Hull: Options, Futures, and Other Derivatives, Ninth Edition Chapter 23: Estimating Volatiliti Volatilities es and Correlations Correlati ons Multiple Choice est !an": #uestions $ith %ns$ers
1. How many parameters are necessary to defne an EWMA model A. B. C. D.
1 2 3 4
Answer: A One parameter lam!da" #s necessary to defne an EWMA model.
2. How many parameters are necessary to defne a $A%CH 1&1" model A. B. C. D.
1 2 3 4
Answer: C '(ree parameters parameters ome)a& alp(a& alp(a& and !eta" !eta" are necessary necessary to defne a $A%CH 1&1" model.
3. At t(e end o* '(+rsday& t(e est#mated ,olat#l#ty o* asset A #s 2- per day. D+r#n) r#day asset A prod+ces a ret+rn o* 3-. An EWMA model w#t( lam!da e/+al to 0. #s +sed. W(at #s an est#mate o* t(e ,olat#l#ty o* asset A at t(e end o* r#day A. 2.0B. 2.10C. 2.12D. 2.14Answer: C '(e ,ar#ance rate rate #s 0.0.02250.10.032 6 0.00047. '(e ,olat#l#ty per day #s t(e s/+are root o* t(#s or 2.12-.
4. At t(e end o* '(+rsday& t(e est#mated ,olat#l#ty o* asset B #s 1- per day. D+r#n) r#day asset B prod+ces a ret+rn o* 8ero. An EWMA model w#t( lam!da e/+al to 0. #s +sed. W(at #s an est#mate o* t(e ,olat#l#ty o* asset A at t(e end o* r#day A. 0.B. 0.7C. 0.2D. 0.0-
Answer: B '(e ,ar#ance rate #s 0.0.01250.10.02 6 0.0000. '(e ,olat#l#ty per day #s t(e s/+are root o* t(#s or 0.7-.
7. At t(e end o* '(+rsday& t(e est#mated co,ar#ance !etween assets A and B #s 0.0001. D+r#n) r#day asset A prod+ces a ret+rn o* 3- and asset B prod+ces a ret+rn o* 8ero. An EWMA model w#t( lam!da e/+al to 0. #s +sed. W(at #s an est#mate o* t(e co,ar#ance at t(e end o* r#day A. 0.00000 B. 0.00001 C. 0.000100 D. 0.00007 Answer: A '(e co,ar#ance #s 0.0.000150.10.030 6 0.00000.
9. W(#c( o* t(e *ollow#n) #s a defn#t#on o* t(e co,ar#ance !etween and ; A. Correlat#on !etween and ; t#mes ,ar#ance o* t#mes ,ar#ance o* ; B.
>. W(at does EWMA stand *or A. B. C. D.
E/+ally we#)(ted mo,#n) a,era)e E/+ally we#)(ted med#an appro?#mat#on E?ponent#ally we#)(ted mo,#n) a,era)e E?ponent#ally we#)(ted med#an a,era)e
Answer: C EWMA stands *or e?ponent#ally we#)(ted mo,#n) a,era)e
. W(#c( o* t(e *ollow#n) #s tr+e w(en t(e parameter lam!da e/+als 0.7 A. '(e we#)(t )#,en to t(e most recent o!ser,at#on #s 0.7 B. '(e we#)(t )#,en to t(e o!ser,at#on one day a)o #s 7- o* t(e we#)(t )#,en to t(e o!ser,at#on two days a)o C. '(e we#)(ts )#,en to o!ser,at#ons add +p to 0.7
D. '(e we#)(ts )#,en to t(e o!ser,at#on two days a)o #s 7- o* t(e we#)(t )#,en to t(e o!ser,at#on one day a)o Answer: D '(e we#)(ts )#,en to o!ser,at#ons )o down !y 7- *or eac( day we )o !ac@ #n t#me.
. W(#c( o* t(e *ollow#n) #s tr+e o* ma?#m+m l#@el#(ood met(ods A. '(ey calc+late t(e ma?#m+m poss#!le ,al+es *or parameters B. '(ey calc+late parameters t(at )#,e t(e (#)(est pro!a!#l#ty o* past data occ+rr#n) C. '(ey calc+late ,al+es *or @ey ,ar#a!les t(at are most l#@ely to occ+r #n t(e *+t+re D. '(ey #n,ol,e a m+lt#,ar#ate re)ress#on analys#s Answer: B Ma?#m+m l#@el#(ood met(ods are concerned w#t( est#mat#n) parameter ,al+es !y searc(#n) *or t(e ,al+es t(at ma?#m#8e t(e c(ance o* o!ser,ed data occ+rr#n).
10.
* t(e ,olat#l#ty *or a port*ol#o #s 20- per year& w(at #s t(e ,olat#l#ty per /+arter A. 20B. 10C. 7D. 2Answer: B
'(e ,olat#l#ty per /+arter #s
11. A. B. C. D.
0.2 ×
0.25
=
0.1
or 10-.
W(#c( o* t(e *ollow#n) #s tr+e $A%CH models #ncorporate mean re,ers#on EWMA models do not EWMA models #ncorporate mean re,ers#on $A%CH models do not Bot( $A%CH and EWMA models #ncorporate mean re,ers#on e#t(er $A%CH nor EWMA models #ncorporate mean re,ers#on
Answer: A $A%CH models #ncorporate mean re,ers#on EWMA models do not.
12.
