Question $2($Binomial model$)(20$ marks$)$

This question has two independent parts, $($a$)$ and $($b$).$

$($a$)$ Use the binomial tree in the following table to price a putable$/$callable bond with the following characteristics: a par value of $$100,9\mathrm{.}5\%$ coupon rate payable annually, $4$ years of maturity, callable at $$102\mathrm{.}5,$$$101\mathrm{.}5,$ and $$100$ in Year $1,$ Year $2,$ and Year $3,$ respectively, and putable at par starting in Year $1$ and thereafter. What is the value of the embedded put option? What is the value of embedded call option? Please show your calculations, including formulas used. $(10$ marks$)$

$\backslash$table$[[$Interest rate,$($in $\%),\backslash$table$[[$Interest$],[$rate$]],($in $\%)],[r_0,3\mathrm{.}5000,r_{2,LL},3\mathrm{.}7492],[r_{1,H},5\mathrm{.}9196,r_{3,HHH},12\mathrm{.}0003],[r_{1,L},3\mathrm{.}9680,r_{3,HHL},8\mathrm{.}0441],[r_{2,HH},8\mathrm{.}3440,r_{3,HLL},5\mathrm{.}3921],[r_{2,HL},5\mathrm{.}5931,r_{3,LLL},3\mathrm{.}6144]]$

$($b$)$ Consider a $9-$month zero coupon bond issued by company $xYZ$ with a face value of $$100.$ Suppose that it can be exchanged for $2$ shares of company XYZ$\text{'}$s stock at any time during the $9$ months. Assume also that it is callable for $110$ at any time. The initial stock price is $$50,$ its volatility is $35\%$ per annum, there are no dividends and risk$-$free rates for all maturities are $5\%.$ What is the value of this convertible bond? $(10$ marks$)$

$u=e^{\sqrt[\phantom{2}]{t}},d=\frac{1}{u},p_u=\frac{e^{rt}-d}{u-d},p_d=1-p_u,t=0\mathrm{.}25$ or $3$ time steps