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) If stock price increases in the first month.

t=1 t=2

By the replicating portfolio

Suppose, we combine Δ stocks with $B invested in Bonds to create the same portfolio which gives the same payoff as that of the $800 strike put option

Then , in case of upside Δ stocks value would be Δ*1080 and B *1.04 , and put options value would be max (800-1080,0) =0

Therefore, Δ*1080+B*1.04=0

Similarly. for downside, Δ*730+B*1.04= 70

Solving these for Δ and B , we get, Δ = -1/5= -0.2 and B = $207.69

So, the replicating portfolio consists of short position in 0.2 stocks and long position of $207.69 in risk free bonds

So, value of put option at t=1 when stock price increases = value of replicating portfolio today

= -0.2*880 + 207.69 = $31.69

b) if stock price decreases in the first month.

Let us construct a riskless portfolio which gives the same value whether the stock goes up or down at t=2

Let the portfolio consists of long position in Δ shares and long position in one put option

portfolio value when stock goes up = Δ*792 + 8

portfolio value when stock goes down = Δ*648 + 152

Solving Δ =1

So, Riskless portfolio has long position in one share and one put option

Thus value of portfolio at t=2

= 792+8 = $800

Value of riskless portfolio today = 800/1.04 = $769.23

Also, Value of riskless portfolio today = 720 + p where p is the value of put option

So, 720+ p = 769.23

=> p = $49.23

c)

For put option from t=0 to t=1

u = 1.1

d= 0.9

risk neutral probability = (1.04-0.9)/(1.1-0.9) =0.7

So, Value of put option today = (0.7*31.69+0.3*49.23)/1.04 = $35.53