Question

Let X be a random variable representing a quantitative daily market dynamic (such as new information about the economy). Suppose that today’s stock price S0 for a certain company is $150 and that tomorrow’s price S1 can be modeled by the equation S1 = S0 · eX. Assume that X is normally distributed with a mean of 0 and a variance of 0.5.

(a) Find the probability that X is less than or equal to 0.1

(b) Suppose the daily dynamics Xi , i = 1, 2, 3, 4, 5 of each of the next five consecutive days are independently and identically distributed as X so that Y ∆= X1 + X2 + X3 + X4 + X5 represents the stock’s five-day logarithmic return.

(i) Describe the distribution of the continuous random variable Y.

(ii) Calculate the probability that Y is less than or equal to 0.1.

(c) Describe the distribution of the random variable S1/S0 representing the ratio of tomorrow’s price to today’s price.

(d) What is the expected value of the random variable S1/S0 ?

(e) What is the variance of the random variable S1/S0

Answer 1