Question

​Eagletron's current stock price is $ 10. Suppose that over the current​ year, the stock price will either increase by 98% or decrease by 60%. ​Also, the​ risk-free rate is 25% ​(EAR).

a. What is the value today of a​ one-year at-the-money European put option on Eagletron​ stock?

b. What is the value today of a​ one-year European put option on Eagletron stock with a strike price of $19.80​?

c. Suppose the put options in parts ​(a​) and ​(b​) could either be exercised​ immediately, or in one year. What would their values be in this​ case?

Answer 1

S0 = 10; u = 1+ 98% = 1.98 d = 1 - 60% = 0.40 r = 25% = 0.25; Su = u x S0 = 1.98 x 10 = 19.8; Sd = d x S0 = 0.4 x 10 = 4

Risk neutral probability of up movement = p = (1 + r - d) / (u - d) = (1 + 0.25 - 0.40) / (1.98 - 0.40) =  0.5380

Part (a)

At the money put option, hence strike price, K = S0 = 10

Pu = max (K - Su, 0) = max (10 - 19.8, 0) = 0

Pd = max (K - Sd, 0) = max (10 - 4, 0) = 6

Hence, value of the put option today = [p x Pu + (1 - p) x Pd] / (1 + r) = [0.5380 x 0 + (1 - 0.5380) x 6] / (1 + 0.25) = $ 2.22

Part (b)

K = 19.80

Pu = max (K - Su, 0) = max (19.80 - 19.8, 0) = 0

Pd = max (K - Sd, 0) = max (19.80 - 4, 0) = 15.80

Hence, value of the put option today = [p x Pu + (1 - p) x Pd] / (1 + r) = [0.5380 x 0 + (1 - 0.5380) x 15.80] / (1 + 0.25) = $ 5.84

Part (c)

Option in part (a) if exercised immediately will give = max (K - S0, 0) = max (10 - 10, 0) = 0 < value of the European option calculated in part (a) = 2.22

Hence, immediate exercise does not provide any incremental value. Hence, the value will still be the value calculated in part (a) = $ 2.22

Option in part (b) if exercised immediately will give = max (K - S0, 0) = max (19.8 - 10, 0) = 9.80 > value of the European option calculated in part (b) = 5.84

Hence, immediate exercise is beneficial. Hence, value of the option in part (b) will now be = max (9.80, 5.84) = $ 9.80