Assume: Current spot S\&P 500 index is at 2,500 , annual risk-free rate \(=4 \%\), zero dividends, logarithmic return's annualized mean \(=\mu^{\mathrm{A}}=0.10\), and logarithmic return's annualized standard deviation \(=\sigma^{\mathrm{A}}=0.25\).

a. Using the single-period BOPM, determine the price of a European S\&P 500 index call with an exercise price of 2,500 and expiring in 30 days.

b. Define the index call's replicating portfolio in terms of a proxy portfolio.

c. Show with a binomial tree the values of the spot index, proxy portfolio, and replicating portfolio.

d. Explain what an arbitrageur would do if the market priced the index call at 85 . Show what the arbitrageur's cash flow at

expiration would be at the two possible index prices at expirations.

e. Explain what an arbitrageur would do if the market priced the index call at 100 . Show what the arbitrageur's cash flow would be at expiration when she closed.

f. Use the single-period BOPM to determine the price of a European S\&P 500 index put with an exercise price of 2,500 and expiring in 30 days.
g. Show with a binomial tree the values of the spot index, proxy portfolio, and replicating portfolio. Include with your tree the values of the index and proxy portfolio.
h. Explain what an arbitrageur would do if the index put were priced at 80 .