#$5$ Non$-$Standard Interest Rate Swaps $-$ Roller Coaster Interest Rate Swaps

At the current date $t_0,$ assume the existence of a term structure of simple and continuously compounded default$-$

free spot interest rates, denoted by $\{R(t_0,t){\}}_t{t}_0$ and $\{r(t_0,t){\}}_t{t}_0$ respectively, with associated capitalisation

and discount factors $\{C(t_0,t){\}}_t{t}_0$ and $\{Z(t_0,t){\}}_t{t}_0$ respectively.

Consider a Roller Coaster Interest Rate Swap $($RCIRS$)$ with fixed simple swap rate $K,$ initiated at date $t_0,$

expiring at date $t_n,$ with reset dates $\{t_0,t_1,t_2,$dots,$t_{n-1}\},$ referencing floating rates $\{R(t_{i-1},t_i)$;$1in\},$

varying nominals $\{N_i$;$1in\}$ and with payment dates $\{t_1,t_2,t_3,$dots,$t_n\},$ i$.$e$.$ a different nominal amount

applies to each payment period. At each $t_{i}in\{t_1,t_2,t_3,$dots,$t_n\},$ the following cash flows occur:

floating cash flow: $N_{i}R(t_{i-1},t_i){}_i,$

fixed cash flow: $N_{i}K{}_i,$

where ${}_i=\frac{t_i-t_{i-1}}{365}$ for iin$\{1,2,$dots,$n\}.$

$($i$)$ Provide one real$-$world scenario where a RCIRS may be used, and preferred to a standard IRS.

$($ii$)$ Derive an expression for the initial fair value of the RCIRS at date $t_0.$ Use no$-$arbitrage arguments to

justify the "replacement" of future floating rates with their forward counterparts, where necessary.

$($iii$)$ Derive an expression for the fair RCIRS rate $K$ within the primary market context.

$($iv$)$ Derive a strategy to replicate the RCIRS with standard and forward$-$starting IRSs.

The following NACC swap zero curve prevails in the South African market on $t_0=13-$Jun$-2014$:

$r(t_0,t)=-0\mathrm{.}0001(\frac{t-t_0}{365}{)}^2+0\mathrm{.}0025(\frac{t-t_0}{365})+0\mathrm{.}055,$

for $t{t}_0.$ This curve gives you access to the complete set discount factors at the current date $t_0.$

$($v$)$ Compute the fair RCIRS rate for a roller coaster payer swap with semi$-$annual payments in arrears, over

the next $2$ years. The initial notional is $R80,000,$ the second exchange of payments is based on a notional

of $R100,000$ and the final two exchange of payments is based on a notional of $R40,000.$ The floating leg

references $6-$month JIBAR $($simple$,$ ACT$/365).$ Generate all reset and payment dates according to standard

South African swap conventions.