Download Interest Rate Variance Swaps and the Pricing of

QUANT PERSPECTIVES
Interest Rate Variance Swaps
and the Pricing of Fixed
Income Volatility
BY ANTONIO MELE
AND
O
ne of the pillars supporting the recent movement
toward standardized measurement and trading of
interest rate volatility (see http://www.garp.org/
risk-news-and-resources/2013/december/a-pushto-standardize-interest-rate-volatility-trading.aspx)
atility pricing. The meaning of this mouthful is best understood
Y O S H I K I O B AYA S H I
and explicitly account for the distinct characteristics of each
plied volatility of forward swap rates. This article provides an
overview of how volatility pricing and indexing methodologies
-
Government Bonds
of spanning variance swap payoffs with those of options on the
same underlying.
The price of volatility derived in this framework carries a
clean and intuitive interpretation as the fair market value of
challenging, mainly because of the high dimensionality of
ward volatility of a three-month future on the 10-year Treasury
note, and that available for trading are American-style options
naturally lends itself as the basis of a benchmark index for
income market and serves as the underlying for standardized
futures and options contracts for volatility trading.
A model-free options-based volatility pricing methodology
prices and exchange rates) was branded and popularized as the
ogy has been carried over to other markets, such as those for
gold, oil and single stocks.
In contrast, to create analogous volatility indexes for the
www.garp.org
desired volatility in a strictly model-free fashion?
The practical answer depends on the magnitude of two isAmerican option prices. A second relates to the mismatch in
maturity between the options and the underlying futures — i.e.,
one-month options are used to span risks generated by threeFortunately, in practice, situations arise in which the modeldependent components may be presumed small enough, such
-
M A R C H 2 0 1 4 RISK PROFESSIONAL
1
value of
of aa variance
variance swap,
swap,value
and to
to
know
whenswap,
the approximation
approximation
is
of aathe
tolerable
magnitude
given
value
ofknow
variance
swap,
and to
to know
know is
when
the
approximation
of aa tt
of
aa variance
and
when
approximation
isis given
of
value
and
when
the
of
tolerable
magnitude
context.
context.
context.
context.
(S,
T)
be
the
forward
price
at
forthe
delivery
atprice
S, of
ofat
a coupon
coupon
bearingat
bond
expiring
Let
F
(S,
T)
be
the
forward
price
for delivery
delivery
at
S, of
of
coupoaa
Letprice
t
T)
forward
t,t, for
S,
aa coupon
Let
FFtt(S,
at
t,t,be
for
delivery
at
S,
aat
bearing
bond
expiring
Let Ft (S, T) be the forward
Bt (T)
B
(T)
t
(T)
tprice
awith
variance
swap,
and
know
when
the where
approximation
is ofprice
a bond
tolerable
BtS
(T)
T) =
where
Pto
(S)
is =
the
at
ofPP
zeroisis
coupon
with tt ≤
≤ SS ≤
≤ T,
T,value
i.e. FFof
(S,
T)
= Bprice
(S)
the
price
atexpi
t of
of
t≤
≤
≤, T,
T, i.e.
i.e. F
Ftt(S,
t (S,
T)
the
at
texp
with
t=
≤
PS
(S)
T)
at
tt of
aatt(S)
zero
coupon
bond
with
i.e.
(S),, where
t (S,
tt(S)
Ptt(S) , where Ptt(S) is the PP
context.
(T) isis the
the
price
atand
ofB
the
underlying
bond.
at SS ≥
≥ t,t, and
and B
Bt(T)
(T)
the price
price
at tt of
of the
the underlying
underlying bond.
bond.
atprice
≥t,
t,
and
B
isis the
at
at
SS ≥
tt(T)
at
tt of
the
underlying
bond.
at
t
value
ofLet
a variance
swap,
and
to forward
knowrate
whenprocess
the approximation
is ofbe
a tolerable
magnitude
given
T)
be
the
price
at
t,
for
delivery
at
S,
of
a
coupon
bearing
Ft (S,
be
the
instantaneous
short-term
and
let
Q
the
risk-neutral
probabilit
Let
r
be
the
instantaneous
short-term
rate
process
and let
let
Qthe
be th
t
Let
r
t
short-term
and
Q
be
Letshort-term
rtt be the instantaneous
instantaneous
rate process
and
let Q berate
the process
risk-neutral
probability
Let rt be the
context.
Bt (T)
(S,
T)
=
,
where
P
(S)
is
the
price
at
t
of
a
zero
co
with
t
≤
S
≤
T,
i.e.
F
(S,
satisfies
well-known (see,
(see,
e.g.,
Mele,
2013, Chapter
Chapter
12)
that
in2013,
diffusion
setting,
t(see, e.g.,
t
is well-known
well-known
Mele,
2013,
Chapter
12)
that
in aT)
aT)
diffusion
set
t(S,
P
(S) in
isT)
(see,
Mele,
12)
that
in
diffusion
tdelivery
satisfies
isis well-known
e.g.,
Mele,
12)
that
aaatdiffusion
setting,
FFtbond
be 2013,
the forward
price ate.g.,
t, for
S, Chapter
of
a coupon
bearing
expiring
at T,sett
Let
Ft (S,
Bt (T)price at t of the underlying bond.
(T)
is
the
at
S
≥
t,
and
B
t
with t ≤ S ≤ T, i.e. Ft (S, T) = Pt (S) , where Pt (S) is the price at t of a zero coupon bond expiring
(S,T)
T)
dFτ (S,
(S,
T) process and let Q be the risk-ne
dFττ (S,
dF
dF
is the
price
t(S,
of the
underlying
at S ≥Let
t, and
instantaneous
rate
rtBbe
t (T)
τthe
= vat
vτ (S,
(τbond.
)T)
, =
τ∈
(t,
S)
T)short-term
·dW
dW
vττ(t,
(τ)),, ττ ∈∈(t,
(t,SS
(S,
T)
dWFFSS (τ
FSS (τ
v∈
(S,
T)
=
)
,
τ=
S)
,, ··dW
T)
·
τ
F
(S,
T)
F
Fττ (S,
(S,and
T) let
τ
rate process
be the risk-neutral
probability. It
rt be theFinstantaneous
FChapter
T)
T)e.g.,short-term
τ (S,
isLet
well-known
(see,
Mele, 2013,
12)Qthat
in a diffusion
setting, Ft (
QUANT PERSPECTIVES
is well-known (see, e.g., Mele, 2013, Chapter 12) that in a diffusion setting, Ft (S, T) satisfies
ter 4) provide further details on how one
may
(S,T)
T) isisthese
the instantaneous
instantaneous
volatility
process
adaptedvolatility
to W
WFSS (τ
(τ
multidimensional
where
to
WFs v(!
S- to
(S,
T) isis the
theprocess
instantaneous
volatility
process
adapted
to W
WFFSBr
(
where
vττ (S,
SB
T)
instantaneous
process
adapted
(τ
where
(S,
the
volatility
adapted
to
),), aa multidimensional
where
vvττestimate
F
dFτ (S, T)dFτ (S, T)
approximation errors. For example, a nian
numerical
experiment
=
v
(τ
)
,
τ
∈
(t,
S)
,
(S,
T)
·
dW
s
S
=
v
(S,
T)
·
dW
(τ
)
,
τ
∈
(t,
S)
,
(1)
forward
probability,
Q
through the
the
Radon-Nikodym
deriva
motion under
under the
the S-forward
S-forward
probability,
S , defined
τ S-forward
FQ
FQ
th
nian motion
motion
under
the
probability,
QRadon-Nikodym
F τ probability,
S , defined through
FS-forward
the
nian
the
, defined
through
derivat
nian motion
probability,
Fτunder
(S, T) F
FFS , defined through
τQ(S,
F S T)
based on Vasicek (1977) model of short-term
interest rate dy- derivative,
as follows:
as follows:as follows:
S
that (1) the early exercise premium
as follows:
!
! SS
!!!), a multidimensional
process
to WF S (τ
where vτ (S, T) is the instantaneous volatility
−! StSadapted
rτ dτ
dτ Browrrττdτ
dQFSS!!!!
e−volatility
dQFFSS!adapted
e−−tott W
process
where vτ (S, T) is the instantaneous
S (τ ), a mu
dQ
e
t rτ dτ
Fderivative,
dQ
e
!
=
F !!
nian motion
probability,
QF S , defined ,through the! Radon-Nikodym
=
,,
=
embedded in American
op- under the S-forward dQ
(2)
,
=
!
!
P
(S)
dQ
(S) the
, defined
through
Radonmotion under the S-forward
QFdQ
S
PPtt(S)
as nian
follows:
dQ !GGSS ! probability,
Ptt(S)
GGSS
!S
dQF S !!
e− t rτ dτ
as
follows:
denotes the
the information
information
set
at time
time
S.
and GGS denotes
(2)
= set
denotes
theS.
information
set
at
time
S.!
and GGSSset
!, time
denotes
the
information
and
denotes
the
information
atat
time
S. S.
at
and
dQ !GS
Ptset
(S)
− tS rτ dτ
S
!
as follows:
!
