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The S&P 500 index current level is 3,000. The dividend yield on the index is equal...

The S&P 500 index current level is 3,000. The dividend yield on the index is equal to the risk- free rate of interest. Given a volatility of the index of 25%: a) Computetheprobabilitythattheindexvaluein6monthsisgreaterthan3,300. b) Compute the probability that the index value in 6 months is less than 2700. c) Computetheprobabilitythattheindexvaluein6monthsisbetween2700and3300.

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Answer #1

Lets first find the standard deviation (252 trading days in a year)

SD = Index Price * Volatility * SQRT (126/252) = 3000 * 0.25 * SQRT (0.5) = +/- 530.33

z-score for 3300 = (3300-3000) / 530.33 = 0.56569

P-value (using z score table) = 0.2858 or 28.58% for values over 3300

P-value (using z score table) = 0.2858 or 28.58% for values below 2700

Probability between 2700 and 3000 = P between 2700 to 3000 + P between 3000 to 3000 = 0.2142 + 0.2142 = 0.4284 which is 42.84%

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