Let $S_t$ be the price of a stock at time $t,$ and let $S_0=$$$100$ be its price today. Suppose

that its up$-$factor and down$-$factor after one month are $u=1\mathrm{.}2$ and $d=0\mathrm{.}8,$ and the

probabilities of the stock price's up and down movements are $p_u=0\mathrm{.}6$ and $p_d=0\mathrm{.}4.$

Assume that the stock pays no dividends, and the rate of a risk$-$free asset is $r=24\%$

compounded monthly.

$($i$)$ Let $K_1$ and $K_2$ such that $0K_1{K}_2$ and $K_1+K_2=200.$ Let $P$ be a portfolio

of derivatives with the stock as its underlying asset, whose whose maturity date

is in one month, and whose gross payout is max$(S_1-K_1,0)-$max$(S_1-K_2,0).$

Using the binomial model, find the price of $P$ in terms of $K_1.$

$($ii$)$ A certain call option on this stock has an expiration date $2$ months from now and

strike price of $$118\mathrm{.}5.$ Determine the theoretical price of this call to $4$ significant

figures.

$($iii$)$ If $p_u=0\mathrm{.}8$ and $p_d=0\mathrm{.}2,$ what are the prices for the derivatives in $($i$)$ and $($ii$)?$

And why?