Consider a portfolio consisting of $6$ stocks and $10$ put options on the stocks. The current stock price is S$0=100$ and the stike price of the options is X $=100.$ The stock pricecan take on only two values at maturity T given by Su $=120$ and Sd $=90.$ The risk$-$free rate is given by $5\%.$

$(1).$ What is the payoff at maturity of the portfolio?

$(2).$ Calculate the $$Delta$$ of the portfolio which is defined as the price change of the put over the price change of the stock, or $=|$Pu$-$Pd$|/$Su$$Sd$.$

$(3).$ How does your answer change in $(1)$ and $(2)$ when considering a put option with strike price X $=110?$