a cap of $\backslash (4\mathrm{.}20\backslash \%\backslash ).$ This floater has a maturity of $3$ years and a par value of $\backslash (\backslash$$ $100\backslash ).$ The coupon rate $($paid annually$)$ on this floater is the $1-$year forward rate flat $($i$.$e$.,$ no margin over the reference rate, which is the $1-$year forward rate$).$ Remember that for floaters, the coupon rate is set at the beginning of the year but paid at the end of the year $($i$.$e$.,$ the beginning of the next year$).$ Finally, what is the value of the capped floater $($option$)?(10$ marks$)$

$($b$)$ We may parallel$-$shift the on$-$the$-$run yield curve by either $10$ bps or $-10$ bps $.$ With a shift of $10$ bps $,$ the resulted binomial tree is in Table I below. With a shift of $-10$ bps $,$ the resulted binomial tree is in Table II below. The bond is $3-$year $\backslash$$$100$ par value coupon rate of $\backslash (7\backslash \%\backslash )$ payable annually callable at $\backslash (\backslash$$ $100\mathrm{.}5\backslash )$ and at par in Year $1$ and Year $2$ from now, respectively. The OAS on this bond is $75$ bps $.$ The bond price before any shift is $\backslash (\backslash$$ $100\mathrm{.}25\backslash ).$ Calculate the effective duration and effective convexity of this bond. $(10$ marks$)$ Note: $\backslash (\backslash$Delta $\backslash )$ Curve $\backslash ((\backslash$Delta $\backslash$mathrm$\{$y$\})=10\backslash$mathrm$\{$bps$\}=0\mathrm{.}001\backslash ).$ Please show your calculations with binomial trees.

Table $1$

Table $2$