Deck 14: Wiener Processes and Ito’s Lemma

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

Answer 1

A
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

Answer 2

B
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.

Answer 3

C
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

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

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

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

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 = 2 and b =3 what is the expected value after 3 years?

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

Question 9

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 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

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

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

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

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

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()

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

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

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

Question 19

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