!
dQ
e
F
Considerin
the
following payoff
payoff
of aa variance
variance
swap:
maturity mismatch is negligible when the difference
maturiConsider
the
following
payoff of
of aa!! variance
variance
swap:,
=
Consider
the
swap:
Consider
the
following
of
swap:
information
setfollowing
at time
S. payoffdQ
and
GS denotes the
Pt (S)
GS
ties is as small as two months, as in the example above. Consider the following payoff of a variance swap:
S
(T,T)
T)
≡information
(T,S,
S,T)
T)−
−PPππat
(t,
T,T)
S,≡
T)VV
≤T)
S,−
(T,
S,
T)
−PP(t,
(t,T,
T,S,
S,T)
T),, TT ≤
≤
(T,
T)
≡
the≡
set
time
S.
and GS denotes
(T,
(t,
T,
S,
T)
,,tt(T,
TTS,
≤
S,
ππ(T,
VVtt(T,
π (T, T) ≡ Vt (T, S, T) − P (t, T, S, T) , T ≤ S,
Maturity and Numéraires Mismatches
Consider the following payoff
swap:
""TT of a variance
2
""TT ∥vwhere
22 dτ is is
"
To illustrate the maturity mismatch issue,where
suppose
that
available
2
where
V
(T,
S,
T)
≡
(S,
T)∥
dτ
is
the
percentage
integrated
variance,
and the
the
fair value
value
o
the
percentage
in-fair
where
V
(T,
S,
T)
≡
∥v
(S,
T)∥
the
percentage
integrated
vari
T
t
τ
2
t
τ
where
V
(T,
S,
T)
≡
∥v
(S,
T)∥
dτ
is the
percentage
integrated
varia
t
t dτ is the percentage
τ
Vt (T, S, T)
≡ Vtt(T,∥vS,τT)
(S,
integrated
variance,
and
of
t
where
≡ T)∥
t ∥vτ (S, T)∥ dτ is tthe percentage integrated variance, and the fair value of the
for trading are European-style options expiring
at
T
on
a
10tegrated
variance,
of the strike (t,T,S, ) isT ≤ S,
strike PP(t,
(t,T,
T,T)
T)
strike
(t,
T,T)
T) isisand
π (T,the
T)fair
≡ Vvalue
strike
T)
is PP(t,
T,
t (T, S, T) − P (t, T, S, T) ,
strike
isis P (t, T,strike
year Treasury note forward expiring at S T. The option span
#
$
$$
" T##1 ""TT − "tT 2rτ dτ
""TT
$$ ## QQ
1
operates under the so-called T-forward probability, whereas
the
1
−
r
dτ
E
e
P
(t,
T,
S,
T)
=
(T,
S,
T)
=−
E−Q
(T,
S, T)) , (3)
(3) Q
V
T
1
r
dτ
t
t
tintegrated
τdτ Vdτ(T,
t FtTT(V
1 ≡ EttPt ∥v
τ
where
VtT)
(T,=S, T)
(S,
T)∥
is
the
percentage
variance,
and
r
dτ
t
FFT
τ
τ
−
r
e
S,
T)
=
E
(V
(T,
S,
T))
,
P
(t,
T,
S,
(T
)
(T,T))
S,T)
T) =
=EEQ
(t,tT,
T,S,
T)
τS,T)
t=
EEtt=eeEtt Ft (Vtt(T,
S,
VVtt(T,
eP(t,
(T,PS,
T)
S,
,
Vt=
P (t, T, -S, T) = Pt (T )Et P
tt
(T
)
Ptt(T )
strike P (t, T, P
T)t (T
is )
the second equality follows by a change of probability, Et denotes conditional expectation under
ability, which generates risks in the forward price that where
cannot
Q
$ the
wherethe
the
second
follows
bysecond
change
of
probability,
denotes
conditional
expectation
under
the
T#
-forward
probability.
That
is,
fair
of uc
the equality
risk-neutral
probability,
Etequality
"EETatadenotes
where
the
second
equality
follows
by
change
of
probability,
denotes
t denotes
be spanned by the set of all available options.
Unless
T=S,
we where
the
change
of
probability,
EEvalue
conditional
expectation
under
the risk-neutral
probwhere
second
equality
follows
by
aaand
change
of
probability,
conditional
expectation
un
t r
1 follows −by
QFtT
t denotes
QF T
dτ
Q
τ
TT
Q
the
variance
swap
is
the
expected
realized
variance
under
the
-forward
probability.
It
is
a
notable
F
Q
t
T
T
Q
E
e
(T,
S,
T)
=
E
(V
(T,
S
V
P
(t,
T,
T)
=
T S,
Fprobability.
under
the
T
-forward
That
is,
the
fair
valu
the
risk-neutral
probability,
and
E
F
t
t
t
F
under
the
T
-forward
probabili
the
risk-neutral
probability,
and
E
t
ability,
andtt t under
under
probability.
is,is,the
under theThat
T -forward
probabilit
the
risk-neutral
probability,
and Ett probability.
the
TT-forward
the
the risk-neutralpoint
probability,
)-forward
Pthe
t (Tcase
of departureand
fromEthe
standard equity
in which the fair value ofThat
a variance
swapfair
is thevalu
This issue is reminiscent of convexity
problems
arising
in
the
variance
swap
is
the
expected
realized
variance
under
the
T
-forward
probability.
It
is
a
not
fair
of
the
variance
swap
is the
expected
realized
variance
the value
variance
swap
therates
expected
realized
variance
under the
the
-forwar
the
variance
swap
isis the
expected
realized
variance
under
the variance swap
is the expectation,
expected
realized
variance
under
the
T -forward
probability.
ItTTis-forwar
a nota
risk-neutral
assuming
interest
are constant.
under
the
T-forward
probability.
It
is
a
notable
point
of
deparwhere
second
equality
follows
by
a
change
of
probability,
E
denotes
condition
point of departure
the
standard
equity
case
in
which
the
fair
value
of
a
variance
swap
is
t
point
of
departure
from
the
standard
equity
case
in
which
the
fair val
va
Forfrom
thethe
case
at
hand,
we
obviously
cannot
assume
constant
interest
rates.
Moreover,
we
still
point of departure from the standard equity case in which the fair
FT
FT
point of departure from the standard equity case inQ which
the fair value of a variance swap is
FT
need
evaluate
the RHS
of expectation,
Eq. (3).
Following
theunder
VIXinterest
Methodology,
one
might
try to link the
the Trates
-forward
probability.
That
thetorisk-neutral
probability,
and
Eassuming
risk-neutral
expectation,
assuming
interest
rates
are
constant.
risk-neutral
rates
are
constant.
t
risk-neutral
expectation,
assuming
interest
are
constant.
risk-neutral
assuming interest rates are constant.
spirit of the standard VIX Methodology
cannot be expectation,
calculated
expectation
in
Eq.
(3)
to
the
value
of
a
log-contract
(Neuberger,
1994)—i.e.,
a
contract
with
a
payoff
variance
swap
is
the
risk-neutral
expectation,
assuming
interest
variance
swap
is case
the cannot
expected
realized
variance
underassume
the T -forward
probab
For the
the case
casethe
at hand,
hand,(S,T)
we
obviously
assume
constant
interest
rates.
Moreover,
we
For
the
case
at hand,
hand,
we obviously
obviously
cannot
assume
constant
inter
For
the
at
we
cannot
intere
For
at
we
obviously
cannot
assume
constant
interest
rates.
Moreover,
we
time
T . In the
standard
equity
case,
the value
of a log-contract
isconstant
indeed minus
equal
to ln FFTtrates
at the strictest level of theoretical rigor.
are
constant.
(S,T) at
point
of
departure
from
the
standard
equity
case
in
which
the
fair
value
of
a
need to
to evaluate the
the RHS
of to
Eq.
(3). Following
Following
the
VIX
Methodology,
one
might
try to
to link
link
need
to evaluate
evaluate
the RHS
RHS
ofVIX
Eq. Methodology,
(3). Following
Followingone
themight
VIX Methodolog
Methodolo
need
the
of
Eq.
(3).
the
VIX
of
try
The silver lining is that, as long as Tneed
and S-Tevaluate
are small, theRHS
ForEq.
the (3).
case at
hand,
wethe
obviously
cannot assume
constant
risk-neutral
assuming
interest
are constant.
expectation
inearly
Eq.
(3) to
toexpectation
theexpectation,
value of
ofin
a log-contract
log-contract
(Neuberger,
1994)—i.e.,
contract with
with
pa
expectation
Eq.
(3) to
to the
the
valuerates
of aa1994)—i.e.,
log-contract
(Neuberger,
1994)—
Eq.
(3)
value
of
log-contract
1994)—
numerical impact of both the maturity expectation
mismatch
and
exin
Eq.
(3)
the
value
ain
(Neuberger,
aa(Neuberger,
contract
aa pa
3
FT (S,T)
F
(S,T)
T
F
(S,T)
For
the
case
at
hand,
we
obviously
cannot
assume
constant
interest
rates
T
(S,T)
T . In
equity
thestandard
value of
of aaequity
log-contract
indeed
m
equalto
tobe
lnFignored,
atequity
time TTcase,
. In
In the
the
standard
equity
case, the
the
value of
of
equal
tothe
ln standard
ercise premium is likely to be small enough
time
.case,
case,
value
aa
equal
to
ln
FT (S,T) atastime
the
value
log-contract
isis indeed
m
equal
to
ln
(S,T) at
FFtt(S,T)
Ftt(S,T) at time T . In the standard
need to evaluate the RHS of Eq. (3). Following the VIX Methodology, one m
expectation in Eq. (3) to the value of a log-contract (Neuberger, 1994)—i.e., a c
(S,T)
3 value of a log-cont
to realize that a VIX-like implementation leads to an approxiat time
time T . In the
at
equal to ln FFTt (S,T)
33 standard equity case, 3the
mation of the true fair value of a variance swap, and to know a log-contract is indeed minus one half the expected realized
half the expected
when the approximation is of a tolerableone
magnitude
given the realized variance. In our case, it follows by Eq. (1) and Itô’s lemma that
of a variance swap,
and
to
know
when
the
approximation
is
of
a
tolerable
magnitude given the
!
"
context.
one half the expected realized variance.
one half
Inthe
our
oneexpected
case,
half the
it follows
realized
expected
realized
Eq. (1) and
variance.
In our
Itô’s
case,
lemma
In it
ourfo
3 byvariance.
FT (S, T)
QF S
QF S
xt.
f a variance swap, and
when
approximation
of delivery
aone
tolerable
magnitude
the
(4)
Let to
Ft know
(S,T) be
thethe
forward
price at t,isfor
S,
of
halfatthe
expected
realized
variance.
