The current price of a non-dividend-paying stock is $30. Over the next six month

Question

The current price of a non-dividend-paying stock is $30. Over the next six months it is expected to rise to $36 or fall to $26. Assume the risk-free rate is zero. I. What long position in the stock is necessary to hedge a short call option when the strike price is $32? Give the number of shares purchased as a percentage of the number of options that have been sold. II. What is the value the call option? III. What long position in the stock is necessary to hedge a long put option when the strike price is $32? Give the number of shares purchased as a percentage of the number of options purchased option. IV. What is the value of the put option. V. What is the risk neutral probability of the stock price moving up?

Explanation / Answer

Assumin no of option purchased = 100

for hedging the investor must choose to buy a share which can be hedged

Heding required for worse condition = 100*(36-32) - 1.60*100 = 240

To hedge no of share to purchase = 240/(36-30) = 40

.

The number of shares purchased as a percentage of the number of options that have been sold = 40/100 = 40%

Hedge ratio = (36 - 32)/(32 - 26)

= 4:6

Thus investor needs to buy 6 shares of this stock in order to hedge the options sold.

2.

Expected Value of upper side = $36

Expected Share value at lower side = $26

Call value at upper side = 36-32 = 4

Call value at lower side = 0

d=26/30 = 0.866667

u =36/30 = 1.20

probability of chances of upperside = e^(r*t)-d/(u-d) = (e^0 - 0.8667)/(1.20-0.8667) = (1-0.8667)/0.33333 = 0.40

Probailty of chances of downside = 0.60

Value of call option = 4*0.40 + 0*0.60 = $1.60

3.

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