Question: The random walk model for a stock price assumes that at each time step the price can either increase or decrease by a fixed amount ∆ > 0.
05 Feb 2023,6:10 PM
Exercise 1. The random walk model for a stock price assumes that at each time step the price can either increase or decrease by a fixed amount ∆ > 0. Suppose that P1, 0 < P1 < 1 is the probability of an increase and that P2 = 1 P1 is the probability of a decrease. Let X be the discrete random variable representing the change in a single step and Y be the discrete random variable representing the number of increases observed in 10 steps.
(a) Find the expected
value of the random variable
X.
(b) Find the variance of the random variable
X.
(c) Describe the distribution of the random variable Y .
(d) 
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Exercise 2. Let X ∼ UNIF (n). Verify that E [X] = a+b .
Exercise 3. Let X ∼ UNIF (n). Verify that V [X] = (b−a+1)2 −1 .
Exercise 4. Suppose that the daily rate of return on a certain stock is normally
distributed with a mean of µ = 0.23% and a standard deviation
of σ = 2.54%. What is the probability that the stock’s price will increase
by more than 5.31% on any given day next week?
Exercise 5. Suppose that the daily rate of return on a certain stock is normally
distributed with a mean of µ = 0.23% and a standard
deviation of σ = 2.54%.
(a)
What is the probability that the stock’s
price will increase
by more than 2% on any given
day next week?
(b) What is the probability that the stock’s
price will decrease
by more than 1% on any given
day next week?
Exercise 6. Let X be a random
variable and let c ∈ R be a real number. Demonstrate that the expectation operator E satisfies E [cX] = c · E [X].
Exercise 7. Let X be a random variable and let c ∈ R be a real number. Demonstrate that the variance
operator V satisfies V [cX] = c2 · V [X].
Exercise 8. The number of defaults in one year within a certain portfolio
of bonds is found to be a Poisson random
variable with parameter
λ
= 7 (i.e., the portfolio has an expectation of 7 defaults per year).
(a)
Find the probability that the bond portfolio will have fewer than 3 defaults during the upcoming
year.
(b) Describe the distribution of the continuous random variable representing the time between
successive defaults.
(c)
Find the expected time between successive
defaults.
(d) Calculate the probability of there being less than 6 months between two successive defaults.
Exercise 9. 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 $120 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 0.25.
(a) Find the probability that X is less than or equal to 0.1.
(b)
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and identically distributed as X so that Y
return.
=∆ X + X + X + X represents the stock’s four-day logarithmic
(i) Describe the distribution of the continuous random variable Y .
(ii) Calculate the probability that Y is less than or equal to 0.1.
1
(c)
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0
representing the ratio
of tomorrow’s price
to
today’s
price.
(d)
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What is the expected value of the random variable S1 ?
0
(e)
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What is the variance of the random variable S1 ?
0
Exercise 10. Let the dynamics Xi, i = 1, 2, . . . , d be
independently and identically distributed as Z ∼
N (0, 1). One approach for modeling the short-term
interest
rate
rt at any time t is given by defining
r =∆ X2 + X2 + . . . + X2.
t 1 2 d
(a)
Describe the distribution of the continuous random variable rt.
(b) Find the probability that rt ∈ (0, 0.02] if d = 3.
(c) Find the probability that rt ∈ (0, 0.02] if d = 7.
Exercise 11 (BONUS). Prove
that the variance
of the sum of two independent random
variables is equal
to the sum of their individual variances.
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