Question $5.$ Options $[9$ marks$]$

Answer the required about S&P $500$ Index Options. Assuming call and put options in the below transactions are European style, with the same time to expiration of $3$ days. The risk$-$free rate is $5\%$ and the S&P $500$ index currently traded at $$4766.$ The below trading strategy has been executed at time $0$ to construct a portfolio:

Transaction $1.$ Writing a call option on the S&P $500$ with a strike price of $$4760,$ receiving $$16\mathrm{.}2$;

Transaction $2.$ Purchasing a put option on the S&P $500$ with a strike price of $$4775,$ costing $$35\mathrm{.}6$;

Transaction $3.$ Acquiring an Exchange$-$Traded Fund $($ETF$)$ that tracks the S&P $500$ performance at the current price of $4766,$ in a quantity matching the notional value of the S&P $500$ index underlying the options in the first two steps.

Required:

a$)$ Plot four payoff diagrams, each depicting the result of the respective transactions outlined above with the fourth diagram depicting the net payoff of all three transactions. Ensure adequate labelling of the diagrams. $[4$ marks$]$

b$)$ Solve the implied volatility for the put option contract $($Transaction $2),$ based on the current S&P index price of $$4766.[2$ marks$]$

c$)$ Assess whether this trading strategy is delta$-$neutral and explain in one sentence or two. $[3$ marks$]$