Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.1
Martingales and Measures
Chapter 21
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.2
Derivatives Dependent on a Single
Underlying Variable
d zσ d tμ
?
d
d zσ d tμ
?
d
??
dzsdtm
d
22
2
2
11
1
1
.
??
??
?
??
?
?
?
S u p p o s e and p r i c e s w i t h
on d e p e n d e n t sd e r i v a t i v e t w o I m a g i n e
p r o c e s s t h e f o l l o w s t h a t s e c u r i t y ) t r a d e d a of
p r i c e t h ey n e c e s s a r i l ( n o t,v a r i a b l e,a C o n s i d e r
21
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.3
Forming a Riskless Portfolio
t??σμ??σμ=
??σ??σ
?σ
?σ
????
???
?
?
)(
)()(
21122121
211122
11
22
d e r i v a t i v e 2nd t h e of
and d e r i v a t i v e 1 s t t h e of +
of c o n s i s t i n g,p o r t f o l i o r i s k l e s s a up s e t c a n We
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.4
or
,gi v es T hi s
,r i s k l es s is por tf ol i o the S i nc e
2
2
1
1
121221
σ
rμ
σ
rμ
r σr σσμσμ
t= r
?
?
?
???
????
Market Price of Risk (Page 485)
? This shows that (m – r )/? is the same for
all derivatives dependent on the same
underlying variable,?
? We refer to (m – r )/? as the market price
of risk for ? and denote it by l
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.5Extension of the Analysis
to Several Underlying Variables
(Equations 21.12 and 21.13,page 487)
t h e n
w i t h
v a r i a b l e s u n d e r l y i n g s e v e r a l on d e p e n d s If
σλrμ
d zσμ d t
?
d
f
n
i
ii
n
i
ii
?
?
?
?
??
??
1
1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.6
Martingales (Page 488)
? A martingale is a stochastic process
with zero drift
? A variable following a martingale has
the property that its expected future
value equals its value today
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.7
Alternative Worlds
dzfdtfrdf
dzσfdtrfdf
??l???
l
??
)(
is
r i s k of pr i c e m ar k e t the w he r e w or l da In
w or l dne utr al-r i s k ltr adi ti on a the In
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.8
A Key Result (Page 489)
i o n )c o n s i d e r a t
under p e r i o d t h e d u r i n g i n c o m e no
p r o v i d e to a s s u m e d a r e and
p r i c e ss e c u r i t y d e r i v a t i v e a l l
f o r m a r t i n g a l e a is t h a t s h o w s
l e m m a sI t o ' t h e n,s e c u r i t y a
of v o l a t i l i t y t h e to equal s e t w eIf
gf
f
gf
g
(
l
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.9
Forward Risk Neutrality
We refer to a world where the market
price of risk is the volatility of g as a
world that is forward risk neutral with
respect to g.
If Eg denotes a world that is FRN wrt g
f
g
E
f
g
g
T
T
0
0
?
?
?
?
?
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.10
Aleternative Choices for the
Numeraire Security g
? Money Market Account
? Zero-coupon bond price
? Annuity factor
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.11
Money Market Account
as the Numeraire
? The money market account is an account that
starts at $1 and is always invested at the
short-term risk-free interest rate
? The process for the value of the account is
dg=rg dt
? This has zero volatility,Using the money
market account as the numeraire leads to the
traditional risk-neutral world
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.12Money Market Account
continued
S i nc e = 1 and =,t he e q ua t i on
be c om es
w he r e denotes e x pe c ta ti on s i n th e
tr ad i ti on a l r i s k - ne ut r al w o r l d
g g e
f
g
E
f
g
f E e f
E
T
r d t
g
T
T
r d t
T
T
T
0
0
0
0
0
0
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.13Zero-Coupon Bond Maturing at
time T as Numeraire
p r i c e bond t h e w r tF R N is
t h a t w o r l da in nse x p e c t a t i o d e n o t e s and
p r i c e bond c o u p o n-z e r o t h e is ),( w h e r e
b e c o m e s
e q u a t i o n T h e
T
TT
T
T
g
E
TP
fETPf
g
f
E
g
f
0
][),0(
0
0
0
?
?
?
?
?
?
?
?
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.14
Forward Prices
In a world that is FRN wrt P(0,T),the
expected value of a security at time T is
its forward price
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.15
Interest Rates
In a world that is FRN wrt P(0,T2) the
expected value of an interest rate
lasting between times T1 and T2 is the
forward interest rate
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.16
Annuity Factor as the
Numeraire
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
)(
)0(
0
0
0
TA
f
EAf
g
f
E
g
f
T
A
T
T
g
b e c o m e s
e q u a t io n T h e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.17
Annuity Factors and Swap Rates
Suppose that s(t) is the swap rate
corresponding to the annuity factor A.
