$-$month European put option with a strike price of $$51?$Problem $15\mathrm{.}4.$Calculate the price of a three$-$month European put option on a non$-$dividend$-$paying stock with astrike price of $$50$ when the current stock price is $$50,$ the risk$-$free interest rate is $10\%$ perannum, and the volatility is $30\%$ per annum.Problem $15\mathrm{.}5.$What difference does it make to your calculations in Problem $15\mathrm{.}4$ if a dividend of $$1\mathrm{.}50$ isexpected in two months?Problem $15\mathrm{.}15.$Consider an American call option on a stock. The stock price is $$70,$ the time to maturity is eightmonths, the risk$-$free rate of interest is $10\%$ per annum, the exercise price is $$65,$ and thevolatility is $32\%.$ A dividend of $$1$ is expected after three months and again after six months.Show that it can never be optimal to exercise the option on either of the two dividend dates. UsePython code to calculate the price of the option.