Question

Consider a model with only one time period. Assume that there exist a stock and a cash bond in the model. The initial price of the stock is $50. The investor believes that with probability 1/3 the stock price will drop to $30 and with probability 2/3 the stock price will rise to $90 at the end of the time period. The cash bond has an initial price of $100 and it will with certainty deliver $110 at the end of the period. Use the replication principle and Önd a price of a European call with a maturity at the end of the time period and a strike price of $60. Is this price fair? Explain your answer. Find the price using risk neutral probabilities. Comment on the equivalence of the two approaches.

Answer 1

Given,

	Stock tree
		Up	90
	up probability	2/3
Stock	50
	down probability	1/3
		down	30

	call option tree
		Up	30
	up probability	2/3
European call	??
	down probability	1/3
		down	0

Strike price = 60, meaning call is exercised when stock reaches 90 at the end of period and value of stock would be 90-60 = 30, if stock goes down to 30, option call is not exercised, hence value of option would be 0.

Now, for cash bond, using FV = PV (1+r) ^n formula, 110 = 100 * (1+r)^1 , r = 10% where r is risk nuetral probability

now value of the option is ((2/3*30)+(1/3*0))/(1+0.1) = 18.18182

hedge ratio = (call up value-call low value)/(stock up value - stock low value) = (30-0)/(90-30) = 0.5