Then:
s(t)=EA[s(T)]
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.18Extension to Several
Independent Factors (Page 492)
??
??
?
?
??
??
?
?
??
?
?
?
?
?
?
?l??
??
?
?
?
?
?
?
?l??
???
???
m
i
iig
m
i
igi
m
i
iif
m
i
ifi
m
i
iig
m
i
iif
dztgtdttgttrtdg
dztftdttfttrtdf
dztgtdttgtrtdg
dztftdttftrtdf
1
,
1
,
1
,
1
,
1
,
1
,
)()()()()()(
)()()()()()(
)()()()()(
)()()()()(
c o n s i s t e n t i n t e r n a l l y a r e t h a t w o r l d so t h e r F o r
w o r l dn e u t r a l-r i s k lt r a d i t i o n a t h e In
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.19Extension to Several
Independent Factors
continued
h o l d, r e s u l t s t h e of r e s t t h e a n d m a r t i n g a l e
a is c a s e,f a c t o r-o n e t h e in As
w h e r e w o r l das
w r tF R N is t h a t w o r l da d e f i n e We
gf
σλ
g
igi,
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.20
Applications
(Section 21.6,page 493)
? Valuation of a European call option
when interest rates are stochastic
? Valuation of an option to exchange one
asset for another
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.21
Change of Numeraire
(Section 21.7,page 495)
wv
wghwv
vhg
q
vqv
a n d b e t w e e n nc o r r e l a t i o t h e is
a n d of v o l a t i l i t y t h e is of
v o l a t i l i t y t h e is w h e r eby i n c r e a s e s
v a r i a b l e a of d r i f t t h e,to f r o m
s e c u r i t y n u m e r a i r e t h e c h a n g e w eW h e n
???
????
,,,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.22
Quantos
(Section 21.8,page 497)
? Quantos are derivatives where the
payoff is defined using variables
measured in one currency and paid in
another currency
? Example,contract providing a payoff
of ST – K dollars ($) where S is the
Nikkei stock index (a yen number)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.23
Diff Swap
? Diff swaps are a type of quanto
? A floating rate is observed in one
currency and applied to a principal in
another currency
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.24
Quantos continued
t w o t h e b e t w e e n nc o r r e l a t i o of
tc o e f f i c i e n t h e is and ),of u n i t per of ( u n i t s
r a t e e x c h a n g e f o r w a r d t h e of v o l a t i l i t y t h e is
,of v a l u e f o r w a r d t h e of v o l a t i l i t y t h e is w h e r e
by i n c r e a s e s v a r i a b l e a of r a t e g r o w t h t h e,
t i m e at m a t u r i n g b o n d s c o u p o n-z e r o to w r t
b e i n g b o t hc u r r e n c y in w o r l dn e u t r a l r i s k
f o r w a r d a to c u r r e n c y in w o r l dn e u t r a l
r i s k f o r w a r d a f r o m m o v e w eWh e n
?
?
?
???
XY
V
VT
X
Y
G
F
GF
)
(
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.25
Quantos continued
t w o t h e
b e t w e e n nc o r r e l a t i o of tc o e f f i c i e n t h e is
and ),of u n i t per of ( u n i t s r a t e e x c h a n g e t h e of
v o l a t i l i t y t h e is,of v o l a t i l i t y t h e is w h e r e
by i n c r e a s e s v a r i a b l e a of r a t e
g r o w t h t h e,c u r r e n c y in w o r l dn e u t r a l r i s k
lt r a d i t i o n a t h e to c u r r e n c y in w o r l dn e u t r a l
r i s k lt r a d i t i o n a t h e f r o m m o v e w eWh e n
?
??
???
XY
V
V
X
Y
SV
SV
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.26
Siegel’s Paradox
A n ex c hang e r ate ( uni ts of c ur r enc y per unti t of
c ur r enc y ) f ol l ow s the r i s k - neutr al pr oc es s
T hi s i mpl i es f r om It o' s l emma th at
G i v en that the pr oc e s s f or S h as a dr i f t
r ate of w e ex pec t the pr oc e s s f or
to hav e a dr i f t of
W hat i s g o i ng on her e?
S Y
X
dS r r Sdt Sdz
d S r r S dt S dz
r r
S r r
Y X S
X Y S S
Y X
X Y
? ? ?
? ? ? ?
?
?
[ ]
( / ) [ ]( / ) ( / )
,
.
?
