Problem $4.(25$p$)($Default probability in Merton's model$)$

At time $t,$ we assume the asset $A_t$ of a company satisfies the stochastic differential equation $($SDE$)$

$d{A}_t=A_{t}dt+A_{t}d{W}_t$

where $$ is the drift parameter, $$ is the volatility, and $\{W_t$:$t0\}$ is a standard Wiener process on

the probability space $(,F,P).$ Let $r$ be the risk$-$free interest rate.

In financial accounting, the asset $A_t$ is a combination of equity $E_t$ and debt $D_t$ so that

$A_t=E_t+D_t$

where at time $F>0A_T>F{A}_T{E}_T{D}_T{E}_t{D}_{t}C(S,t)=S_{t}N(d_1)-K{e}^{-r(T-t)}N(d_2)d_1=\frac{log(\frac{S_t}{K})+(r+\frac{{}^2}{2})(T-t)}{\sqrt[\phantom{2}]{T-t}},$

$d_2=d_1-\sqrt[\phantom{2}]{T-t},N(*)C-P=S_t-K{e}^{-r(T-t)}t$

under the Black$-$Scholes framework.

Hints: You can use the facts below

European Call Option Price under the Black$-$Scholes Model:

$C(S,t)=S_{t}N(d_1)-K{e}^{-r(T-t)}N(d_2)$

where

$d_1=\frac{log(\frac{S_t}{K})+(r+\frac{{}^2}{2})(T-t)}{\sqrt[\phantom{2}]{T-t}},$

$d_2=d_1-\sqrt[\phantom{2}]{T-t},$

and $N(*)$ denotes the cumulative distribution function $of$ the standard normal distribution.

Put$-$Call Parity:

$C-P=S_t-K{e}^{-r(T-t)}$

This formula states that the difference between the call price and the put price equals the

difference between the current stock price and the discounted strike price using the risk$-$free

interest rate.