$(3)$ You hold a portfolio made of a long position in $1000$ put options with strike price $30$ and

maturity of three months, on a non$-$dividend$-$paying stock with lognormal distribution with

volatility $30\%,$ a long position in $400$ shares of the same stock, which has spot price $$25,$

and $$10,000$ in cash. Assume that the risk$-$free rate is constant at $2\%.$

$($i$)$ How much is the portfolio worth?

$($ii$)$ How do you adjust the stock position to make the portfolio Delta$-$neutral?

$($iii$)$ The spot prices of the underlying asset at the end of the next four weeks are

$$30,$$$26,$$$22,$$$27.$

You rebalance the hedge at the end of each week, to make the portfolio Delta$-$neutral.

Record the action you take and the portfolio and the action you are taking to do so in a table

like the one below $("$BH$"$ means before balancing the hedge; $"$AH$"$ means after balancing

the hedge$).$ Recall that cash accumulates interest over each time period.