W(#c( o* t(e *ollow#n) #s tr+e A. All opt#on #mpl#ed ,olat#l#t#es tend to mo,e !y t(e same a!o+t *rom one day to t(e ne?t B. '(e #mpl#ed ,olat#l#t#es o* lon)dated opt#ons tend to mo,e !y more t(an t(e #mpl#ed ,olat#l#t#es o* s(ortdated opt#ons C. '(e #mpl#ed ,olat#l#t#es o* s(ortdated opt#ons tend to mo,e !y more t(an t(e #mpl#ed ,olat#l#t#es o* lon)dated opt#ons
D. omet#mes C #s tr+e and somet#mes D #s tr+e Answer: C W(en t(ere #s a s(oc@ ca+s#n) an #ncrease or decrease #n ,olat#l#ty #ts #mpact tends to d#sappear o,er t#me. '(#s #s t(e eFect o* mean re,ers#on." '(e res+lt #s t(at t(e #ncrease or decrease ,olat#l#ty (as most eFect on t(e #mpl#ed ,olat#l#t#es o* s(ortdated opt#ons.
13.
W(#c( o* t(e *ollow#n) #s tr+e o* a pos#t#,e sem#defn#te ,ar#ance co,ar#ance matr#? A. All elements o* t(e matr#? are pos#t#,e B. '(e determ#nant o* t(e matr#? #s pos#t#,e C. '(e matr#? #s symmetr#c D. '(e matr#? #s #nternally cons#stent Answer: D '(e correlat#ons !etween many d#Ferent ,ar#a!les are not #nternally cons#stent +nless t(e ,ar#anceco,ar#ance matr#? #s pos#t#,e sem#defn#te.
14. A. B. C. D.
W(#c( o* t(e *ollow#n) #s tr+e
Answer: B As t(e *o+r#nde? e?ample #n t(e c(apter s(ows& !ot( ,olat#l#t#es and correlat#ons #ncreased #n t(e second (al* o* 200
17.
'(e parameters #n a $A%CH 1&1" model are: ome)a 60.000002& alp(a 6 0.04& and !eta 6 0.7. '(e c+rrent est#mate o* t(e ,olat#l#ty le,el #s 1- per day. * we o!ser,e a c(an)e #n t(e ,al+e o* t(e ,ar#a!le e/+al to 2-& (ow does t(e est#mate o* t(e ,olat#l#ty c(an)e A. 1.29B. 1.19C. 1.09D. 1.03Answer: C '(e new ,ar#ance rate #s 0.00000250.040.02 250.70.012 6 0.000113. '(e new ,olat#l#ty #s t(e s/+are root o* t(#s or 1.09-
19.
'(e parameters #n a $A%CH 1&1" model are: ome)a 60.000002& alp(a 6 0.04& and !eta 6 0.7. W(#c( o* t(e *ollow#n) #s t(e closest to t(e lon) r+n
a,era)e ,olat#l#ty A. 1.1B. 1.2C. 1.3D. 1.4Answer: D '(e lon) r+n a,era)e ,ar#ance rate #s 0.000002G10.040.7"60.0002. '(e lon) r+n a,era)e ,olat#l#ty per day #s t(e s/+are root o* t(#s or 1.4-
1>.
'(e parameters #n a $A%CH 1&1" model are: ome)a 60.000002& alp(a 6 0.04& and !eta 6 0.7. '(e c+rrent est#mate o* t(e ,olat#l#ty le,el #s 1- per day. W(at #s t(e e?pected ,olat#l#ty #n 20 days A. 1.0B. 1.10C. 1.11D.1.12Answer: A '(e lon) r+n a,era)e ,ar#ance rate #s 0.000002G10.040.7"60.0002. '(e e?pected ,ar#ance rate #n 20 days #s 0.000250.0450.7" 200.012 0.0002" 6 0.00011. '(e ,olat#l#ty per day #s t(e s/+are root o* t(#s or 1.0-.
1.
W(#c( o* t(e *ollow#n) #s tr+e A. $A%CH 1&1" w#ll always )#,e a ma?#m+m l#@el#(ood at least as (#)( as
EWMA B. EWMA w#ll always )#,e a ma?#m+m l#@el#(ood at least as (#)( as $A%CH 1&1" C. '(e ma?#m+m l#@el#(ood #s t(e same *or $A%CH 1&1" and EWMA D. omet#mes A #s tr+e and somet#mes B #s tr+e.
Answer: A EWMA #s t(e part#c+lar case o* $A%CH 1&1" w(en t(e ome)a parameter #s 8ero. t *ollows t(at $A%CH 1&1" m+st )#,e at least as )ood a ft to t(e data as EWMA.
1. '(e parameters #n a $A%CH 1&1" model are: ome)a 60.000002& alp(a 6 0.04& and !eta 6 0.7. W(at #s t(e re,ers#on rate *or t(e ,ar#ance rate #mpl#ed !y t(e model A. B. C. D.
0.7- per 1.0- per 1.7- per 2.0- per
Answer: B
day day day day
'(e re,ers#on rate #s 1α−β. n t(#s case #t #s 10.040.7 6 0.01 or 1- per day.
20. A. B. C. D.
W(#c( o* t(e *ollow#n) #s tr+e EWMA #s a part#c+lar case o* $A%CH 1&1" w(ere t(e re,ers#on rate #s 8ero EWMA (as a lower re,ers#on rate t(an $A%CH 1&1"& !+t #t #s not 8ero EWMA (as a (#)(er re,ers#on rate t(an $A%CH 1&1" omet#mes EWMA (as a (#)(er re,ers#on rate t(an $A%CH 1&1" and somet#mes #t (as a lower re,ers#on rate t(an $A%CH 1&1".
Answer: A '(e re,ers#on rate #s 1α−β #n $A%CH 1&1". EWMA #s a part#c+lar case o* t(e $A%CH 1&1" model w(ere ω60& and α+β = 1. '(#s means t(at t(e re,ers#on rate #s 8ero.
