XYZ Corp. will pay a $2 per share dividend in 2 months. Its stock price currentl
ID: 2786267 • Letter: X
Question
XYZ Corp. will pay a $2 per share dividend in 2 months. Its stock price currently is $88 per share. A call option on XYZ has an exercise price of $80 and 3-month time to expiration. The risk-free interest rate is 0.5% per month, and the stock's volatility (standard deviation)-23% per month. Find the pseudo-American option value. (Hint: Try defining one "period" as a month, rather than as a year.) (Round your answer to 2 decimal places. Omit the "$" sign in your response.) Pseudo-American option valueExplanation / Answer
Current Stock Price, S0=$88
Strike Price, X= $80
Time to Expiry,T = 3months or 3 periods
Dividend expected, D= $2
Volatitlity , = 23% per month
Rate =0.5% per month
Time, t = 2 months =2 periods
PV of dividend =De-rt = 2 x e-0.005x2 = 2 x e-0.010 = 1.98
Discounting dividend from current stock price, we get S0' =$88 - 1.98 =$86.02
C = S0'N(d1) - Xe-rTN(d2)
d1 = {log(S0' /X) + (r + 2/2)T} / x T0.5
= {log(86.02/80) + (0.005 + 0.232/2)x3} / (0.23 x 30.5)
= {0.03151 + 0.09435} / 0.39837
= 0.3159
d2 = d1 - ( x T0.5 ) = 0.2120 - (0.23 x 30.5) = 0.3159 - 0.39837 = -0.08244
N(d1) = 0.62397
N(d2) = 0.46715
Thus, C = 86.02x0.62397 - 80xe-0.005x3x0.46715 =53.68 - 36.82 = 16.86
Now, we need to calculate the price setting maturity just before the last dividend is paid i.e.
T = 2months = 2
C = S0'N(d1) - Xe-rTN(d2)
S0' = S0
d1 = {log(S0' /X) + (r + 2/2)T} / x T0.5
= {log(88/80) + (0.005 + 0.232/2)x2} / (0.23 x 20.5)
= {0.04139 + 0.0629} / 0.32527
= 0.3206
d2 = d1 - ( x T0.5 ) = 0.3206 - (0.23 x 20.5) = 0.3206 - 0.3527 = -0.00463
N(d1) = 0.62576
N(d2) = 0.49815
Thus, C = 88x0.62576 - 80xe-0.005x2x0.49815 = 55.06 - 39.45 = 15.61
Pseudo American option price will be greater of abve two calculated prices, i.e. 16.86 > 15.61
Hence, Option price =16.86
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