Consider a stock worth $$25$ that can go up or down by $15$ percent per period. The risk$-$free rate is $10$ percent. Use one binomial period. Assume the stock pays no dividends.

a$.$ Determine the two possible stock prices for the next period.

b$.$ Determine the intrinsic values at expiration of a European call option with an exercise price of $$25.$

c$.$ Find the value of the option today.

d$.$ Construct a hedge by combining a position in stock with a position in the call. Show that the return on the hedge is the risk$-$ free rate regard$-$less of the outcome, assuming that the call sells for the value you obtained in part $c.$

e$.$ Determine the rate of return from a riskless hedge if the call is selling for $$3\mathrm{.}50$ when the hedge is initiated.

Consider a two$-$period and two$-$state world. Let the current stock price be $45$ and the riskfree rate be $5$ percent. Each period the stock price can go either up by $10$ percent or down by $10$ percent. A call option expiring at the end of the second period has an exercise price of $40.$

a$.$ Find the stock price sequence.

b$.$ Determine the possible prices of the call at expiration.

c$.$ Find the possible prices of the call at the end of the first period.

d$.$ What is the current price of the call?

Find the value of an American put option using the binomial option pricing model with $n=2$ periods. The parameters are $S=62,x=70,r=0\mathrm{.}08,u=1\mathrm{.}10,$ and $d=0\mathrm{.}95.$ There are no dividends.

How would your answer change for a European put option?

$4.$ Use the Excel spreadsheet BlackScholesMertonBinomial$10$e$.$xlsm and determine the value of a European call option and a European put option on a stock currently priced at $100$ where the risk$-$free rate is $5$ percent $($compounded annually$),$ the exercise price is $100,$ the volatility is $30$ percent, the option expires in one year and there are no dividends on the stock. Let the number of binomial periods be $25.$ Verify that put$-$call parity holds.