? ?1 1 1
1
2
21.1
Martingales and Measures
Chapter 21
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.2
Derivatives Dependent on a Single
Underlying Variable
d zσ d tμ
?
d
d zσ d tμ
?
d
??
dzsdtm
d
22
2
2
11
1
1
.
??
??
?
??
?
?
?
S u p p o s e and p r i c e s w i t h
on d e p e n d e n t sd e r i v a t i v e t w o I m a g i n e
p r o c e s s t h e f o l l o w s t h a t s e c u r i t y ) t r a d e d a of
p r i c e t h ey n e c e s s a r i l ( n o t,v a r i a b l e,a C o n s i d e r
21
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.3
Forming a Riskless Portfolio
t??σμ??σμ=
??σ??σ
?σ
?σ
????
???
?
?
)(
)()(
21122121
211122
11
22
d e r i v a t i v e 2nd t h e of
and d e r i v a t i v e 1 s t t h e of +
of c o n s i s t i n g,p o r t f o l i o r i s k l e s s a up s e t c a n We
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.4
or
,gi v es T hi s
,r i s k l es s is por tf ol i o the S i nc e
2
2
1
1
121221
σ
rμ
σ
rμ
r σr σσμσμ
t= r
?
?
?
???
????
Market Price of Risk (Page 485)
? This shows that (m – r )/? is the same for
all derivatives dependent on the same
underlying variable,?
? We refer to (m – r )/? as the market price
of risk for ? and denote it by l
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.5Extension of the Analysis
to Several Underlying Variables
(Equations 21.12 and 21.13,page 487)
t h e n
w i t h
v a r i a b l e s u n d e r l y i n g s e v e r a l on d e p e n d s If
σλrμ
d zσμ d t
?
d
f
n
i
ii
n
i
ii
?
?
?
?
??
??
1
1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.6
Martingales (Page 488)
? A martingale is a stochastic process
with zero drift
? A variable following a martingale has
the property that its expected future
value equals its value today
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.7
Alternative Worlds
dzfdtfrdf
dzσfdtrfdf
??l???
l
??
)(
is
r i s k of pr i c e m ar k e t the w he r e w or l da In
w or l dne utr al-r i s k ltr adi ti on a the In
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.8
A Key Result (Page 489)
i o n )c o n s i d e r a t
under p e r i o d t h e d u r i n g i n c o m e no
p r o v i d e to a s s u m e d a r e and
p r i c e ss e c u r i t y d e r i v a t i v e a l l
f o r m a r t i n g a l e a is t h a t s h o w s
l e m m a sI t o ' t h e n,s e c u r i t y a
of v o l a t i l i t y t h e to equal s e t w eIf
gf
f
gf
g
(
l
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.9
Forward Risk Neutrality
We refer to a world where the market
price of risk is the volatility of g as a
world that is forward risk neutral with
respect to g.
If Eg denotes a world that is FRN wrt g
f
g
E
f
g
g
T
T
0
0
?
?
?
?
?
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.10
Aleternative Choices for the
Numeraire Security g
? Money Market Account
? Zero-coupon bond price
? Annuity factor
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.11
Money Market Account
as the Numeraire
? The money market account is an account that
starts at $1 and is always invested at the
short-term risk-free interest rate
? The process for the value of the account is
dg=rg dt
? This has zero volatility,Using the money
market account as the numeraire leads to the
traditional risk-neutral world
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.12Money Market Account
continued
S i nc e = 1 and =,t he e q ua t i on
be c om es
w he r e denotes e x pe c ta ti on s i n th e
tr ad i ti on a l r i s k - ne ut r al w o r l d
g g e
f
g
E
f
g
f E e f
E
T
r d t
g
T
T
r d t
T
T
T
0
0
0
0
0
0
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.13Zero-Coupon Bond Maturing at
time T as Numeraire
p r i c e bond t h e w r tF R N is
t h a t w o r l da in nse x p e c t a t i o d e n o t e s and
p r i c e bond c o u p o n-z e r o t h e is ),( w h e r e
b e c o m e s
e q u a t i o n T h e
T
TT
T
T
g
E
TP
fETPf
g
f
E
g
f
0
][),0(
0
0
0
?
?
?
?
?
?
?
?
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.14
Forward Prices
In a world that is FRN wrt P(0,T),the
expected value of a security at time T is
its forward price
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.15
Interest Rates
In a world that is FRN wrt P(0,T2) the
expected value of an interest rate
lasting between times T1 and T2 is the
forward interest rate
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.16
Annuity Factor as the
Numeraire
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
)(
)0(
0
0
0
TA
f
EAf
g
f
E
g
f
T
A
T
T
g
b e c o m e s
e q u a t io n T h e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.17
Annuity Factors and Swap Rates
Suppose that s(t) is the swap rate
corresponding to the annuity factor A.
