This is a Finmath question.Problem $1$

As an employee of Stark Industries, you participate in the employee stock purchase plan. The

share price of Stark today is $S_0=40.$ One year from today, the company will deduct $$3000$

from your paycheck, and in exchange you will acquire shares of Stark at the below$-$market price of

min$(30,0\mathrm{.}75S_1)$ per share $($a $25\%$ discount to the price today or to the price in a year, whichever

is lower$).$ The entire $$3000$ will be spent on Stark shares $($including fractional shares if necessary$).$

There is no risk that Stark will default on this plan, and no risk that you will leave the company

in the next year.

Stark stock follows a geometric Brownian motion with $=0\mathrm{.}22.$ It pays no dividends. The

interest rate on the bank account is $r=0\mathrm{.}05.$ Markets are frictionless.

If this question asks you for a value or a position, your final answer must be a number. When

you state a position, we will assume it is a long position, unless you either use a $"-"$ sign or say

"short".

$($a$)$ The stock purchase plan $($consisting of the debit of $$3000$ and the receipt of shares$)$ is an

asset to you. Find its time$-0$ value.

$($b$)$ The value of the plan will fluctuate as Stark stock fluctuates. Suppose you don't want to be

subject to this risk. What time$-0$ position, in vanilla options, should you take, in order to

hedge perfectly that risk? $($Assume that European calls and puts on Stark are available at

any strikes and expiries that you desire.$)$ Does this position need to be rebalanced over the

course of the year?

$($c$)$ Now suppose that options on Stark are not available. What time$-0$ position in Stark stock

should you take, in order to hedge perfectly that risk? Does this position need to be rebalanced

over the course of the year?

By "hedge perfectly", I mean that the stock purchase plan together with your proposed hedge $($a

self$-$financing portfolio strategy$)$ should have a total time$-1$ value which is non$-$random.