To answer this question, please start by builiding and calibrating a $10-$period Black$-$Derman$-$Toy model for the

short$-$rate, $r_{i,j}.$ You may assume that the term$-$structure of interest rates observed in the market place is:

Period $1,2,3,4,5,6,7,8,9,10$

As in the video modules, these interest rates assume per$-$period compounding. For example, the market$-$price of a

zero$-$coupon bond that matures in period $6$ is $Z_0^6=\frac{100}{(1+\mathrm{.}035{)}^6}=81\mathrm{.}35$ assuming a face value of $100.$

Assume $b=0\mathrm{.}05$ is a constant for all $i$ in the BDT model as we assumed in the video lectures. Calibrate the $a_i$

parameters so that the model term$-$structure matches the market term$-$structure. Be sure that the final error

returned by Solver is at most ${10}^{-8}.($This can be achieved by rerunning Solver multiple times if necessary, starting

each time with the solution from the previous call to Solver.$)$

Once your model has been calibrated, compute the price of a payer swaption with notional $$1M$ that expires at

time $t=3$ with an option strike of $0.$ You may assume the underlying swap has a fixed rate of $3\mathrm{.}9\%$ and that if

the option is exercised then cash$-$flows take place at times $t=4,$dots,$10.($The cash$-$flow at time $t=i$ is based

on the short$-$rate that prevailed in the previous period, i$.$e$.$ the payments of the underlying swap are made in

arrears.$)$

Submission Guideline: Give your answer rounded to the nearest integer. For example, if you compute the answer

to be $10,456\mathrm{.}67,$ submit $10457.$

To answer this question, please continue using the calibrated model from the last question.

Repeat the previous question but now assume a value of $b=0\mathrm{.}1.$

Submission Guideline: Give your answer rounded to the nearest integer. For example, if you compute the answer

to be $10,456\mathrm{.}67,$ submit $10457.$

Please refer to the material on defaultable bonds and credit$-$default swaps $($CDS$)$ to answer this question.

Construct a $n=10-$period binomial model for the short$-$rate, $r_{i,j}(i=0,1,2$dots$9).$ The lattice parameters are:

$r_{0,0}=5\%,u=1\mathrm{.}1,d=0\mathrm{.}9$ and $q=1-q=\frac{1}{2}.$ This is the same lattice that you constructed in

Assignment $5.$

Assume that the $1-$step hazard rate in node $(i,j)$ is given by $h_{ij}=a{b}^{j-\frac{1}{2}}$ where $a=0\mathrm{.}01$ and $b=1\mathrm{.}01.$

Compute the price of a zero$-$coupon bond with face value $F=100$ and recovery $R=20\%.$

Submission Guideline: Give your answer rounded to two decimal places. For example, if you compute the answer

tPlease answer the following questions:

o be $73\mathrm{.}2367,$ submit $73\mathrm{.}24.$