Bank of Nowhereland has just issued a market$-$linked CD that is linked to the performance of S&P

$500$ index. The CD has the following payout structure:

$$ it will mature in $5$ years;

$$ if the percentage change in S&P $500$ index over the $5-$year period is between $0\%$ and $20\%,$ it

pays $10\%$ net return;

$$ if the percentage change in S&P $500$ index over the $5-$year period is more than $20\%,$ it

participates in $80\%$ of the S&P $500$ index appreciation, for each percentage return of S&P

$500$ that surpasses $20\%.$

$7$

$$ if S&P $500$ index depreciates over the $5-$year period, it has a $50\%$ participation, starting from

a net return of $10\%.$

The structure is illustrated below, where rCD is the net return in percentage of the CD and rS is

the net return in percentage of the index.

$($a$)[2$ points$]$ Determine the slopes of the red and green lines respectively.

$($b$)[2$ points$]$ Draw the profit diagram of the CD$,$ assuming a principal of $L has been invested and

the current index level is S$0$

$$ y$-$axis: dollar return of the CD

$$ x$-$axis: ST $($index level in T$=5$ years$)$

$($c$)[2$ points$]$ Draw the payoff diagram of the CD$,$ assuming a principal of $L has been invested

and the current index level is S$0$

$$ y$-$axis: dollar return of the CD

$$ x$-$axis: ST $($index level in T$=5$ years$)$

$($d$)[3$ points$]$ Based on the result in $($c$),$ decompose the payoff of the CD into $3$ parts

$$ part $1$: a long position in x$1$ units of risk$-$free zero$-$coupon bonds with face value of L

$$ part $2$: a short position in x$2$ units of European puts with a strike price of y$2$

$$ part $3$: a long position in x$3$ units of European calls with a strike price of y$3$

Express x$1,$x$2,$ x$3,$ y$2,$ y$3$ precisely with L and S$0,$ assuming a principal of $L has been invested and

the current index level is S$0.$

$8$

$($e$)[6$ points$]$ The current index level is unknown, the risk$-$free rate is r $=5\%$ per annum in continuous compounding, volatility of the index is $15\%$ per annum. If the principal is $$10,000.$ Determine

the value of the $5-$year market$-$linked CD according to an option$-$based model and show that the

value is irrelevant to the current value of the index.