Question

Today, the continuous compound interest rate is 0.1% and one share of Amazon is $2367.92.

1. What price do you expect Amazon to be 6 months from now? What European style option should you buy in this case? Your future price can’t be a whole dollar, everyone should have different future stock price.

2. Assume the volatility of Amazon is 27% and you set your own strike price to 2 decimal places, find out how much your option in part 1 costs by using the Black-Scholes formula. You strike price can’t be a whole dollar, everyone should have different strike price.

3. Draw the profit diagram for the straddle with the strike price in part 2 above (you may need to use the put-call parity formula).

4. Use the binomial model with 4 steps to find the American style option price in part 2 above.

Answer 1

1.

S0: Present share price = $2367.92

r: interest rate = 0.1%

time = 6 months = 0.5 years

s: standard deviation = 27%

z: is a random number between -1.96 to 1.96

Assume z = 1

e: natural exponent

St = 2367.92*e^(0.1%*0.5+1*27%*(0.5^0.5)) = $2880.41 (expected stock price after 6-months)

Since the expected stock price has increased, I will prefer to buy a European call option

2. K: strike price = 2870.41 (assumed)

So current stock price = 2367.91

r risk free rate = 0.1%

s: standard deviation = 27%

t: time to maturity = 6month = 0.5 year

d1 = -0.90993

d2 = -1.10085

N(d1) = normsdist(d1) = 0.1814

N(d2) = normsdist(d2) = 0.1355

C: value of call option

c = $40.92 (price of European call option)

3. p: price of put option

p + So = c + K*e^(-r*t)

p + 2367.92 = 40.92 + 2870.41*e^(-0.1%*0.5)

p = 541.98

Pay-off diagram

dt: time lengths = 6months/4 = 1.5 months = 1.5/12 or 0.125 years

u: up factor = e^(s*(dt^0.5)) = e^(27%*(0.125^0.5)) = 1.1

d: down factor = 1/1.1 = 0.91

p: probability of up movement

p = (e^(r*t)-d)/(u-d) = (e^(0.1%*0.5)-0.91)/(1.1-0.91) = 47.6%

q: probability of down movement = 1-47.6% = 52.4%

Price of American style call option = $30.79