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Hull: Options, Futures, and Other Derivatives, Ninth Edition
Chapter 14: Wiener Processes and Ito’s Lemma
Multiple Choice Test Bank: Questions with Answers
1. 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
Answer: C
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

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Hull: Options, Futures, and Other Derivatives, Ninth EditionChapter 14: Wiener Processes and Ito’s e!!a ultiple Choice #est $an%: &uestions 'ith (ns'ers
1.
A variable
x
starts at 10 and follows the generalized Wiener process
dx
=
a dt
+
b dz
where time is measured in years. If
a
= 2 and
b
= what is the e!pected valueafter years A.12#.1$%.1&'.1(Answer) % *he drift is 2 per year and so the e!pected increase over three years is 2+ = & andthe e!pected value at the end of years is 10,& = 1&.
2.
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
= and
b
=$ what is the standard deviation of the value in $ years A.$#.(%.12'.1&Answer) # *he variance per year is $
2
or 1&. *he variance over four years is 1&+$ = &$. *he standard deviation is
64 8
=
.
3.
A variable
x
starts at 10 and follows the generalized Wiener process
dx
=
a dt
+
b dz
If
a
= and
b
=$ what is the standard deviation of the value in three months A.1#.2%.'.$Answer) # *he variance per year is $
2
or 1&. *he variance over three months is 1&+0.2- = $. *he standard deviation is
4 2
=
.
4.
*he variance of a Wiener process in time
t
is
A. tB. t squared
C.
the suare root of t
D.
t to the power of 4
Answer) A *he variance of a Wiener process is 1 per unit time or
t
in time
t
.5.
*he process followed by a variable
X
is
dX = mX dt+sX dz
What is the coe/cient of
dz
in the process for the suare of
X
.
A.
sX
B.
sX
2C.
2
sX
2
D.
msX
Answer) %rom Itos lemma the coe/cient of
dz
is
X f sX
∂∂
where
f
=
X
2
. #ecause
X X f
2
=∂∂
the coe/cient of
dz
is
2
sX
2
.
6.
*he process followed by a variable
X
is
dX = mX dt+sX dz
What is the coe/cient of
dt
in the process for the suare of
X
.
A.
2
mX
2
+
s
2
X
2B.
2
mX
2
C.
mX
2
,2
s
2
X
2
D.
mX
2
+s
2
X
2
Answer) Arom Itos lemma the coe/cient of
dt
is
2222
21
X f X s X f mX
∂∂+∂∂
where
f
=
X
2
. #ecause
X X f
2
=∂∂
and
2
22
=∂∂
X f
the coe/cient of
dt
is
2
mX
2
+
s
2
X
2
7.
Which of the following is true when the stoc3 price follows geometric #rownian motionA.*he future stoc3 price has a normal distribution#.*he future stoc3 price has a lognormal distribution%.*he future stoc3 price has geometric distribution'.*he future stoc3 price has a truncated normal distributionAnswer) #
Itos lemma show that the log of the stoc3 price follows a generalized Wiener process. *his means that the log of the stoc3 price is normally distributed so that the stoc3 price is lognormally distributed.
8.
If a stoc3 price follows a 4ar3ov process which of the following could be trueA.Whenever the stoc3 price has gone up for four successive days it has a 506 chance of going up on the 7fth day.#.Whenever the stoc3 price has gone up for four successive days there is almost certain to be a correction on the 7fth day.%.*he way the stoc3 price moves on a day is una8ected by how it moved on the previous four days.'.#ad years for stoc3 price returns are usually followed by good years.Answer) %A 4ar3ov process is a particular type of stochastic process where only the current value of a variable is relevant for predicting the future. 9toc3 prices are usually assumed to follow 4ar3ov processes. *his corresponds to a wea3 form mar3et e/ciency assumption.
9.
A variable
x
starts at zero and follows the generalized Wiener process
dx
=
a dt
+
b dz
where time is measured in years. 'uring the 7rst two years
a
= and
b
=$. 'uring the following three years
a
=& and
b
=. What is the e!pected value of the variable at the end of - yearsA.1&#.20%.2$'.0Answer) %'uring the 7rst two years the drift per year is and so the total drift is +2 or &. 'uring the ne!t three years the drift per year is & and the total drift is &+ = 1(. *he total drift over the 7ve years is &,1( =2$. :iven that the variable starts at zero its e!pected value at the end of the 7ve years is therefore 2$.
10.
A variable
x
starts at zero and follows the generalized Wiener process
dx
=
a dt
+
b dz
where time is measured in years. 'uring the 7rst two years
a
= and
b
=$. 'uring the following three years
a
=& and
b
=. What the standard deviation of the value of the variable at the end of - yearsA.&.2#.&.5%.5.2'.5.5
Answer) ' *he variance per year for the 7rst two years is $
2
or 1&. *he variance per year for the ne!t three years is
2
or ;. *he total variance of the change over 7ve years is 2+1&,+;= -;. *he standard deviation of the value of the variable at the end of the 7ve years is therefore
59 7.7
=
11.
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
= e!p<!
. A.dy = by dz B.dy = 0.5b
2
y dt+by dz C.dy =
(
y+0.5b
2
y
dt+by dz D.dy = 0.5b
2
y dt+b dz
Answer) #Itos lemma shows that the process followed by
y
is
dy = 0.5b
2
e!p(
x
dt +b
e!p(
x) dz.
9ubstituting
y =
e!p(
x
we get the answer in #.
12.
If the ris3>free rate is
r
and price of a nondividend paying stoc3 grows at rate
m
with volatility
s
at what rate does a forward price of the stoc3 grow for a forward contract maturing at a future time
.
A.m
B.
m
?
s
2
@2
C.
m
?
r D.
r
?
s
2
@2Answer) % *his is the application of Itos lemma in 9ection 1$.&.
13.
When a stoc3 price
!
follows geometric #rownian motion with mean return
m
and volatility
s
what is the process follows by
X
where
X
= ln
!.
A.
dX = m dt + s dz
#.
dX = m#r) dt + s dz
%.
dX = m
#s
2
) dt + s dz

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