The current stock price is $800. The stock price will either increase by 10% or decrease by 10% in the first month. If the price increases in the first month, it will go up by $200 or down by $150 in the second month. If the price decreases in the first month, it will go up by 10% or down by 10% in the second month. The risk-free interest rate is 4% per month. Assume there is a 2-month put option with an exercise price of $800.

a) Use the replicating portfolio approach to calculate the value of the put if stock price increases in the first month.

b) Use the risk neutral probability approach to calculate the value of the put if stock price decreases in the first month.

c) Use either approach to calculate the 2-month put premium today.

 

Answer:

a) Replicating portfolio approach

The Equations are as follows,

(800 + 200)*(A) + (1.04)B = 800

(800 - 150)*(A) + (1.04)B = 0

where, A is call option value, B is put option value

solving both equations we get,

1000A + 1.04B = 800

650A + 1.04B = 0

Subtracting we get,

350A = 800

A = 2.285

B = ( 2285 - 800 ) / 1.04

B = 1427.88

Therefore put option price is calculated as below,

= 800*2.285 - 1427.88

= 400.12

Therefore, put option price is $ 400.12.

b) Risk neutral probability approach

p = (r - d) / ( u - d)

p = 200 - 80 / 150 - 80

p = 120 / 70

p = 1.714 is the probability

Price of put option

= 800*(1.714/1.04) - 800

Price of put option = $ 518.46

c) The put premium today as per the above mentioned two approaches are as follows,

Replicating portfolio

approach put option value

= $ (400.12 - ( 800 - 800))

= $ 400.12

Risk neutral approach put option value

= $ 518.46