Consider a single-period binomial model of a non-dividend-paying stock with current price S = $100. The risk-free cumulative return over the period is R = 1.08. All options considered below are European style, they have strike $50, and they expire at the end of the single period.

(a). What are the restrictions on u and d that preclude arbitrage opportunities?

(b). Suppose that u = 1.20 and d = 1.05. What is the price of the call? What is the replicating portfolio? Derive the results mechanically using the standard binomial procedure.

(c). Suppose now that u = 1.4 and d = 0.6. Can you tell by inspection what the price of the call and corresponding replicating portfolio are going to be in this case? Confirm your result using the formal binomial calculations.

(d). Finally, let u = 1.4 and d = 0.4. How do you think the call option price compares to (b) and (c)? Is it higher or lower? How do you account for this? Compute the actual price and the replicating portfolio using the binomial formulas.