In
our
case,
it
follows
by
Eq.
(1)
and
Itô’s
lemm
Egiven
(V
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t
t
t
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! (S, T)
F
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atF ST,
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S, T))
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Bt (T)
F
(S,
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T
S
S
F
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,
where
P
(S)
is
the
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at
t
of
a
zero
coupon
(S,
T)
=
,
where
(S)
is
the
price
at
t
of
a
zero
coupon
bond
expiring
≤
S
≤
T,
i.e.
F
t
t
t
F
F
t
at S, of a coupon
bearing
bond
expiring
at
T,
Ft (S, T) be the tforward price
Pt (S)at t, for delivery
. the fair value
lnin Eq. (3) is
S, T))
= expectation
−2Et
Et to (V
However, Eq. (4) does not link
Eq.
(3).
The
t (T,
Ft (S, T)
t (T)
bond
expiring
at Sthet,underlying
and Bt (T) isbond.
the priceatatt tofofathe
underlyT,
(T)Fist (S,
theT)
price
at
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≥≤t,Sand
Bti.e.
= B
zero
coupon
bond
expiring
≤ T,
contract
expiring
at
T
,
whereas
the
expectation
on
the
LHS
of
Eq.
(4)
is
the
value
Pt (S) , where Pt (S) is the pricevariance
However, Eq. (4) does not link to
However,
Eq. (3).
However,
Eq.The
(4)expectation
does
Eq. not
(4) does
link
in Eq.
to
notEq.
(3)
link
(3).
istofair
the
Eq.
The
fair(3)
exv
ing
bond.
be
the
instantaneous
short-term
rate
process
and
let
Q
be
the
risk-neutral
probability.
It
tt,rand
t
Bt (T) is the price at t of the underlying bond. variance contract
expiring
at
S.
In
other
words,
the
market
numéraire
for
government
bond
cont
However,
Eq.
(4)
does
not
link
to
Eq.
(3).
The
expectation
in
(3)
is
the
fair
variance
contract
expiring
at
T
,
whereas
variance
the
contract
variance
expectation
expiring
contract
on
the
at
expiring
T
LHS
,
whereas
of
at
Eq.
T
,
the
whereas
(4)
expectation
is
the
the
fair
ex
Let r be theChapter
instantaneous
short-term
rate process
and
let T)
Q satisfies
of a variance contract expiring at S. In other words, the market
-known
(see,
e.g., Mele, t2013,
that inand
a diffusion
Ft (S,maturity
instantaneous
short-term rate12)
process
letexpiring
Q besetting,
the
probability.
ItTprice
rt be the
atrisk-neutral
ae.g.,
given
Y is
of zeros
expiring
at Y,numéraire
where
Y other
=
TS.
in
and
Y
variance
contract
expiring
at
, variance
whereas
the
expectation
on
the
LHS
Eq.
(4)(3)
is words,
the
fai
variance
contract
atthe
S.for
In
other
words,
contract
variance
the expiring
market
contract
at
expiring
S. In
at
for
words,
InEq.
other
the
bond
mark
be the risk-neutral probability. It is well-known
(see,
Mele, expiring
numéraire
government
bond
contracts
expiring
atof
agovernment
given
(S,
T)
satisfies
known (see, e.g., Mele, 2013, Chapter 12) that in a diffusion
setting,
F
t
in Eq.
(4). atcontract
variance
expiring
In other
words,
the
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for
Fexpiring
t (S,T
maturity at
is the
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ofexpiring
expiring
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,iswhere
where
=T
inin
a given maturity
Y S.
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expiring
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azeros
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maturity
a given
at
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maturity
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the
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Eq.expirin
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ofbon
zea
dFτ (S, T)
= vτ (S, T) · dWF S (τ ) , τ ∈To(t,evaluate
S) ,
(1)
the
RHS
of
Eq.
(3),
we
need
something
more
than
Itô’s
lemma.
We
have
to
ackn
S
expiring
at
a
given
maturity
Y
is
the
price
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zeros
expiring
at
Y,
where
Y
=
T
in
Eq.
(3)
Fττ (S,
(S, T)
T)
in Eq. (4).
in Eq. (4).in Eq. (4).
dF
= vτ (S, T) · dWF S (τ ) , edge
τ ∈ (t,
S)Eq.
,the dynamics
under the “variance swap” pricing measure are those under Q
that
in
(4). (1) of FT (S, T)(1)
To evaluate the RHS of Eq. (3), we
Toneed
evaluate
something
Tothe
evaluate
RHS
more
of
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Eq.
RHS
(3),
Itô’s
ofwe
Eq.
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need
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m
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volatility process adapted not
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),
aevaluate
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,
as
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by
Girsanov’s
theorem,
we
have
the
following:
under
Q
S
the
RHS
of
Eq.
(3),
we
need
something
more
than
Itô’s
lemma.
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have
F the dynamics of FT (S, T)
under
(S, T)
under
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T) are
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those
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through
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is thethe
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),Radon-Nikodym
athat
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Browvτ (S, T)under
S edge
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T)
under
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pricing
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are
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dynamics
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T
,
as
in
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Girsanov’s
,
as
theorem,
in
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,
as
(1).
we
in
have
Now,
Eq.
the
(1).
by
following:
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by
Girsano
theore
Qthe
not
not
Q
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FS
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FS
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ows:
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otion under the S-forward probability, !QF S , defined
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T))
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v
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· dW
) , following:
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,
as
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have
the
not
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Q
S−v
τ
τ
τ
τ
F T (τ
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=
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+τv(S,
T)T)
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T (τ
τ (S,
τ (S,
!
FdF
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e Ptt (S)
τ (S,
τ (S,
F S ! GS
where
W
That
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the
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Brownian
motion
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Q
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T)
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T)
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T))
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.
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lem
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a
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a
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under
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T
FT
FFT
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ed variance. In our case, it follows by Eq. (1) and Itô’s lemma that(2001),
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#
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F
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whereas
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!
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alf the expected realized variance.
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variance contract
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not link to Eq. expiring
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1
and
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,
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FY
FtT)
(S, T)price
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Q F S Et
. of zeros expiring at
(4) Y, −where FY =
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S, T))maturity
=Q−2E
T ln
at
given
is
the
T
in
Eq.
(3)
and
Y
=
S
FS t
(K − FT (S, T))
dK +
(FT (S, T) − K)
dK ,
(4)
(Vt (T, S, T)) = −2Et
Et
Ft (S, .T)
K2
K2
wever,
Eq. (4)
not
link on
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Eq. LHS
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in value
Eq. and
(3)
the
fair value
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a QT , T ,
expectations
under
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0
Ft (S,T)
F ,Q
t (S,expectation
t T , whereas
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ofThe
(4) is the fair
ofoftake
ais
and
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expectations
under
expectations
However, Eq.
(4) does
not link to Eq. (3). The expectation in Eq. (3) is the fair valueand
atake
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QF T F F
inexpectation
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eS.
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However,
Eq.
(4) does
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expectation
Eq.
(3)is is
the
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value
a(S,
expiring
at
T , whereas
the
expectation
on
the LHS
Eq. (4)
the
fair
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of a expectations
Invariance
other
words,
the
numéraire
government
bond
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$fair
and
Q%
, (F
owever,
Eq.
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does not
link
Eq. (3).
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1 acknowlln
E
QFTQ
T)
T)) − to
FT$(S,under
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FET %
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t take
To
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RHS
Eq.
(3),
we
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something
more
We
(F(S,
(S,
Q
T)
T))
EE
FTF(S,
=T)tFt (S,
1 T))
lnF TT)Itô’s
Et of
QtFa(S,
T (S,
T−
F
variance
contract
expiring
at
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words,
the of
market
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government
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T
t (FT
variance
contract
expiring
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T
,
whereas
the
expectation
on
the
LHS
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is
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value
=t t(S,T)
ln
E
eY
contract
expiring
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other
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the
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bond
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=F
− 1− 1
ln
E
F
(S,
T)
T)
expected
realized
variance.
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our
case,
it
follows
by
Eq.
(1)
and
Itô’s
lemma
that
nce
contract
expiring
at
T
,
whereas
the
expectation
on
the
LHS
of
Eq.
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is
the
fair
value
of
a
t
!
#
t$ t
is the
price
of
zeros
expiring
at
Y,
where
Y
=
T
in
Eq.
(3)
and
Y
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S
%
Q
F
F
(S,
T)
(S,
T)
!
#
"
"
F
F
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(S,
T)
T
expiring
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given
maturity
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price
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where
Y
=
T
in
Eq.
(3)
and
Y
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S
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the “variance
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pricing
are
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edge
that
dynamics
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FFtt (S,T)
variance contract expiring
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T))
Et 1F (F
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QF T
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##
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1F T ," ∞ ∞
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nce
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other
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government
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g at a in
given
maturity Y is the price of zeros expiring
at Y, where
Y = T Ein
Y=− = S 1 1
ln (3) and
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t Eq.
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Put
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tt
t
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121
xpiring at a given maturity
Y is the
price
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zeros
expiring
at
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Y = T in Eq. (3)
and Y = S t (S, T)the
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2 2 1 1
Eq.
Now,
Girsanov’s
under
Q(3),
S ,zeros
Pt−
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K
KCall
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ng at a given
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=lemma.
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F (S,T)
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=F −2E
Etsomething
F (Vt (T, more
Ft (S,T)
n Eq.(3),
(4). we need
0 0
Ft (S,T)
Eq.
Itô’s
have .to acknowl1
1
t lemma.
swap” pricing
measure
under QF ,
that the dynamics
of FT (S, T) under the “variance
(4). edge
FtItô’s
(S,
T) are those
− at K, the
Put
dKon
+
Callt (K)
dKwe ,have(8)
t (K)
2
and
calls
struck
weofof
have
relied
pricand to
Call
prices
European
andthe
callsfollowing
struck
at K,K
and
where
Put
theorem,
the
following:
To T)
evaluate
the
RHS
ofwe
Eq.have
(3),
we
need
something
morewe
than
lemma.