Then:
s(t)=EA[s(T)]
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.18Extension to Several
Independent Factors (Page 492)
??
??
?
?
??
??
?
?
??
?
?
?
?
?
?
?l??
??
?
?
?
?
?
?
?l??
???
???
m
i
iig
m
i
igi
m
i
iif
m
i
ifi
m
i
iig
m
i
iif
dztgtdttgttrtdg
dztftdttfttrtdf
dztgtdttgtrtdg
dztftdttftrtdf
1
,
1
,
1
,
1
,
1
,
1
,
)()()()()()(
)()()()()()(
)()()()()(
)()()()()(
c o n s i s t e n t i n t e r n a l l y a r e t h a t w o r l d so t h e r F o r
w o r l dn e u t r a l-r i s k lt r a d i t i o n a t h e In
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.19Extension to Several
Independent Factors
continued
h o l d, r e s u l t s t h e of r e s t t h e a n d m a r t i n g a l e
a is c a s e,f a c t o r-o n e t h e in As
w h e r e w o r l das
w r tF R N is t h a t w o r l da d e f i n e We
gf
σλ
g
igi,
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.20
Applications
(Section 21.6,page 493)
? Valuation of a European call option
when interest rates are stochastic
? Valuation of an option to exchange one
asset for another
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.21
Change of Numeraire
(Section 21.7,page 495)
wv
wghwv
vhg
q
vqv
a n d b e t w e e n nc o r r e l a t i o t h e is
a n d of v o l a t i l i t y t h e is of
v o l a t i l i t y t h e is w h e r eby i n c r e a s e s
v a r i a b l e a of d r i f t t h e,to f r o m
s e c u r i t y n u m e r a i r e t h e c h a n g e w eW h e n
???
????
,,,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.22
Quantos
(Section 21.8,page 497)
? Quantos are derivatives where the
payoff is defined using variables
measured in one currency and paid in
another currency
? Example,contract providing a payoff
of ST – K dollars ($) where S is the
Nikkei stock index (a yen number)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.23
Diff Swap
? Diff swaps are a type of quanto
? A floating rate is observed in one
currency and applied to a principal in
another currency
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.24
Quantos continued
t w o t h e b e t w e e n nc o r r e l a t i o of
tc o e f f i c i e n t h e is and ),of u n i t per of ( u n i t s
r a t e e x c h a n g e f o r w a r d t h e of v o l a t i l i t y t h e is
,of v a l u e f o r w a r d t h e of v o l a t i l i t y t h e is w h e r e
by i n c r e a s e s v a r i a b l e a of r a t e g r o w t h t h e,
t i m e at m a t u r i n g b o n d s c o u p o n-z e r o to w r t
b e i n g b o t hc u r r e n c y in w o r l dn e u t r a l r i s k
f o r w a r d a to c u r r e n c y in w o r l dn e u t r a l
r i s k f o r w a r d a f r o m m o v e w eWh e n
?
?
?
???
XY
V
VT
X
Y
G
F
GF
)
(
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.25
Quantos continued
t w o t h e
b e t w e e n nc o r r e l a t i o of tc o e f f i c i e n t h e is
and ),of u n i t per of ( u n i t s r a t e e x c h a n g e t h e of
v o l a t i l i t y t h e is,of v o l a t i l i t y t h e is w h e r e
by i n c r e a s e s v a r i a b l e a of r a t e
g r o w t h t h e,c u r r e n c y in w o r l dn e u t r a l r i s k
lt r a d i t i o n a t h e to c u r r e n c y in w o r l dn e u t r a l
r i s k lt r a d i t i o n a t h e f r o m m o v e w eWh e n
?
??
???
XY
V
V
X
Y
SV
SV
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
21.26
Siegel’s Paradox
A n ex c hang e r ate ( uni ts of c ur r enc y per unti t of
c ur r enc y ) f ol l ow s the r i s k - neutr al pr oc es s
T hi s i mpl i es f r om It o' s l emma th at
G i v en that the pr oc e s s f or S h as a dr i f t
r ate of w e ex pec t the pr oc e s s f or
to hav e a dr i f t of
W hat i s g o i ng on her e?
S Y
X
dS r r Sdt Sdz
d S r r S dt S dz
r r
S r r
Y X S
X Y S S
Y X
X Y
? ? ?
? ? ? ?
?
?
[ ]
( / ) [ ]( / ) ( / )
,
.
?
? ?1 1 1
1
2
