Exercise $5\mathrm{.}7($Hedging a short position in the perpetual American

put$).$ Suppose you have sold the perpetual American put of Section $5\mathrm{.}4$ and

are hedging the short position in this put. Suppuse that at the current time

the stock price is $s$ and the value of your hedging portfolio is $v(s).$ Your hedge

is to first consume the amount

$c(s)=u(s)-\frac{4}{5}[\frac{1}{2}u(2s)+v(\frac{s}{2})]$

and then take a position

$(s)=\frac{v(2s)-v(\frac{s}{2})}{2s-\frac{s}{2}}$

in the stock. $($See Theorcm $4\mathrm{.}2\mathrm{.}2$ of Chapter $4.$ The processes $C_n$ and ${}_n$ in

that theorm arc obtained by replacing the dummy variable $s$ by the stock

pricc $S_n$ in $(5\mathrm{.}7\mathrm{.}3)$ and $(5\mathrm{.}7\mathrm{.}4)$; i$.$e$.,C_{v_2}=c(S_{,})$and ${}_n=(S_n).)$ If you hedge

this way, then regardless of whether the stock goes up or down on the next

step, the value of your hedging portfolio should agree with the value of the

perpetual American put.

$($i$)$ Compute $c(s)$ when $s=2^j$ for the three cases $j0,j=1,$ and $j2.$

$($ii$)$ Compute $(s)$ when $s=2^j$ for the three cases $j0,j=1,$ and $j2.$

$($iii$)$ Verify in each of the three cases $s=2^j$ for $j0,j=1,$ and $j2$ that

the hedge works $($i$.$e$.,$ regardless of whether the stock goes up or down,

the value of your hedging portfolio at the next time is equal to the value

of the perpetual American put at that time$).$