We lemma.
have
to acknowlt (K)
t (K) denote
Pt (Tthe
) and
K 2putsputs
evaluate
the
RHS
Eq.
(3),
we
need
something
more
We
have
acknowlasof
in
Eq.
(1).
by
Girsanov’s
theorem,
haveare
thethan
following:
notthe
under
Qthe
(S,
T)
dF
0
Fand
t (S,T)calls struck at K, and we have
(S,
under
“variance
swap”
pricing
measure
those
under
Q
,
Fof, Eq.
τNow,
T
oT
evaluate
RHS
(3),
we
need
something
more
than Itô’s
lemma.
WeItô’s
have
to
acknowl(K)
and
Call
(K)
denote
prices
European
where
Put
tF
tand
(K)
and
Call
(K)
denote
the
prices
of
European
puts
and
calls
struck
and
have
where
Put
(K)
Call
(K)
denote
the
prices
of
European
puts
and
calls
struck
at at
K,K,
and
wewe
have
where
Put
t
t
t
t
−vτ (S,swap”
T) (vpricing
− vτare
(S,those
T))
dτ on
+Qthe
vFτT (S,
dWF Tequations:
(τ ) , τ ∈ (t, T ) ,
(5)
relied
following
τ (T, T)
measure
under
, T) · pricing
dge that the dynamics of FT (S, T) under the=“variance
under
the
“variance
swap”
pricing
measure
are
thosemeasure
under
Q
, fair
hat
the dynamics
of FT)
T
T)
under
the
“variance
swap”
pricing
are
those
under
Qdenote
at
the
dynamics
of
Flink
T , equations:
Fto
T)
T (S,
relied
on
the
pricing
equations:
Ffollowing
TT)(S,
dF
τ (S,
F
,not
Eq.
(4)
does
Eq.
(3).
The
expectation
in
Eq.
(3)
is
the
value
of
a
).
Now,
by
Girsanov’s
theorem,
we
have
the
following:
relied
on
the
following
pricing
τ (S,not
relied
on
the
following
pricing
equations:
(K)
and
Call
(K)
the
prices
of
European
puts
and
calls
struck
at
K,
and
we
have
where
Put
t
−vτ (S,
T) (vby
T) − vτ (S, theorem,
T)) dτ + vτ we
(S, T)
· dW
) , τ ∈ (t, T ) ,
(5) t
Now,
Girsanov’s
have
the
following:
under QF S , as in Eq.=(1).
τ (T,
F (τ
&
'
&
'
Callt (K)
Putt (K)
Q
Q
T) Now, by Girsanov’s theorem, we have the following:
Eq.
nder Q
Q SS, ,asasin F
τ (S, (1).
in
Eq.
Now, the
by Girsanov’s
we have
the(4)
following:
der
= pricing
Eof
= Et
− FT (S, T))+ ,
(FT (S, T) − K)+ .
relied
on the
following
equations:
t
tract FFexpiring
at
T ,(1).
whereas
expectationtheorem,
on the LHS
of Eq.
isPut
the
fair
value
aQ(K
&
'
&
&
'
&
' '
P
(T
)
P
(T
)
(K)
(K)
Put
Call
&
'
&
Q
Q
(K)
(K)
Put
Call
Q
t
t
t
t
(K)
(K)
Call
T
T
t
t
+
Q
Q
T
T
+
+
F the forward
t
F F
+ T))T))
+ '+. .
WF T (τ ) is
is,
a multidimensional
Brownian
motionis not
under
QFET .F TThat
F TE
dF
=
(K
−
(F(S,
(S,
T)
− K)
τ (S,
=(K
Et EF−
E
−F
, , t is not
T)−
− K)
K)
where
WT)
)where
is a multidimensional
motion under QF . That
is, the forward
T (S,
T T)
TF(S,
=
E=t =
, price
.
T))
t F(K
F (τ
t t (F(F
T)
dF(v
T (S,
T T(S,
tt (T
τ (S,
=
−v
(S,
T)
(vdτ
T)vBrownian
−(S,
vτ (S,
T))
dτ + vTτ (τ
(S,)T)
· dW
(τ ) T
, government
τ, ∈price
(t, T(5)
)P, (Tbond
(5)=
T (t,
P
P
)
(T
)
τ(S,
τ (T,
P
P
(T
)
(T
)
expiring
at
S.
In
other
words,
the
market
numéraire
for
contracts
&
'
&
'
F
t
,tract
T)
(T,
T)
−
v
T))
+
T)
·
dW
,
τ
∈
)
t
t
(K)
(K)
Put
Call
Q
Q
P
)
(T
)
τ
τ
τ
t
t
=
−v
(S,
T)
(v
(T,
T)
−
v
(S,
T))
dτ
+
v
(S,
T)
·
dW
(τ
)
,
τ
∈
(t,
T
)
,
(5)
T
T
T
+ one, but
(S,(K) −isFTunder
not
t Ft (S, T) is=
τunder
τ Itô’s Q
τ F
(S, T)where
F RHSQ
W
(! thatunder
by
lemma,
aFmartingale
Q
FT
Because
not
QaF+ ,martingale
onunder
Eq.T)(8)
is not
τ T)
(S,
dF
F τF, Tsuch
a martingale
EF)tat Fmartingale
, the firstt term =
Ethe
(FTofF(S,
.
(S, T))
− K)
(
Fττ (S,
T)
t
F T , such that by Itô’s lemma,
Q
Pt (7).
(T
)(t, T
(T ) Utilizing this expression and combining
ℓ̃(t,T,S,T)
T)the
(v
(T,ofT)zeros
−price
vτexpiring
(S,isT))
(S, T)Y
·under
dW
(τBecause
) , Et (3)
τPFt ∈
) not
,is, where
(5)
τ forward
a given
maturity
Yτ (S,
is the
price
Y,
= TF Tin
Eq.
Y
Sa )martingale
"
!vτwhere
eand
ℓ̃ (t, T, S, T)
isunder
as Q
in Q
Eq.
equals
Q=T−v
. That
is,
notdτat
aQQ +
martingale
Q
T)
not
, the
first
term
RHS
is not
one,
(S,
T)
is =
first
term
onon
thethe
RHS
of of
Eq.Eq.
(8)(8)
is not
one,
butbut
t((S,
F T ,Because
tF
F T F, Tthe
FT (S,
Fτ (S,
Q
Q
) a martingale
where
W T)
(τ ) isF a multidimensional
under
That
is,T)the
forward
is not
"under
!
(S,
T)
is
not
a(
martingale
under
Q
Because
F
T , the first term on the RHS of Eq. (8) is not one, but
Q
F T . ln
tprice
QFEq.
,
(6)
(Vt (T,Brownian
S, T))Brownian
= 2Emotion
(ℓ̃motion
(t,under
T, S, T))
2E
Et
F
T F T)
(
ℓ̃(t,T,S,T)
WF T (τ F) Tis a multidimensional
Q−
.
That
is,
the
forward
price
is
not
Eq.
(8)
with
(6),
we
find
that
the
fair
value
of
the
variance
swap
in
Eq. (3)
is,
ℓ̃(t,T,S,T)
T
t
t
F
is
as
in
Eq.
(7).
Utilizing
this
expression
and
combining
e
,
where
ℓ̃
(t,
T,
S,
equals
E
E
e
,
where
ℓ̃
(t,
T,
S,
T)
is
as
in
Eq.
(7).
Utilizing
this
expression
and
combining
equals
Ft (S,
T)
(S,
T)
is
not
a
martingale
under
Q
the
first
term
on
the
RHS
of
Eq.
(8)
is
not
one,
but
Because
F
T , T)
F
(S,
T)
Q
Q
Q
Q
t
t
mensional
Brownian
motion
under
Q
.
That
is,
the
forward
price
is
not
T
T
F
t
T
T
T
T
(
)
F
FF
F, where ℓ̃ (t, T, S, T) is as in Eq. (7). Utilizing this expression and combining
a martingale under QF T , such that by Itô’s
eQℓ̃(t,T,S,T)
EtS,
,fair
(6)
ln
(Vt (T, S, T)) = 2Et F equals
(ℓ̃ (t, T,
T))
Etlemma,
T 2E
(the
))
tingale under QF T , such that by Itô’s lemma,
ℓ̃(t,T,S,T)
F−
tEq.
Eq.
(8)
Eq.
we
the
of
variance
swap
ewith
,(6),
where
ℓ̃find
(t,Q
T,that
S,
T)
is fair
asvalue
invalue
Eq.of (7).
Utilizing
this
expression
and
combining
equals
Et(8)
Eq.
with
(6),
we(is
find
that
thethe
variance
swap
in in
Eq.Eq.
(3)(3)
is, is,
W
That
is,
the
forward
not
(τwhere
)RHS
is we
aItô’s
multidimensional
Brownian
motion
QF T .lemma.
Tthe
(S,
have
defined
ate
of Eq.
(3), we need
something
moreunder
than
We
have
to
acknowleℓ̃(t,T,S,T)
− ℓ̃the
(t, T,variance
S, T)
P
(t,
T,
S,price
T) = that
2F
1t−
Et T)
F that
! Itô’s
"
ch
by
lemma,
(8)Eq.
with
Eq.
(6),
we
find
the
fair
value
of
swap
in
Eq.
(3)
is,
"Eq.
!Q T
(S,
T)
F
(8)
with
Eq.
(6),
we
find
that
the
fair
value
of
the
variance
swap
in
Eq.
(3)
is,
QF T
QF#T
))
(
(
))
(
(
T
!"
