Question 2. Please show all the derivations in your calculations. There are two
ID: 1170169 • Letter: Q
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Question 2. Please show all the derivations in your calculations. There are two stocks, A and B, with spot prices $50 and $40 respectively Further, the T-bill rate is 5% per annum with continuous compounding. An investor obtains the following information for expected prices of these two stocks 1-month from today under two scenarios of market returns (suppose these are the only potential cases): Market Returns Stock AStock B Probability $62.5 $32.5 25% $45 40% 5% $42 60% Suppose neither of the stocks will pay dividend during this month. Find: i. The betas of Stock A and Stock B ii. The expected rate of return of Stock A and Stock B ii. The value of a 1-month European call option of Stock A with a strike price of $55.5.Explanation / Answer
I am going to answer the first 4 parts of the question:
i) Beta is slope of the line. So it can be calculated as:
Y2-Y1/X2-X1
Return of Stock A at 62.5= 62.5/50 -1= 25%
Return of Stock A at 32.5= 32.5/50 -1= -35%
Therefore, Beta (A)= 25-(-35)/25-(-5)=60/30= 2
Return of Stock B at 45=45/40-1= 12.5%
Return of Stock B at 42= 42/40 -1= 5%
Therefore, Beta (B)= 12.5-5/(25-(-5)= 0.25
ii) Expected Return of:
A= 0.40*25+0.60*(-35) = -11%
B= 0.40*12.50 +0.60*5= 8%
iii) Expected Payoff from the call option when price at 62.50 = Max (62.50-55.5, 0)*0.40= $ 2.80
Expected Payoff from the call option when price at 32.50 = Max (32.50-55.5, 0)*0.60= $ 0
Discount Factor for 1 month= e-0.05*(1/12)= 0.9958
Therefore, value of the call option: (2.80+0)*0.9958= $ 2.788
iv) Discount factor for 3 months= e-0.05*3/12= 0.9875
Discount factor for 6 months= e-0.05*6/12= 0.9753
Therefore, sum of present value of dividends:
1.5*0.9875+1.5*0.9753= 2.944
Hence, value of the forward contract:
(50-2.944)e0.05*9/12=$ 48.8541
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