Deck 14: Wiener Processes and Itos Lemma

Question 1

For what value of the correlation between two Wiener processes is the sum of the processes also a Wiener process?

A) 0.5
B) −0.5
C) 0
D) 1

Answer 1

B
The variance of each process is 1 per unit time.The variance of the sum is 1+1+2 where  is the correlation.This is 1 when =−0.5.

Question 2

If the risk-free rate is r and price of a nondividend paying stock grows at rate m with volatility s,at what rate does a forward price of the stock grow for a forward contract maturing at a future time T.

A) m
B) m−s2/2
C) m−r
D) r−s2/2

Answer 2

C
This is the application of Ito's lemma in Section 14.6.

Question 3

The process followed by a variable X is dX = mX dt+sX dz
What is the coefficient of dz in the process for the square of X.

A) sX
B) sX2
C) 2sX2
D) msX

Answer 3

2sX2

Question 4

A variable x starts at 10 and follows the generalized Wiener process dx = a dt + b dz
Where time is measured in years.If a = 2 and b =3 what is the expected value after 3 years?

A) 12
B) 14
C) 16
D) 18

Question 5

A variable x starts at zero and follows the generalized Wiener process dx = a dt + b dz
Where time is measured in years.During the first two years a=3 and b=4.During the following three years a=6 and b=3.What the standard deviation of the value of the variable at the end of 5 years

A) 6.2
B) 6.7
C) 7.2
D) 7.7

Question 6

A variable x starts at 10 and follows the generalized Wiener process dx = a dt + b dz
If a = 3 and b =4 what is the standard deviation of the value in three months?

A) 1
B) 2
C) 3
D) 4

Question 7

A variable x starts at zero and follows the generalized Wiener process dx = a dt + b dz
Where time is measured in years.During the first two years a=3 and b=4.During the following three years a=6 and b=3.What is the expected value of the variable at the end of 5 years

A) 16
B) 20
C) 24
D) 30

Question 8

A variable x starts at 10 and follows the generalized Wiener process dx = a dt + b dz
Where time is measured in years.If a = 3 and b =4 what is the standard deviation of the value in 4 years?

A) 4
B) 8
C) 12
D) 16

Question 9

If a variable x follows the process dx = b dz where dz is a Wiener process,which of the following is the process followed by y = exp(x).

A) dy = by dz
B) dy = 0.5b2y dt+by dz
C) dy = (y+0.5b2y) dt+by dz
D) dy = 0.5b2y dt+b dz

Question 10

The variance of a Wiener process in time t is

A) t
B) t squared
C) the square root of t
D) t to the power of 4

Question 11

Which of the following is true when the stock price follows geometric Brownian motion

A) The future stock price has a normal distribution
B) The future stock price has a lognormal distribution
C) The future stock price has geometric distribution
D) The future stock price has a truncated normal distribution

Question 12

The process followed by a variable X is dX = mX dt+sX dz
What is the coefficient of dt in the process for the square of X.

A) 2mX2+s2X2
B) 2mX2
C) mX2+2s2X2
D) mX2+s2X2

Question 13

A stock price is $20.It has an expected return of 12% and a volatility of 25%.What is the stock price that has a 2.5% chance of being exceeded in one day? (For this question assume that there are 365 days in the year.)

A) $20.41
B) $20.51
C) $20.61
D) $20.71

Question 14

If a stock price follows a Markov process which of the following could be true

A) Whenever the stock price has gone up for four successive days it has a 70% chance of going up on the fifth day.
B) Whenever the stock price has gone up for four successive days there is almost certain to be a correction on the fifth day.
C) The way the stock price moves on a day is unaffected by how it moved on the previous four days.
D) Bad years for stock price returns are usually followed by good years.

Question 15

When a stock price,S,follows geometric Brownian motion with mean return m and volatility s what is the process follows by X where X = ln S.

A) dX = m dt + s dz
B) dX = (m−r) dt + s dz
C) dX = (m −s2) dt + s dz
D) dX = (m − s2/2) dt + s dz

Question 16

If e is a random sample from a standard normal distribution,which of the following is the change in a Wiener process in time dt .

A) e times the square root of dt
B) e times dt
C) dt times the square root of e
D) The square root of e times the square root of dt

Question 17

Which of the following gives a random sample from a standard normal distribution in Excel?

A) =NORMSINV()
B) =NORMSINV(RAND())
C) =RND(NORMSINV())
D) =RAND()

Question 18

A stock price is $20.It has an expected return of 12% and a volatility of 25%.What is the standard deviation of the change in the price in one day.(For this question assume that there are 365 days in the year.)

A) $0.20
B) $0.23
C) $0.26
D) $0.29

Question 19

Which of the following defines an Ito process?

A) A process where the drift is non-constant and can be stochastic
B) A process where the coefficient of dz is non-constant and can be stochastic
C) A process where either the drift or the coefficient of dz or both are non-constant and can be stochastic
D) A process where proportional changes follow a generalized Wiener process