Topic: Binomial Tree

Please help, I can't do any question

Section B $(41\%)$

A modified index amortizing swap $($henceforth$,$ the swap$)$ is a swap whose notional value

decreases over time depending on the interest rate scenario. Consider the index amortizing

swap with initial notional $N_0=100$ with the following characteristics:

i$.$ Maturity $i=2$

ii$.$ Amortization schedule:

a$.$ If $r_i<mn\; id="EditorElement\_21873">0</mn><mn\; id="EditorElement\_21874">.</mn><mn\; id="EditorElement\_21875">0</mn><mn\; id="EditorElement\_21876">2</mn>$ then $100\%$ reduction in notional;

b$.$ If $0\mathrm{.}02r_i<mn\; id="EditorElement\_21783">0</mn><mn\; id="EditorElement\_21784">.</mn><mn\; id="EditorElement\_21785">0</mn><mn\; id="EditorElement\_21786">4</mn>$ then $50\%$ reduction in notional;

c$.$ If $0\mathrm{.}04r_i<mn\; id="EditorElement\_21688">0</mn><mn\; id="EditorElement\_21689">.</mn><mn\; id="EditorElement\_21690">0</mn><mn\; id="EditorElement\_21691">6</mn>$ then $20\%$ reduction in notional;

d$.$ If $r_{i}0\mathrm{.}06$ then no reduction in notional;

iii. At $i=0$ no amortization takes place $($lockout period$).$

iv$.$ In Period i $(i=1,2),$ Firm A pays Bank B $c{N}_i$ with a fixed swap rate $c,$ while

Bank B pays Firm A $r_i{N}_i$ with a floating rate $r_i.N_i$ is the notional in Period i;

$$ is the time passage between two periods; $r_i$ is the $1-$period interest rate in Period i$.$

For example, if at time $i=1$ the interest rate went up to $4\mathrm{.}5\%(r_1=0\mathrm{.}045),$ then the notional

to apply to the current payment is not $N_1=100$ but $N_1=(1-20\%)=80$; if at time $i=2$

the interest rate went down to $3\mathrm{.}5\%(r_2=0\mathrm{.}035),$ then the notional $N_2=80$

$(1-50\%)=40$

Given that $r_{i,j}$ either increases to $r_{i,j}+m_i+\sqrt[\phantom{2}]{}$ or decreases to $r_{i,j}+m_i-\sqrt[\phantom{2}]{}$

in the next period, where $$ is the passage of time in years between two periods and

$\sqrt[\phantom{2}]{50}=7\mathrm{.}071,$ calculate the $$ implied by the above interest rate tree $($round to $2$

decimals$)$ and explain what $$ means in fixed income securities$.(2\%)$

Following Question B$-1,$ extend the above binomial tree of interest rates to $3$ steps

given $E(r_3)=5\mathrm{.}1\%.(2\%)$

What is the risk$-$neutral probability in period $i=0?($Round to $2$ decimals$)(4\%)$

What is the risk$-$neutral probability in period $i=1?($Round to $2$ decimals$)(5\%)$

What is the risk$-$neutral probability in period $i=2?($Round to $2$ decimals$)(6\%)$

What are the possible $N_1$ and $N_2$ for the swap? $(3\%)$

What is the fair swap rate $c,$ such that the value of the swap is $0$ at inception? $(6\%)$

What is the swap's spot rate duration in Period $i=1$ after the exchange of cashflow

when interest rate goes up$?(3\%)$

Describe the dynamic replication strategies of the swap from Firm A$\text{'}$s perspective

using the short$-$term bond that matures in the next period and the long$-$term bond that

matures in period $i=3(10\%)$