Question

A ?rm’s stock sells at $50. The stock price will be either $65 or $45 three months from now. Assume the 3-month risk-free rate is 1%.

a. What is the price of a European call with a strike price of $50 and a maturity of three months?

b. What is the price of a European call with a strike price of $55 and a maturity of three months?

c. What is the price of a European put with a strike price of $50 and a maturity of three months?

d. Find a portfolio of the stock and bond (a position in risk-free borrowing) such that buying the European call with a strike $50 is equivalent to holding this portfolio. Compare the cost of this portfolio with the call price.

Answer 1

We shall use the Binomial Model to find the Options Price.

According to the model, Price of Call = [p x C_u + (1-p) x C_d] / R

p - probability of price going up = (R - d) / (u - d)

1-p - probability of price going down

C_u = Upside price (uS) - Strike price of call

C_{d -} In the case of a call option, this is always equal to 0, since if the downside price is less than strike price, the call option will not be exercised.

R = 1 + Risk free rate = 1 + 0.01 = 1.01

a) Strike price of $50:

Price of the Call = $ 4.08

b. Strike price of $55:

Price of the Call = $ 2.72

c. Put Option - Strike price of $50: For a Put Option, P_u = 0 since, when the upside price is more than the strike price, the put option is not exercised.

P_d = Strike Price - dS = $50 - 45 = $ 5

Price of Put = $ 3.59

d. First, find Delta or Hedge ratio (number of shares required in the replicating portfolio) = (Cu - Cd) / (uS - dS) = $15 - 0 / $65 - $45 = 15 / 20 = 0.75

Hence, number of shares to be bought for every call option = 0.75

Position in risk-free borrowing (Bond) = (u x Cd - d x Cu) / (u - d) x R

= (1.30 x 0) - (0.90 x 15) / (1.30 - 0.90) x 1.01 = -13.50 / 0.404 = $ -33.42

Value of Call = Current stock price x Delta + Risk free borrowing position (investing in Bond) = $50 x 0.75 - $33.42 = $ 37.50 - 33.42 = $ 4.08

Hence, Cost of this portfolio = Price of the call.