Question $1$

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 $12345678910$

Spot Rate $3\mathrm{.}0\%3\mathrm{.}1\%3\mathrm{.}2\%3\mathrm{.}3\%3\mathrm{.}4\%3\mathrm{.}5\%3\mathrm{.}55\%3\mathrm{.}6\%3\mathrm{.}65\%3\mathrm{.}7\%$

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

$6$

$6$ is

$0$

$6$

$=$

$100$

$/$

$($

$1$

$+$

$.$

$035$

$)$

$6$

$=$

$81\mathrm{.}35$

Z

$0$

$6$

$$

$=100/(1+\mathrm{.}035)$

$6$

$=81\mathrm{.}35$ assuming a face value of $100.$

$------------------------------------------------------------------------------------------------------------------$

Assume

$=$

$0\mathrm{.}05$

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

$1$

$0$

$$

$8$

$10$

$8$

$.($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 $$1$M that expires at time

$=$

$3$

t$=3$ with an option strike of

$0$

$0.$ You may assume the underlying swap has a fixed rate of

$3\mathrm{.}9$

$\%$

$3\mathrm{.}9\%$ and that if the option is exercised then cash$-$flows take place at times

$=$

$4$

$,$

$...$

$,$

$10$

t$=4,...,10.($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 $10457.$