Question 1

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

Accepted Answer

The variance of a Wiener process is 1 per unit time or t in time t.

Question 2

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

Accepted Answer

Ito's lemma shows that the process followed by y is 
dy = 0.5b²exp(x)dt +bexp(x)dz.Substituting y = exp(x)we get the answer in B.

Question 3

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.

Accepted Answer

A Markov process is a particular type of stochastic process where only the current value of a variable is relevant for predicting the future.Stock prices are usually assumed to follow Markov processes.This corresponds to a weak form market efficiency assumption.

Question 4

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

Accepted Answer

From Ito's lemma, the coefficient of $d z$ is $s X \partial f / \partial X$ where $f = X ^ { 2 }$. Because $\partial f / \partial X = 2 X$, the coefficient of $d z$ is $2 s X ^ { 2 }$.

Question 5

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

Accepted Answer

The standard deviation of the change in one day is 
$20 \times 0.25 \times \sqrt { 1 / 365 } = \$ 0.26$

Question 6

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

Accepted Answer

The change is $e \sqrt { d t }$ .This result is used when the process is simulated.

Question 7

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

Accepted Answer

The variance of each process is 1 per unit time.The variance of the sum is 1+1+2$\rho$ where $\rho$ is the correlation.This is 1 when $\rho$=-0.5.

Question 8

A variable x starts at 10 and follows the generalized Wiener process 
$\quad$$\quad$$\quad$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

Accepted Answer

The drift is 2 per year and so the expected increase over three years is 2×3 = 6 and the expected value at the end of 3 years is 10+6 = 16.

Question 9

A variable x starts at 10 and follows the generalized Wiener process 
$\quad$$\quad$$\quad$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

Accepted Answer

The variance per year is 4² or 16.The variance over four years is 16×4 = 64.The standard deviation is $\sqrt { 64 } = 8$

Question 10

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

Accepted Answer

During the first two years,the drift per year is 3 and so the total drift is 3×2 or 6.During the next three years,the drift per year is 6 and the total drift is 6×3 = 18.The total drift over the five years is 6+18 =24.Given that the variable starts at zero,its expected value at the end of the five years is therefore 24.

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

Accepted Answer

Ito's lemma show that the log of the stock price follows a generalized Wiener process.This means that the log of the stock price is normally distributed so that the stock price is lognormally distributed.

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

The variance per year for the first two years is 4² or 16.The variance per year for the next three years is 3² or 9.The total variance of the change over five years is 2×16+3×9= 59.The standard deviation of the value of the variable at the end of the five years is therefore $\sqrt { 59 } = 7.7$

Question 14

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

Accepted Answer

This is the example in Section 14.7

Question 15

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()

Accepted Answer

The correct instruction in Excel is =NORMSINV(RAND())

Question 16

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

Accepted Answer

From Ito's lemma,the coefficient of dt is $m X \frac { \partial f } { \partial X } + \frac { 1 } { 2 } s ^ { 2 } X ^ { 2 } \frac { \partial ^ { 2 } f } { \partial X ^ { 2 } }$
where $f = X ^ { 2 }$. Because $\partial f / \partial X = 2 X$ and $\partial ^ { 2 } f / \partial X ^ { 2 } = 2$ the coefficient of $d t$ is $2 m X ^ { 2 } + s ^ { 2 } X ^ { 2 }$

Question 17

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

Accepted Answer

In a generalized Wiener process the drift and coefficient of dz are both constant.In an Ito process they are not both constant .

Question 18

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

Accepted Answer

From the previous question the standard deviation of the change in one day is $0.26.There is a 2.5% chance that the stock price will increase by more than 1.96 standard deviations.The answer is therefore 20+1.96×0.26 = $20.51.The expected return in one day is small and can be ignored.

Question 19

A variable x starts at 10 and follows the generalized Wiener process 
$\quad$$\quad$$\quad$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 three months?
A)1
B)2
C)3
D)4

Accepted Answer

The variance per year is 4² or 16.The variance over three months is 16×0.25 = 4.The standard deviation is $\sqrt { 4 } = 2$

Question 20

Which of the following is NOT a property of a Wiener process?
A)The change during a short period of time dt has a variance dt
B)The changes in two different short periods of time are independent
C)The mean change in any time period is zero
D)The standard deviation over two consecutive time periods is the sum of the standard deviations over each of the periods

Accepted Answer

Variances of Wiener processes are additive but standard deviations are not.

Quiz 14: Wiener Processes and Ito’s Lemma

The variance of a Wiener process in time t is

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) .

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

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 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.)

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 .

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

A variable x starts at 10 and follows the generalized Wiener process $\quad$ $\quad$ $\quad$ 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 variable x starts at 10 and follows the generalized Wiener process $\quad$ $\quad$ $\quad$ 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 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

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

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 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

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.

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

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.

Which of the following defines an Ito process?

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 variable x starts at 10 and follows the generalized Wiener process $\quad$ $\quad$ $\quad$ 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 three months?

Which of the following is NOT a property of a Wiener process?