Replicating portfolios.
A stock sells for $50 today. The risk-free rate over the period is 10% with annual compounding. Assume that next period (in one year) the stock will either go up by 30% (to $65) with probability 0.7 or go down by 20% (to $40) with probability 0.3.
Suppose you own an out-of-the-money European call option on the stock with a strike price X equal to the futures price for delivery in one year.
In what follows, always use annual compounding to compound (or discount).
a) What is the futures price for delivery in one year?
b) Show that a portfolio long D shares of the stock and short D bonds with face value $1 can be used to replicate the payoffs of the option at maturity.
c) Using the result in b) and the law of one price, what is the price of the call?
d) What are the “risk-neutral probabilities” of up and down moves? (Hint: use the fact that, given the risk-neutral probabilities, the expected rate of return on the stock is the risk-free rate).
e) Calculate the price of the option using risk-neutral probabilities and verify that it is the same as what you found in point (c).