Question

​Eagletron's current stock price is $ 10 $10. Suppose that over the current​ year, the stock price will either increase by 95 % 95% or decrease by 57 % 57%. ​Also, the​ risk-free rate is 25 % 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 $ 19.50 $19.50​? c. Suppose the put options in parts ​(a​) and ​(b​) could either be exercised​ immediately, or in one year. What would their values be in this​ case? a. What is the value today of a​ one-year at-the-money European put option on Eagletron​ stock? The value today of the​ one-year at-the-money European put option on Eagletron stock is ​$ nothing . ​ (Round to the nearest​ cent.)

Answer 1

We can calculate the desired results using the provided information as follows

Current Stock Price (Sp) = $ 10

Up rate (u) = 1 + 95% = 1.95

Down rate (d) = 1 - 57% = 0.43

Risk free rate (r) = 25%

Increase in Stock price (Su) = Current price * (1+increase percentage)

= 10 * (1+95%) = 10+9.5 = $ 19.50

Decrease in Stock price (Sd) = Current price * (1-decrease percentage)

= 10 * (1-57%) = 10 - 5.70 = $ 4.30

Risk neutral probability (p) = (1 + r - d) / (u - d)

= ( 1 + 0.25 - 0.43) / ( 1.95 - 0.43 )

= 0.82 / 1.52 = 0.5394

a) The value today of a​ one-year at-the-money European put option is

Given the money put option, the strike price is current price of stock

K = Sp = $ 10

Put price in case of increase (Pu) = max (K - Su, 0) = max (10 - 19.50,0) = 0

Put price in case of decrease (Pd) = max (K - Sd, 0) = max (10 - 4.30,0) = 5.70

Hence, value of the put option today = [ Risk neutral probability * Pu + (1 - Risk neutral probability) * Pd] / (1 + r)

= [0.5394 * 0 + (1 - 0.5394) * 5.70] / (1 + 0.25)

= [ 0 +(0.4606)* 5.70] / (1.25) = 2.625 / 1.25

= $ 2.10

b) The value today of a​ one-year European put option at strike price of $ 19.50 is

Put price in case of increase (Pu) = max (K - Su, 0) = max (19.50 - 19.50,0) = 0

Put price in case of decrease (Pd) = max (K - Sd, 0) = max (19.50 - 4.30,0) = 15.20

Hence, value of the put option today = [ Risk neutral probability * Pu + (1 - Risk neutral probability) * Pd] / (1 + r)

= [0.5394 * 0 + (1 - 0.5394) * 15.20] / (1 + 0.25)

= [ 0 +(0.4606)* 15.20] / (1.25) = 7.001 / 1.25

= $ 5.60

c) If the put options are to be excercised immediately then the value will be

i) Given Strike price in option a is $ 10 and exercising it immediately will result in

= max (K - S0, 0) = max (10 - 10, 0) = 0

So the value is less than the value of put option of $2.10 as calculated in the option a, so it is not beneficial to exercise the put option immediately.

ii) Given Strike price in option b is $ 19.50 and exercising it immediately will result in

= max (K - S0, 0) = max (19.50 - 10, 0) = 9.50

So the value is more than the value of put option of $9.50 as calculated in the option b, so it is beneficial to immediately use the put option.

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