#
F F (S, T)
"
Q
Q
Q
Q
T
Q
E
(V
(T,
S,
T))
=
2E
(
ℓ̃
(t,
T,
S,
T))
−
2E
ln
,
(6)
T
T
T
T
T
(6)those
such
lemma,
under
t T) that
(S,T)
T ,(S,
Fby
F
tF
t Itô’s
t pricing
the
“variance
swap”
measure
are
under
,2 =
engale
dynamics
F
T
ℓ̃ (t,
eℓ̃(t,T,S,T)
P (t,
T,
−
EFtT F eℓ̃(t,T,S,T)
1 − ℓ̃−(t,
1
,T)
(6)
(V
S,
T))
=under
T))
Et F ofQ
"(S,dτ.
!−
T,∞T,
S, S,
T)T)
P (t,
T, Q
S,(FS,
T)TT)
=
2 21 −1FE
t (T,
Ft (S,
T)
(vτ2E
(T,t T) − vln
(7)
ℓ̃ (t,
T,2E
S,tT) ≡(ℓ̃−(t, T,vτS,(S,
τ (S,
+ ( QFQTF T((tℓ̃(t,T,S,T)
Putt (K) 2 dK + ))))Callt (K) 2 dK .
FtT))
T)
ℓ̃(t,T,S,T)
(S,
T)
F
#
QF Twhere we have defined
!e"!
# (9)
t QF T
T
P
(T
)
K
K
P
(t,
T,
S,
T)
=
2
1
−
E
e
−
ℓ̃
(t,
T,
S,
T)
"
"
t
−
ℓ̃
(t,
T,
S,
T)
P
(t,
T,
S,
T)
=
2
1
−
E
"
0
F
(S,T)
"
!
as in
Now,
byT))
Girsanov’s
we have
S ,T))
Ft (S,T)
Ft (S,T)
∞∞
t 2t 2
, the following:
(6)
ln
,FS,
= Eq.
2Et (1).
(ℓ̃ (t,
T, S,
− 2Et theorem,
1
1 1
1(9)
where
we haveQdefined
!
#
(S,
T)
F
#
Q
Q
#. .
T to the
" dK
Ft (S, term
T) (t, T,TFS,TT) in addition
T
+" " Ft (S,T)
Put
Call
dK
+!
Put
Call
(9)(9)
we have defined
t (K) 2 dK
t (K) 2 dK
F T expressions in Eqs. (4) and (6) F
t (K)
t (K)
The
"+∞+
, a formulation
ln Accordingly,
(Vt (T, S, T)) = #2Et differ
(ℓ̃by(t,theT,“tilting”
S, T)) − ℓ̃2E
Et two
11 K
1
P
)of0government
K2 ∞
t)(S,T)
P(6)
(TF
K K2
t (T
of2 an
bond
is
t index
0
Ft (S,T)
FCall
t (S,T)
Put
dKvolatility
+ (7) index
dK 1 .
(9)
+2T))
t (K)
t (K)
two
The F
tilting
T)
(vterm
(T, T) − v+τ (S,
dτ.
(t, T,
S,different
T) ≡ probabilities.
− t vτ (S,
# TRHS Tare takenℓ̃under
2
2
t (S,
τ T)
(S, T)fact that the expectations on the
Put
(K)
dK
+
Call
(K)
dK
.
(9)
P
(T
)
K
K
t
t
t
0
Ft (S,T)
vτ(S,
(S,T))
T) (v
vτ (S,
dτ. F T under
ℓ̃impact
(t,(v
T, S,
T) T)
≡ −− vτmismatch:
* K2
τ (T,
= −vτ ℓ̃(S,
T)
(T,
dτ
+T)
(S,
· tdW
(τ ) ,QFAccordingly,
τand
∈ (t,(7)T ) a, formulation
(5)
Pt (T
)
K2
encapsulates
theT,
the
the
forward
aT))
martingale
τ−price
Ft (S,T)
vτ t(S, T)
(vτ (T,
T)
−
vτv(S,
T))isT)
dτ.
(7)
(t,
S, T)τof≡
− maturity
Accordingly,
a formulation
index
government
bond
volatility
index
of0of
anan
index
of of
government
bond
volatility
index
is is
1
(7)
(S, T) yet we are insisting
P (t, T, S, T)
GB-VI(t, T, S, T) ≡
in evaluatingt the expectation of its realized variance under Q , motivated as we
we have defined
a formulation of
index of government
bond
index is
T−
t volatility
# T for the fair value of the original variance swap P (t, T, S, T) in FEq. (3). Accordingly,
Accordingly,
a an
formulation
of an
index
of government
bond
**
are in
our search
We
expect
Accordingly,
a
formulation
of
an
index
of
government
bond
volatility
index is
expressions
(4)
and
the
“tilting”
term
ℓ̃
(t,
T,
S,
T)
in
addition
to1 the
The two
expressions
inThe
Eqs.two
(4) and
(6) differ byin
theEqs.
“tilting”
term
ℓ̃ (6)
(t, T,differ
S, T) inby
addition
to the
1
he
two
expressions
in
Eqs.
(4)
and
(6)
differ
by
the
“tilting”
term
ℓ̃
(t,
T,
S,
T)
in
addition
to
the
*T)
volatility
is
ℓ̃ (t,
T,−S, T) to vbe
zero
only
when
T =T)
S.# −
.
That
is,
the
forward
price
is
not
(τ
)
is
a
multidimensional
Brownian
motion
under
Q
T
(S,
T)
(v
(T,
v
(S,
T))
dτ.
(7)
T,
S,
T)
≡
P
(t,
T,
S,
T)
GB-VI(t,
T,
S,
≡
P
(t,
T,
S,
T)
GB-VI(t,
T,
S,
T)
≡
T
τ
τ
τ
F probabilities. Thewhere
P (t,
T, S, T) is as in Eq. (9).
act that the expectations
on thethe
RHSexpectations
are taken underon
twothe
different
tilting
term
1 Tterm
T T,
−
t− tS, T)
that
RHS
arev taken
under
two
different probabilities.
tilting
hat the expectations
the
RHST)
are≡taken
under
two T)
different
probabilities.
The tilting
term
t fact
P (t,
GB-VI(t,
T) ≡ *
−mismatch:
vτto
(S,
(vτ that
(T,
dτ.
(7) T, S,The
ℓ̃on(t,of
T,
term
in addition
the
fact
the
on
τ (S, T))
such
that
by Itô’s
lemma,
encapsulates
underSpanning
QFthe
T −1t
T ,impact
theS,maturity
the
forward
priceT)
is −
a expectations
martingale
under
QF S and
sulates the impact ofencapsulates
the maturity mismatch:
is a martingale
under Q
t the forward
F S and
the impact
of theprice
maturity
mismatch:
the
forward
price
isisGB-VI(t,
aas
martingale
(t, T, S, T)
T,
S, T)under
≡5 QF S Pand
where
P (t,
(9).
where
(t,
T, T,
S, S,
T)T)
is as
in in
Eq.Eq.
(9).
asP we
yet we are insisting in evaluating the expectation of its realized variance under
QF T , motivated
T −t
"
!Qaddition
asItPwe
eqs.
are insisting
evaluating
theasexpectation
realized
variance
We in
are
notdiffer
done yet,
we still
needoftoits
derive
the
value
ofS,
theunder
log-contract
in Eq. where
(6).the
is(t, T, S, T) is as in Eq. (9).
(4) and
(6)
by
the
“tilting”
term
ℓ̃
(t,
T,
T)
in
to
F T , motivated
yet
we value
are
insisting
in evaluating
the
expectation
ofKamal
its
realized
(S,
T)
F
Q
Qthe
are in our
for
the
fair
of
original
variance
swap
PQ
(t,
T,
T)Derman,
inTEq.
(3).
We
expect variance under QF T , motivated as we
encapsulates
the
impact
ofswap
the
maturity
the
T term
T S, mismatch:
Fsearch
F T “spanning”
FT)
natural
atthe
this
juncture
toof
onoriginal
the
approach,
by
Demeterfi,
and
for
fair
the
P
(t,
T, S,
inln
Eq.
(3).ℓ̃ (t,
Weterm
expect
e˜nour
twosearch
in
Eqs.
(4)
and
(6)
by
“tilting”
term
T,
S,
, T) in addition
(6)to the
(Vare
(T,
S,value
T))
=rely
2E
ℓ̃ variance
(t,differ
T, S,
T))the
−led
2E
Eexpressions
taken
under
two
different
probabilities.
The
tilting
t S. (
t
5
(t,the
T, S,tRHS
T) to forward
bet zero
only
when
T
=
T, (9).
S, T) in Eq. (3).5 We5 expect
are
in
our
search
for
the
fair
value
of
the
original
variance
swap
F
where
(S,
T)
S
where
P
(t,
T,
S,
is asPin(t,
Eq.
price
, S, T) to be zero only when
T =isS.a martingale under Q F and yet we tare insist- T)
at
the
expectations
on
the
RHS
are
taken
under
two
different
probabilities.
The
tilting
term
4
he maturitying
mismatch:
the
forward
price
is
a
martingale
under
Q
S and
F
inℓ̃ (t,
evaluating
expectation
its realized
T, S, T) the
to be
zero only of
when
T = S. variance under
Spanning
ulates
theexpectation
impact
of the
maturity
mismatch:
the forward
price
a martingale
under
QF S and
ve defined
, motivated
as of
we Time
ating
the
of its
realized
variance
Qfor
ning
Deposits
5
Q FT, motivated
as
we are
in our under
search
fair isvalue
F T the
We
are
not
done
yet,
as
we
still
need
to
derive
the
value
of
the
log-contract
in
Eq.
(6).
It
is
,
motivated
as
we
are
insisting
in
evaluating
the
expectation
of
its
realized
variance
under
Q
T
#
F
Time
deposit
variance
contracts
share an interesting feature
P
(t,
T,
S,
T)
in
Eq.
(3).
We
expect
re value
theyet,
original
variance
are notof
done
as we still
need toT swap
derive the
value
of
the
log-contract
in
Eq.
(6).
