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

I. Purchases (.4) 40% Shares for each call option sold.

II. Value of each call option is (e^-.06x5) [.435x4]= $ 1.6

III. Delta is the number of the shares that should be purchased as long position to hedge a long put option or say (0-5)/(36-26)= -.5 shares

Iv value of put option when risk free rate is Zero= (e^-0x.5) [.5x5]= $2.825

V Formula is Probability = (1-d)/(u-d)

Here d= down Price/ Current Price i.e. (26/30)= .8667

u= up price/ Current Price i.e (36/30) = 1.2

Risk Neutral Probability is (1-.8667)/(1.2-.8667) = 0.4

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