Please show all work for every part, thank you

$(10\%$;$2\%$ for each subquestion$)$ A firm has non$-$dividend$-$paying equity $($Et: equity value at time $t$ and$(10\%$;$2\%$ 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 vield $(\%).$

$($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, $=10\%.$

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

zero$-$coupon debt $($Bt: debt value at time $t$; promised payment at time $T$ is $300).A_t$ is the asset value of the

firm at time t$.A_0=260,r_c=2\%($continuous time$),$ asset volatility $=20\%,T=5$ years.

$($a$)$ If the asset value at time $T$ is $350,$ what is the payoff received by shareholders at time $T?$ Which group

of investors $($bondholders or shareholders$)$ is the seller of the call option on the firm's assets?

$($b$)$ What is the current market value of equity?

$($c$)$ What is the current market value of risky debt?

$($d$)$ The Black$-$Scholes formula for pricing a European call option on a stock is $C_0=S_{0}N(d_1)-x{e}^{-rT}N(d_2),$

where

$d_1=\frac{ln(\frac{S_o}{x})+(r+\frac{{}^2}{2})T}{\sqrt[\phantom{2}]{T}},d_2=d_1-\sqrt[\phantom{2}]{T}$

Find the formula for pricing a European put option with the same strike price $x($hint: using put$-$call parity

and $N(-z)=1-N(z)).$

$($e$)$ If the government is willing to provide debt guarantee to protect the firm from bankruptcy, what is the

current value of debt guarantee?