It
is
Time
Deposits
Spanning
natural
at thisfor
to rely
the
approach,
led byswap
and (3). We expect
our
fair
of“spanning”
the
original
PT))
(t,Derman,
T, Kamal
S, T)Kamal
in
al
at search
this
to
onvalue
the
“spanning”
approach,
led
Demeterfi,
andEq.
T)
≡onbe
−
vτonly
(S,
T)
(vτvariance
(T,byT)
− vDemeterfi,
dτ.
(7) bonds: they can be priced based on the same
ℓ̃juncture
(t,the
T,rely
S,
with
government
to
zero
when
T=S.
τ (S,Derman,
when
T juncture
= S.
t done yet, as we still need to derive the value of the log-contract in Eq. (6). It is
are Tnot
change-of-numéraire
set forth infeature
the previous
A pointbonds: they can
S, T) to be zero onlyWe
when
= S.
4
Time deposit variance contracts
share an interesting
with section.
government
4
natural at this juncture
to rely on the “spanning” approach,
led
by Demeterfi,
Derman,
Kamaltime
anddeposit volatility in
of
departure
arises
when
expressing
Spanning
expressions in Eqs. (4) and (6) differ by the “tilting”
term ℓ̃on
(t,the
T, S,same
T) in change-of-numéraire
addition to the
priced based
set forth
inasthe
previous
section.
A point of depart
terms of basis point volatility
of rates,
opposed
to the
more faing
e we
expectations
on
the
RHS
are
taken
under
two
different
probabilities.
The
tilting
term
still need to derive the value of the log-contract
in Eq.
(6).
is miliar
arises when
expressing
deposit
volatility
in terms
of basis
point volatility
of rates, as oppose
notion
of percentage
volatility
of prices.
To accommo4 It time
and
sely
theon
impact
ofyet,
the maturity
the
forward
priceofisthe
a Kamal
martingale
under
Q
S (6).
are
not
done
as we
stillmismatch:
need
the value
log-contract
in
Eq.
It
is
F
the
“spanning”
approach,
ledtobyderive
Demeterfi,
Derman,
and
this practice,
we needoftoprices.
consider
arguments
dif-practice, we nee
the more familiar notion ofdate
percentage
volatility
Tospanning
accommodate
this
as those
we and
nsisting
evaluating
the on
expectation
of its realized
variance
QF T , motivated
at this in
juncture
to rely
the “spanning”
approach,
led byunder
Demeterfi,
Derman,
Kamal
ferent from
in the previous section.
consider spanning arguments different from those in the previous section.
earch for the fair value4 of the original variance swap P (t, T, S, T) in Eq. (3). We expect
to be zero This
only approach
when T =links
S. the value of 4the log- (and other) contracts The Underlying Risks
QUANT PERSPECTIVES
t
FS
FS
t
T
FT
FT
t
t
T
S
T
FT
T
FT
T
T
T
FT
FT
FT
FT
FT
t
t
S
T
The Underlying Risks Let lt
for the time period from t to t
style options.
be the
simplytion,
compounded
rate on a deposit for the time period from t to t+
Let
l (∆)
we refer to interest
lt
To apply these spanning arguments to
our
context,
consider
As a non-limitative illustration, we refer to lt (∆) as the LIBOR.
t, one
Define a forward contract as one where at time t, one party agrees to pay a counterparty a pa
100×(1–l
S
t (S,S
at time S. The forward LIBOR price, Zt (S, S + ∆
equal to 100 × (1 − l (∆)) − Z (S, S + ∆)
Z (S,S
agreed at time t such that trage
in the absence of arbitrage
Zou (1999); Bakshi and Madan (2000); Britten-Jones and Neuberger (2000); and Carr and Madan
Zou
(1999);
Bakshi
andThis
Madan
(2000);links
Britten-Jones
and
and Carr to
andthat
Madan
(2001),
among
others.
approach
the value of
theNeuberger
log- (and (2000);
other) contracts
of a
(2001),
others. This approach
links the value options.
of the log- (andt other) contracts to that of a
portfolioamong
of out-of-the-money
(OTM) European-style
not done
yet,
wespanning
still need
the
valuea Taylor’s
of theexpansion
log-contract
in Eq. (6). It is
portfolio
ofas
out-of-the-money
(OTM)to
European-style
options.
To
apply
these
arguments
toderive
our context,
consider
with remainder,
To apply these spanning arguments to our context, consider a Taylor’s expansion with remainder,
his juncture
to
rely
on
the
“spanning”
approach,
led
by
Demeterfi,
Derman,
Kamal and
FT (S, T) − Ft (S, T)
FT (S, T)
=
ln
−F
FT (S,FT)
FFTt (S, T)
T)t (S, T)
t (S,
= !"
ln
s
#
" ∞
Ft (S, T)
Ft (S, T)
Ft (S,T)
t
S
1
1
− !" Ft (S,T) (K − FT (S, T))
4++ K12 dK + " ∞ (FT (S, T) − K)++ K12 dK# ,
t
0
−
(K − FT (S, T))
dK + Ft (S,T) (FT (S, T) − K)
dK ,
K2
K2
0
Ft (S,T)
and take expectations under QF T ,
FT
and take expectations under QF T ,
%
$
Q T
Et F (FT (S, T))
FT (S, T)
QF T
t
$ ln
% = QF T
−1
Et
FFT (S, T)
E Ft(F
Q T
(S,
T) T))
T (S,
−
1
Et F
ln t
= t
!"
#
" ∞
Ft (S, T)
(S, T)
Ft (S,T)
1 Ft !
1
1
" Ft (S,T) Putt (K) 2 dK + " ∞
−
Callt (K) 2 dK# ,
(8)
t
1
1
Pt 1(T )
K
K
0
Q Fs
−
Putt (K) 2 dK + Ftt (S,T) Callt (K) 2 dK ,
(8)
Pt (T )
K
K
t
0
Ft (S,T)
S puts and calls S
struck at K, and we havet
where Putt (K) and Callt (K) denote the prices of European
and Call
the prices of European puts and calls struck at K, and we have
where
Putthe
t (K)
t (K) denote
relied on
following
pricing
equations:
relied on the following pricing equations:
'
'
Putt (K)
Callt (K)
Q T &
Q T &
= Et F & (K − FT (S, T))+' ,
= Et F & (FT (S, T) − K)+' .
(K)
(K)
Put
Call
QF T
QF T
Pt t(T
Pt t(T
)
)
+
+
= Et
= Et
(K − FT (S, T)) ,
(FT (S, T) − K) .
Pt (T )
Pt (T )
τ
T) is not)a martingale under QF T , the first term on the RHS of Eq. (8) is not one, but
Because Ft (S,
(
QF T
ℓ̃(t,T,S,T)
under
first(7).
term
on the RHS
of Eq. (8) and
is not
one, but τ
Because
where ℓ̃ (t, T,
S, T)QisF Tas, the
in Eq.
Utilizing
this expression
combining
equals EtFt (S,
( eT) is not)a, martingale
QF T
eℓ̃(t,T,S,T)
where
ℓ̃ (t,the
T, S,
is as of
in the
Eq. variance
(7). Utilizing
thisEq.
expression
equals
Eq. (8)Ewith
Eq.
(6), we, find
that
fairT)value
swap in
(3) is, and combining
t
and take expectations under Q ,
(10)
Z (S, S + ∆) = 100 × (1 − ft (S, S + ∆)) ,
www.garp.org
(
(8)
Q S
where fLIBOR,
(S,S
where f (S, S + ∆) is the forward
which satisfies: ft (S, S + ∆) = Et F (lS (∆)). Beca
f (S,S
(lS
ls
fs (S,S
ft (S,S
l (∆) = f (S, S + ∆), f (S, S + ∆)t is a martingale under QF S . Therefore, assuming that
information in this market is driven by Brownian motions, the forward price, Zt (S, S + ∆), satis
the following:
dZ (S, S + ∆)
M A∆)
RC
H F S2 (τ
0 1) ,4 RISK
PROFESSIONAL
3
= vτz (S,
dW
τ ∈ (t,
S) ,
(
Z (S, S + ∆)
z
er spanning arguments different from those in the previous section.
t
We define the basis point LIBOR integrated
rate-variance as,
such that, by arguments similar to those leading
Section 2, "the fair value of the time
! T to Eq. (3) in "
ider spanning Risks
arguments different from those in the previous section.
2
f,bp
Underlying
f 2,following:
!
suchdeposit
that, by
arguments swap
similar
those
leading
Eq.
(3)
in
Section
the"fair value of the time
"
"2the
T
rate-variance
generated
at
t,
and
paying
offSat
, is"
Vto
(T,
S,
∆)
≡ to
fτ2 (S,
+T
∆)
"v
t
"
" τ (S, ∆)" dτ,
f,bp
2
f
Vt generated
(T, S, ∆) at
≡ t, and
fτ (S,
St + off
∆) at
(S,
∆)
dτ,
"v
"
deposit
rate-variance
swap
paying
T
,
is
the
following:
τ
Underlying Risks
t
f,bp
simply compounded interest rate on a deposit for the time period
to t+∆.Vsimilar
etUnderlying
lt (∆) be the Risks
(T,toS,those
∆) − leading
Pbp
S,Eq.
∆) , (3)Tin≤Section
S,
t
f (t, T,to
such from
that, byt arguments
2, the fair value of the time
f,bp
V
(T,
S,
∆)
− Pbp
(t,
T, S,
T ≤ S,2, the fair value of the time
such that,deposit
by arguments
similarswap
leading
to
Eq.
(3)∆)
in, Section
tto those
f
rate-variance
generated
at
t,
and
paying
off
at
T
,
is
the following:
compounded
rate
deposit for the time period from t to t+∆.
tnon-limitative
lt (∆) be the simply
as on
theaLIBOR.
illustration,
we referinterest
to lt (∆)
is rate-variance swap generated at t, and paying off at
deposit
$
# T , is the following:
Q
be the
simply compounded
rate
onLIBOR.
a deposit
for to
thepay
time
period from tatopayoff
t+∆.
