Financial Economics

Consider a three period binomial model (one initial date and two steps) as in class. Let the initial stock price be $100. Let the upward increments be u = 1.2 and the downward be d = .9. Thus, the up-up state should have Suu = u ∗ u ∗ S0 = 1.2 ∗ 1.2 ∗ 100 = 144 and so on. The risk free rate is ten percent per period. The call option with strike price of $100 was priced in class: call price C=19.1.

1. Price the put option with strike price of $100 using replicating portfolios. Does put-call parity hold (remember the interest accrues in discrete steps)? Explain.

2. Suppose we had an Asian call option whose payoff is max{S−K, 0} where S is the average stock price, with strike K = 100. So, if the state is up-up the payoff would be (100 + 120 + 144)/3 − 100 = 21.33. Use replicating portfolios to price this option.

3. Suppose we had a futures contract to buy one share of the stock at the final date at price F0. Use the spot-futures parity theorem to derive the futures price. Now compute the futures price at the intermediate date following an up or down step, Fu and Fd. At each of these states (0, u, or d) is the market in contango or normal backwardation?

4. (Not using the binomial model above) Characterize the behavior of the hedge ratio (for a European call option) in the limit as it gets farther in the money and as it gets farther out of the money.