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

Solution : Here FMP means Future market price "S" Denote Spot market price of stock. "X" is strike price. Given information : S-0 = 30 FMP-L = 26 FMP-H = 36 Rf = 0 DIVIDEND = 0 As portfolio consists of "1" Short call & "h" Long spot stock. h = Differece in Payoff / Diff. in FMP (Always in absolute terms) Calculation of Payoff: Position : Short call X = 32 FMP-L FMP-H FMP 26 36 Exercise Y/N N Y Payoff - -4 So now h = 0- (-4) / 36-26 = 0.4 So number of shares purchased as a percentage of the number of options that have been sold is 40% As Rf is 0 so formula accordingly : Vc = h * [ S-0 -FMP-L] = 0.4 * [ 30 - 26] = 1.6 As portfolio consists of "1" long Put & "h" Long spot stock. h = Differece in Payoff / Diff. in FMP (Always in absolute terms) Calculation of Payoff: Position : Long put X = 32 FMP-L FMP-H FMP 26 36 Exercise Y/N Y N Payoff 6 0 So now h = 6 - 0/ 36-26 = 0.6 number of shares purchased as a percentage of the number of options purchased option Is 60 % Vp = h * [FMP-H - S-0] = 0.60 * [36-30] = 3.6 risk neutral probability of the stock price moving up We know that FMP-H = S-0 * H FMP-L = S-0 * D SO H = 36/30 = 1.2 D = 26/30 = 0.866 P = (1+ Rf - D) / (H-D) = (1+0 - 0.866)/ (1.2 - 0.866) = 0.4 = 40% So probablity is 40% As H = (1+h) So h = 1.2 - 1 = 0.2 = 20% here "h" denotes % increase in spot price.

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