Let
lt (∆)
f,bp
isa counterparty
bp
non-limitative
illustration,
tointerest
lat
T, S,
∆)
(T,∆)
S,$,∆) T . ≤ S,
(13)
Pbp
efine
a forward
contract
aswe
onerefer
where
timeas
t,the
one
party
agrees
t (∆)
t #P V
t T, S,
f (t,
Vtf,bp
(T,
S,=
∆)E−
(t,
Q
f
bp
f,bp
(t, T,
=bpE(t,
Vt∆) , (T,TS,≤∆)
(13)
Pf (T,
Vtf,bp
S, S,
∆)∆)
−P
S, .
t T, S,
(∆)
as
the
LIBOR.
non-limitative
illustration,
we
refer
to
l
t
f
fine
a forward
time
t, one
partyforward
agrees to
pay a price,
counterparty
a+payoff
S + ∆)atat
time
S. The
LIBOR
∆), is The first is the same maturity mismatch arising in the government
to 100
× (1 − contract
lS (∆)) −asZone
We Z
face
twoScomplications.
t (S,where
t (S,
is two complications. The first is the same maturity
# mismatch
$ ofinbasis
We
face
arising
the point
government
Define
a×forward
as(S,
one
where
at
time S.
t, one
party
agrees
to pay
abond
counterparty
a∆),
payoff
case. The second
complication
is that we areQdealing
with a notion
variance,
f,bp
is
(∆))
−Z
S
+
∆)
at
time
The
forward
LIBOR
price,
Z
(S,
S
+
is Pisbp
(1
− lS contract
dtoat100
time
t such
that in
the
absence
of
arbitrage
t
tThe second complication
$a(T,
#=dealing
T, we
S,
∆)
Et
V
S, ∆) of. basis point variance,
(13)
t case.
f (t,
bond
case.necessitates
that
are
with
notion
Q the
bp treatment
f,bp
which
a different
from
percentage
(t,
T,
S,
∆)
=
E
(T,
S,
∆)
.
(13)
V
P
− Zabsence
∆)arbitrage
at time S. The forward LIBORwhich
price,
Zt (S,a S
+ ∆),
is from the
al at
totime
100 ×
(1 − that
lS (∆))
t
t
f
t (S, S + of
d
t such
in the
necessitates
different
percentage
case.
In the
case, treatment
it is by now
well-understood
that
its formulation
relates
to ainlog-contract.
Wepercentage
face two
complications.
The
first is
the same
maturity
mismatch
arising
the government
We
twocase.
complications.
The
firstwell-understood
is the on
same
mismatch
ina the
Inface
the
percentage
case,
it(10)
isthe
by
now
formulation
toofagovernment
log-contract.
ed at time t such that inZthe
of 100
arbitrage
The
contract
we The
shall
link
expectation
the maturity
RHS
of its
Eq.
(13) with
is, arising
instead,
“quadratic”
contract
S + ∆) =
× (1 − ft (S, S + ∆)) ,
bond
second
complication
is
that
wethat
are
dealing
arelates
notion
basis
point
variance,
t (S,absence
2 (S,
bond
case.
Thewe
second
complication
istreatment
dealing
with
notion
basis
point variance,
Thedelivering
contract
shall link
the
onwethe
(13)
instead,
a “quadratic”
contractnote
Sthat
+ ∆)
−are
ftRHS
Sof+Eq.
∆).
Toa is,
see
howofthis
contract
is useful,
anecessitates
payoff
equal
toexpectation
fT2 (S,
which
a different
from
the
percentage
case.
(10)
Zt (S, S + ∆) = 100 × (1 − ft (S, S + ∆)) ,
2
2
which
necessitates
apercentage
different
percentage
S+
∆) theorem,
−the
ft well-understood
(S,
S + ∆). case.
To that
see how
this contract
is useful,
note
delivering
payoff
equaland
to treatment
fcase,
byaIn
Itô’s
the
Girsanov
T (S,
Qthat
thelemma
it
isfrom
by
now
its formulation
relates
to a log-contract.
FS
S + ∆) =
100 ×
(1 − ft (S,
S +S∆))
, =E
(10)
Zt (S,LIBOR,
e ft (S, S + ∆) is the forward
which
satisfies:
ft (S,
+ ∆)
(l
Because
In
percentage
it
by
now well-understood
that its formulation relates to a log-contract.
that
by Itô’s
lemmacase,
and
theis Girsanov
theorem,
S (∆)).
t the
QUANT PERSPECTIVES
FT
FT
FT
FT
shall link &the expectation on the QRHS#of Eq. (13) $is, instead, a “quadratic” contract
% we
Q
Q S The contract
martingale under QF . Therefore, assuming that the informabp
2
2
bp
F
we
theS
expectation
on
RHS
Eq.
is,(t,instead,
a “quadratic”
contract
Eshall link
fBecause
+
(S,Sthe
S++∆)
∆)−=fof22E
T,$S)
+
Pf (t, T, S, ∆)
, (14)(14)
T (S,
t #S(13)
+ ∆)
the fforward
satisfies:
ft (S,
= The
Et contract
(ldelivering
(∆)).
+ ℓ̃∆).
To
see how
equal
f 2ft(S,
&∆)to −
)ft=(S,fSS (S,
S +is∆),
∆) is a which
martingale
under
QFSS .+ ∆)
Therefore,
assuming
that
the
Q t % a2 payoff
Q
t (S,
t (S, S +LIBOR,
bp this contract is useful, note
2T
QF SaSE
2 (S,−
ℓ̃bpsee
fequal
(S, −
S+
=S 2E
(t, T,
S) this
+ Pcontract
,
(14)
S f+t Girsanov
∆)
ft2∆)
(S,
+ ∆).
To
how
is ∆)
useful,
note
delivering
payoff
to+fT∆)
T (S, S
t by Itô’s
t
f (t, T, S,
that
lemma
and
the
theorem,
re
f
(S,
S
+
∆)
is
the
forward
LIBOR,
which
satisfies:
f
(S,
S
+
∆)
=
E
(l
(∆)).
Because
t forward
S+ ∆),
tt assuming
= tfS (S,
S + market
∆),price,
ft (S,
S + ∆)byisBrownian
a martingale
under
that
the ! theorem,
(S,
satisfies
mation
in this
isZdriven
motions,
theQ
price,
where
F S . Therefore,
thatZby
Itô’sS
lemma
and
the Girsanov
where
#
$
T
t (S,S
#
$
% 2 !the &2 2
Qbp that
Q f
f
∆)
= finS (S,
+ ∆), fist (S,
S +by
∆)Brownian
is a martingale
QF S . Therefore,
(t,fT,
S)
≡T+ ∆)fτ −(S,
+S
∆)+vτ∆)
(S,
∆)
vτfS)
(S,$∆)
dτ.
(15) (14)
#
$ (t,−T,
ftS(S,
=#∆)
2E vτ (T,ℓ̃bp
+ Pbp
Etℓ̃ satisfies
SQ +% ∆),
mation
thisSmarket
driven
motions,under
the forward
price, where
Zt (S,assuming
& S
ollowing:
T (S,
f (t, T, S, ∆) ,
bpf t
2
f Q
f bp
ℓ̃
fℓ̃T2bp(S,
∆)≡ − ft2ft(S,
S
+
∆)
=
2E
(t,
T,
S)
+
P
(t,
T,dτ.
S, ∆) ,
(14)
Et
(t,ST,+S)
(S,
S
+
∆)
v
(S,
∆)
v
(T,
∆)
−
v
(S,
∆)
(15)
τ
τ t
τ
τ f
(S,
S
+
∆),
satisfies
rmation
in
this
market
is
driven
by
Brownian
motions,
the
forward
price,
Z
dZ
(S,
S
+
∆)
t On the other hand, by taking
τ
t
lowing:
under the T -forward probability of a Taylor’s expansion
(11)
where
= vτz (S, ∆) dWF S (τ ) , τ ∈ (t, S) ,
(11) expectations
! T
$ expansion
2
Onfthe
other hand, byremainder,
taking
expectations
under
the T -forward#probability of a Taylor’s
whereof
we obtain
the following:
Zττ (S,
(S, S
S+
+ ∆)
∆)
dZ
following:
2
!
T (S, S + ∆) withℓ̃bp
z
T S) ≡
T- (15)
(t, T,
fτ (S, S + ∆)#vτf (S, ∆) vτf (T, ∆) −$vτf (S, ∆) dτ.
= vτ (S, ∆) dWF S (τ ) , τ ∈ (t, S) , of fT2 (S, S + ∆)ℓ̃bpwith
(11)
2 obtain
f
f
remainder,
we
thev ffollowing:
t
(S,
S
+
∆)
dZ
(t,
T,
S)
≡
f
(S,
S
+
∆)
(S,
∆)
v
(T,
∆)
−
v
(S,
∆)
dτ.
