Calculating option prices by using true probabilities.

A stock sells for $60 today. The risk-free rate over the period is 20% with annual compounding. Assume that next period (in one year) the stock will either go up to $90 with probability 0.8 or go down to $30 with probability 0.2.

In what follows, always use annual compounding to compound (or discount).

a). What is the expected stock price at the end of the period, i.e., in one year? What is the expected gross return for the stock?

b). Consider a call option with strike price 60. What is the expected payoff to a call option holder at the end of the period?

c). Use, as in the previous question, a portfolio long D share of the stock and short D bonds with face value $1 to replicate the payoffs of the option at maturity (in one year). What are D and D?

d). What is the expected gross return for the portfolio and, therefore, for the option?

e). Price the option using (i) discounted expected cash flows with true probabilities, (ii) discounted expected cash flows with risk-neutral probabilities, and (iii) the replicating portfolio.

f). Assume the true probabilities of the stock going up or down are now 0.9 and 0.1, respectively. What would the option price be now? Briefly explain.