Question 1

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

Accepted Answer

A) 0.5 
B) -0.5 
C) 0 
D) 1 
A) 0.5 
B) -0.5 
C) 0 
D) 1 B

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.

Accepted Answer

A) m 
B) m-s2/2 
C) m-r 
D) r-s2/2 
A) m 
B) m-s2/2 
C) m-r 
D) r-s2/2 C

Question 3

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.)

Accepted Answer

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

Question 4

The variance of a Wiener process in time t is

Accepted Answer

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

Question 5

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 .

Accepted Answer

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 
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 6

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.)

Accepted Answer

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

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 the standard deviation of the value of the variable at the end of 5 years

Accepted Answer

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

Question 8

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.

Accepted Answer

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 
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 9

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

Accepted Answer

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 
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 10

Which of the following defines an Ito process?

Accepted Answer

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 
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 11

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?

Accepted Answer

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

Question 12

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).

Accepted Answer

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 
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 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 is the expected value of the variable at the end of 5 years

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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 
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

Question 16

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?

Accepted Answer

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

Question 17

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.

Accepted Answer

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

Question 18

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

Accepted Answer

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

Accepted Answer

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

Question 20

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.

Accepted Answer

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

Exam 14: Wiener Processes and Ito’s Lemma

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

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 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.)

The variance of a Wiener process in time t is

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

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 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 is true when the stock price follows geometric Brownian motion

Which of the following defines an Ito 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 = 3 and b =4 what is the standard deviation of the value in 4 years?

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 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 gives a random sample from a standard normal distribution in Excel?

Which of the following is NOT a property of 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 = 3 and b =4 what is the standard deviation of the value in three months?

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.

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

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?

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.

Introduction

Mechanics of Futures Markets

Hedging Strategies Using Futures

Interest Rates

Determination of Forward and Futures Prices

Interest Rate Futures

Swaps

Securitization and the Credit Crisis of 2007

OIS Discounting, Credit Issues, and Funding Costs

Mechanics of Options Markets

Properties of Stock Options

Trading Strategies Involving Options

Binomial Trees

The Black-Scholes-Merton Model

Employee Stock Options

Options on Stock Indices and Currencies

Futures Options

The Greek Letters

Volatility Smiles

Basic Numerical Procedures

Value at Risk

Estimating Volatilities and Correlations

Credit Risk

Credit Derivatives

Exotic Options

Filters

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 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 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 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?