$(10\%$;$2\%$ for each subquestion$)$ Company ABC has non$-$dividend$-$paying equity $($Et: equity value at time

t$)$ and zero$-$coupon debt $($Bt: debt value at time t; promised payment at time $T$ is $$500$ million$).A_t$ is the

asset value of the firm at time $t.$ The firm has $20$ million shares outstanding in the stock market, and the

stock price is $$\frac{3}{?}$ share. $A_0=$$$400$ million, $r_c=2\%($continuous time$),$ and $T=5$ years.

$($a$)$ Find the $($implied$)$ asset volatility.

$($b$)$ Find the continuously compounded bond yield $(\%).$

$($c$)$ What is the current value of risk$-$free debt?

$($d$)$ If the government is willing to provide debt guarantee $($i$.$e$.,$ a put option$)$ to protect the firm from

bankruptcy, what is the current value of debt guarantee?

$($e$)$ Find the implied probability of default, which is calculated as $N(-D{D}_0).$ Also, $=10\%.$

$D{D}_0=\frac{ln(\frac{A_0}{B})+(-\frac{{}_A^2}{2})T}{{}_A\sqrt[\phantom{2}]{T}}$