Question

1) In a two period model, a stock starts at 150. It either goes up by a factor of u=1.2, or down by a factor of d 1/1.2. The interest rate isr=1/39 between period 0 and 1, and r1/19 between period 1 and 2There originally was only one interest rate here, but the review session put the idea in my head.) A(15 points) What is a call option at 153 expiring in 2 periods on this stock worth? B(5 points) using Put-Call parity, what is a put option at 153 expiring in 2 periods on this stock worth?

Answer 1

Probability downward factor=1/1.2=0.8333

Interest rate factor for period 1(R1)=1+1/39=1.025

Interest rate factor for period 2(R2)=1+1/19= 1.052

Probability of upward price movement for period 1= R1-d/(u-d)=(1.025-0.8333)/(1.2-0.8333)=52.27%

Probability of downward price movement for period 1=(1-52.27%)=47.73%

Probability of upward movement for period 2= R2-d/(u-d)=(1.052-0.8333)/(1.2-0.8333)=59.64%

Probability of downward movement for period 2= 1-59.64%=0.4036=40.36%

Price of stock after period 1 is either 150*1.2=180 or 150*0.8333=125

Price of stock after period 2 is either 180*1.2=216 or 180*0.8333=150 or 125*1.2=150 or 125*0.8333=104.15

So, expected payoff for a call option with strike price of 153= 52.27%*59.64%*(216-153)+0+0=$19.63

Present value of expected value of call option= 19.63/(1.025*1.052)=$18.20

Value of call option is $18.20

b. Similarly, expected payoff for a put option with 153 strike price= 0+(153-150)*52.27%*40.36%+(153-150)*47.73%*59.64%+(153-104.15)*47.73%*40.36%=$10.89

Present Value of expected payoff= 10.89/(1.025*1.052)=$10.099

Value of put option is $10.099