$1.$ A European put option written on stock has strike price K $=$ $$27$

and expires at time t $=1.$ At the current time t $=0$ the underlying

stock has price S$(0)=$ $$26$ and at expiry the price will be either

S$(1,)=$ $$30$ or S$(1,)=$ $$24.$ The interest rate over one timestep

is r $=5\%.$

$($a$)$ Show that there is no arbitrage opportunity.

$($b$)$ Construct a replicating portfolio for this put option. Calculate

H$0$ and H$1($as defined initial value of replicating portfolio V$(0)=$H$0+$H$1$S$(0)$ for this replicating portfolio and thus find the value of the put at the current time P$(0).$

$($c$)$ By considering the values of H$0$ and H$1$ you have obtained,

explain what your replicating portfolio contains.

$($d$)$ Verify your solution for P$(0)$ by calculating the risk neutral

probabilities and using the general pricing formula.

$($e$)$ A European call based on the same underlying stock, with the

same strike price and expiry $($at t $=1)$ is available at t $=0$ for

a premium of C$(0)=$ $$1\mathrm{.}64.$ Show that this call has not been

correctly $($rationally$)$ priced.

$($f$)$ Explain, by constructing a cash flow table, how you could

take advantage of the arbitrage opportunity presented by the

mispriced call described above and make a guaranteed profit.