Question

Binomial model Over the coming year Ragwort’s stock price will halve to $50 from its current level of $100 or it will rise to $200. The one-year interest rate is 10%.

a. What is the delta of a one-year call option on Ragwort stock with an exercise price of $100?

b. Use the replicating-portfolio method to value this call.

c. In a risk-neutral world what is the probability that Ragwort stock will rise in price?

d. Use the risk-neutral method to check your valuation of the Ragwort option.

e. If someone told you that in reality there is a 60% chance that Ragwort’s stock price will rise to $200, would you change your view about the value of the option? Explain.

Answer 1

S = 100; Su = 200; Sd = 50; u = Su / S = 200 / 100 = 2; d = Sd / S = 50 / 100 = 0.5; r = 10%

a. What is the delta of a one-year call option on Ragwort stock with an exercise price of $100?

K = 100

Cu = max (Su - K, 0) = max (200 - 100, 0) = 100

Cd = max (Sd - K, 0) = max (50 - 100, 0) = 0

Delta = (Cu - Cd) / (Su - Sd) = (100 - 0) / (200 - 50) = 2/3 = 0.6667

b. Use the replicating-portfolio method to value this call.

Let the replicating portfolio comprises of buying A number of stocks and borrowing an amount B at risk free rate.

Value of the portfolio in up state = A x Su - B x (1 + r) = A x 200 - B x (1 + 10%) = 200A - 1.1B = Cu = 100

Value of the portfolio in down state = A x Sd - B x (1 + r) = A x 50 - B x (1 + 10%) = 50A - 1.1B = Cd = 0

Subtract the two, we get 200A - 50A = 150 A = 100; Hence A = 100 / 150 - 2/3 = 0.6667

and B = 50A / 1.1 = 50 x 0.6667 / 1.1 = $ 30.30

Hence, value of the call today = A x S - B = 0.6667 x 100 - 30.30 = $ 36.3636

c. In a risk-neutral world what is the probability that Ragwort stock will rise in price?

p = (1 + r - d) / (u - d) = (1 + 10% - 0.5) / (2 - 0.5) = 40.00%

d. Use the risk-neutral method to check your valuation of the Ragwort option.

Value of the option = [Cu x p + (1 - p) x Cd] / (1 + r) = [100 x 40% + 0 x (1 - 40%)] / (1 + 10%) = 36.3636

e. If someone told you that in reality there is a 60% chance that Ragwort’s stock price will rise to $200, would you change your view about the value of the option? Explain.

No, the valuation is taking in a risk neutral world and none of the parameters of the risk neutral world is impacted by the actual probability of price occurrence. Hence, I will not change my view about the valuation.