$2.$ Finite dierence method for Black Scholes PDE $($in re$$ected log coordinates$)$: Consider the

double knock$-$out power option discussed in class. The underlying stock price at time t is St$.$

The initial stock price is S$02(50$; $100),$ and the option's maturity date is T $=1,$ measured

in years. The risk$-$free rate is r $=0$:$05$ per year, and the stock's volatility is $=0$:$4$ per year.

At expiration, if the stock price lies in the interval S $2(50$; $100),$ and its price has never been

outside of this interval between t $=0$ and T the option pays out:

P$($S; T$)=($S $50)(100$ S$)$:

If the stock price has been outside of the interval, the payout is P$($S; T$)=0,$ i$.$e$.,$ the option

is worthless. Since the price path of the stock is continuous $($almost surely$),$ this occurs if

min$0$tT St $>50,$ and max$0$tT St $<100$:

$($a$)$ PDE: Write the PDE for the price of this option, V $($s; x$),$ including boundary and initial

conditions, in log space coordinates, x $=$ ln$($S$),$ and re$$ected time coordinates, s $=$ T $$t$.$

$($b$)$ Solver: Implement a nite dierence solver for the PDE, using the nite dierence

method with operators D$+$s$,$ D$0$x$,$ and D$+$xD$$x for the dierent derivative estimates, as

discussed in class.

You may use your favorite programming language $($Matlab$,$ Python, etc.$),$ or even Ex$-$

cel. Please include your source code $($or in case of Excel, your spreadsheet$)$ with your

submission.

$($c$)$ Solution: Using the ratio between time and space step$-$length k $=$ t

x$2=2,$ calculate

the approximate solution at t $=0,$ with x $=100$ points.

$$ Please include graphs of V $($x; $0),$ V $($x; $1),$ P$($S; $1),$ and P$($S; $0).$

$$ Where does the value $($in S coordinates$)$ reach its maximum?