PI Hedge Fund has formed a proxy portfolio that it is using to identify arbitrage opportunities using put-call parity relations. The proxy portfolio is highly correlated with the S\&P 500 with a beta of 1 , is currently worth \(\$ 25\) million, and is expected to pay dividends worth \(\$1.25\) million at the end of one year. The current S\&P 500 index is at 2,500 and the risk-free rate is \(5 \%\).

a. Define the PI Hedge Fund's proxy portfolio as an investment in hypothetical shares in the S\&P500.What is the dividend per index share?

b. Determine the equilibrium price of a \(2,500 \mathrm{S \& P} 500\) index put expiring in one year given a 2,500 S\&P 500 call expiring in one year trading at 50 .

c. Describe the arbitrage strategy PI could employ if the S\&P 500 put were trading at 40 .

d. Evaluate PI's arbitrage strategy from \(c\) at the index's expiration by first assuming the market increases by \(10 \%\), then assuming it decreases by \(10 \%\). Assume the portfolio is perfectly correlated with the market.