Given the following:

- The S\&P 500 futures option expiring at the end of 60 days.

- The S\&P 500 futures contract expiring at the end of 120 days.

- The current spot index is at \(S_{0}=3,000\).

- The estimated annualized volatility and mean of the spot index's logarithmic return are \(\sigma^{\mathrm{A}}=0.25\) and \(\mu^{\mathrm{A}}=0\).

- The annual risk-free rate is \(R_{f}=3 \%\).

- The futures price is determined by the carrying-cost model:

\(f_{0}=S_{0} e^{\left(R_{f}-\psi\right) n_{f} \Delta t}\)

- \(\Psi=\) continuous annual dividend yield \(=5 \%\). Determine the following:

a. Show the spot index and futures prices at each node for a two-period binomial tree with a length of each step equal to 60 days \((\Delta t=60 / 365=0.164384)\). In generating the tree, use the up and down parameters defining the spot index and those defining the futures.

b. Explain the relationship between the spot and futures prices that you obtained in 3 a.

c. Using the BOPM, determine the equilibrium price for the 3,000 index futures call expiring in 60 days \((n=1)\).

d. Explain the arbitrage strategy you would employ if the price of the 3,000 index futures call were priced at 135 .

e. Using the BOPM, determine the equilibrium price for the 3,000 index futures put expiring in 60 days ( \(n=1\) ).

f. Explain the arbitrage strategy you would employ if the price of the 3,000 index futures put were priced at 155 .