(15)
τ S + ∆)
% 2
& τ2
τ
τ
τ
2
Q
Zτ (S,
z S)
where
WF (! Brownian
f
(S,S
t − ft (S, S + ∆)
= vτz (S,
∆) dW
(τ
)
,
τ
∈
(t,
,
(11)
motion
under
Q
,
and
v
(S,
∆)
is
a
vector
of
instaneWF S (τ ) is a multidimensional
S
E
f
(S,
S
+
∆)
S
T
T
F
F
τ
taking expectations
% 2the other hand,
&# by
Q tOn
2 #
$
$ under the T -forward probability of a Taylor’s expansion
zS + ∆)
τ (S,
E - other
fremainder,
S by
+ ∆)
−Q
fexpectations
S + ∆) the
taking
underfollowing:
the T -forward probability of a Taylor’s expansion
Thand,
t (S,
2(S,
QF ,Zand
obtain
(S,
S +S∆)
withEwe
remainder,
fT22f
+instan−
eℓ̃ (t,T,S)
=
istheatof
vector
motion under QF S , and vτz (S, ∆)On
W
S (τ ) is a multidimensional
#∆)
$we obtain
$ 1 the following:
(τ ).
usFvolatilities,
adapted
to vW F SBrownian
t (S, of
t#
2
Q
(t,T,S) the following:
2 with'
ℓ̃ obtain
S+
∆)
remainder,
we
of fT (S, =
z
(
(S,
S
+
∆)
E
e
−
1
2f
! ∞
t
ed to
W
().S (τBrownian
a adapted
multidimensional
motiondesigns
under Qreferencing
(S, ∆)instead
is at vector
instanreW
% !2of
&
f (S,S+∆)
F S (τ
F S , and vτ rates
2Qof
). swap security
us
volatilities,
toFW
ecause
we) is
shall
deal
with
variance
(
f f 2 (S, S + ∆)
F
f(S,S+∆)
∆)
! fprices,
% 2 Et'
& S +2we
T (S,
t f , T, S, ∆) dKf!+∞
+
(16)
Put−
(K
Callft (Kf , T, S, ∆) dKf ,
Q
t
#f S
#
$
$
) + ∆)
Et + f2TP(S,
− fPut
t (TS
).
ous
volatilities,
adapted
to
WF S (τswap
0
f (S,S+∆)
t (S,
Q + ∆)
(16)
T,eℓ̃S,(t,T,S)
∆) dK−
Callft (Kf , T, S, ∆) dKf ,
2 #
f +
cause
we
shall deal
withequivalent
variance
security
designs
referencing rates instead
of2fprices,
we
tE(Kf ,$
der
the
forward
LIBOR
to Eq.
(11), as
follows:
$
#
(S,
S
+
∆)
1
(16)
P2t (T )=
t
t
Q
0
f (S,S+∆)
(t,T,S)
referencing
rates instead
of
prices, designs
we consider the forward LI∆)prices,
E
−1
eℓ̃we
=instead
2ft (S, S +of
't!
(
Because
we shallLIBOR
deal
with
variance
security
! ∞
f (S,S+∆)
er the forward
equivalent
toswap
Eq. (11),
as follows: referencing rateswhere,
'! 2
(
f! T
#
!
f
(S,S+∆)
∞
where,
+
Put
(Kf f, T, S, ∆) dK
Call$ft (Kf , T, S, ∆) dKf ,
(16)
f f +
f
t
2
(S,
S
+
∆)
df
!
f
f
τ equivalent to fEq. (11), as follows:
ider the forward LIBOR
ℓ̃f (t, (K
T, S)
≡T
vdK
vτ(K
(S,$
∆) dτ,
Pt (T )
#∆) vτ (T, ∆)
τ (S,
0Put
f−
(S,S+∆)
+
(16) (17)
Call
f , T, S, ∆)
f +
f , T, S, ∆) dKf ,
t
t
= vτ (S, ∆) dWF S (τ ) , τ ∈ (t, S) ,
Pt (T )and:
T, S) ≡
vtτf (S, ∆) vτff(T,
∆) − vτf (S, ∆) dτ,
(17)
ℓ̃f (t,(12)
0
(S,S+∆)
(S,SS +
+ ∆)
∆)
dffττ (S,
f
(12)where,where, (12) t
∆) dWF S (τ ) , τ ∈ (t, S) ,
!
S+
∆)= vτ (S,
7
τ (S,
where,
f
fτdf(S,
S+
∆)
(1#− K ) , T, S, ∆)$
CallztT #(100
Pu
! T,
f
!
" F Sz (τ ) , τ ∈ (t, S) ,
=
vτ (S, ∆) dW
(12)
T S) ≡
(t,
v7τf (S, ∆) vτf (T,f∆) −$vτf (S, ∆) ,dτ, Callf (K , T, ∆)
(17) =
ℓ̃f∆)
f
(K
,
T,
=
Put
−1
f
f
f
t
t (17)f
f
(S,
S
+
∆)
t
e, by Itô’s lemma, vτ (S,
vτ (S, ∆)
vτ (T, ∆)100
− vτf (S, ∆) dτ,
ℓ̃f (t, T, S) ≡
τ ∆) ≡ 1 − fτ (S, S + ∆) vτ (S, ∆).
(17)
!
"
t
, by Itô’s lemma, vτf (S, ∆) ≡ 1!− fτ−1 (S, S + ∆) v"τz (S, ∆).
7
z
z
f
Put (Kz , T, S, ∆) and Call
and:
7 t (Kz , T, S, ∆) are the prices of OTM puts and ca
re, by
Itô’s lemma,
vτ (S, ∆)
≡ Volatility
1 − fτ−1 (S, Indexes
S + ∆) vτz (S, ∆).
and: t
OR
Variance
Contracts
and
and:
f
T ; Putft (KPut
S,(1 ∆)
with
strike price
Kz(1 and
f , ztT,
− Kf ) maturity
, T, S, ∆)
(100
− Kfand
) , T, S,Call
∆) t (Kf , T, S
Callzt (100
f
and:
, Callft (Kf , T, ∆)z =
.
Put
R Variance Contracts and Volatility Indexes
z t (Kf , T, ∆) =
S, ∆)100 on fthe forward Put
− K100
Callt (100 (1 − Kf ) , T,options
f ) , T, S, ∆)
t (100 (1
rate.
Call
. (1 − K ) , T,
= LIBOR
Putft (Kf , T, ∆) = European-style
z,
OR Variance Contracts and Volatility Indexes
z t−(K
K f), ,T,
T,∆)
S, ∆)
Putz (100
zCallt (100 (1
z z f
f
100
100
and:
Put
(K
S,
∆)
and
Call
(Kz , T, S, ∆)tare thezfprices
of OTM, puts
andfcalls
written
on=Zt (S, St + ∆)
t
tPut
LIBOR
Variance
Contracts
and
Volatility
Indexes
,T,S
Call
(K
,
T,
∆)
=
(K
,
T,
∆)
t z t, T,
z,T,S
f
f we are
t
Finally,
by
matching
(16)
to
Eq.
(14),
able
to use
optio
fEq.
f
e define the basis point LIBOR integrated rate-variance as,
100
100
z
and
maturity
T
;
Put
(K
,
T,
S,
∆)
and
Call
(K
,
T,
S,
∆)
are
hypothetical
OTM
with
strike
price
K
z
z
f
f
t
z
define
(1 −
Kf ) ,t T,
∆) puts
T,+
S, ∆)
∆)
Callare
Putt (100
define the
the basis
basis point
point LIBOR
LIBOR integrated
integrated rate-variance
rate-variance as,
as, rate-variance
f ) ,S
t (100
(K
, T,calls
S,
the
prices
ofZS,OTM
and
calls written
on(1Z−t K(S,
Putzt (Kz , T, S,
Call
as,∆) andEuropean-style
puts
and
written
on
t (S,S
z and
, Callft (K
.
= ∆) on
Putftt (K
f ,zT, ∆)
f , T, ∆) =
options
the forward
LIBOR
rate.
point
volatility
, 100
aszt (K
follows:
100
f of
Putzt (K
T, S,Call
∆) fare
the
and calls
written on Zt (S,
T,; Put
S, ∆)
zandfCall
z prices of OTM puts
z ,T
z ,and
! T
6
"
"
and
maturity
(K
,
T,
S,
∆)
(K
,
T,
S,
∆)
are
hypothetical
OTM
with
strike
price
K
z
f
f
Finally,
by
matching
Eq.
(16)
to
Eq.
(14),
we
are
able
to
use
options
on
Z
to
price
the
basis
tt z
maturity
T
2
z,T,S
t
z,T,S
!
!
ft
f
z
"
"
"
"
T
"
"
T
f,bp
Put
(Kz!
,maturity
T, S, ∆) areTthe
prices
of
puts
calls
written
Zt "
(S,
S + are
∆) hypothetica
(Kzstrike
, T, S, ∆)
and K
Call
f
; Put
, T, S,
∆)
and
Call
S, ∆)
with
price
2 dτ,
z tand
f , T,
!fOTM
!and
!
" on
t (K
t (K
of f , asLIBOR
follows:
"" "v
""2European-style
optionspoint
ont volatility
the forward
rate. f
V
(T, S, ∆) ≡
fτ2 (S,
S + ∆)
∆)
f,bp
f
QF T
τ (S,
t (T,
bpstrike price Kz and
maturity
;2Putforward
∆) and Q
Call
∆) are hypothetical OTM
with
(t,T,S)
ℓ̃fS,
VVtf,bp
ff22(S,
∆)
F tT(Kf , T,
(T,S,
S,∆)
∆)≡≡
(S,6SS++∆)
∆)"v
(S,
∆)"" "dτ,
dτ,
t (Kf , T, S,
"vτfτf(S,
European-style
options
onfT!the
LIBOR
rate.
t
""
!
!
"
"
!
(S,
S
+
∆)
E
(t,
T,
S,
∆)
=
2
e
ℓ̃bp (t
−
1
−
E
P
t ττ
t Qare
Finally, by matching
to2 Eq.
we
ablerate.
to use −options
on ZT, S)
to price the basis t
t Q
6
bpfEq. (16) options
2 on(14),
European-style
the forward
tt
S, ∆) = by
fmatching
Et LIBOR
eℓ̃ (t,T,S)
Etwe are
ℓ̃bp (t,
P
t (S, S + ∆)Eq.
f (t, T,
Finally,
(16)
to Eq.− 1(14),
able to use options on Z to price th
#
% $
Finally, by matching#Eq.
(16) to Eq. $(14), we are able to$ use options on Z to price the basis
point volatility of f , as follows:
$ f (S,S+∆)
∞
ft (S,S+∆)
∞
point
volatility
of fprice
, as follows:
2thePutbasis
f as follows:
f
options
on
Z2to
point!volatility
of
hat, by arguments similar to those leading to Eq. (3) in Section 2, the fair value
ofvolatility
the
time
f
+
(K
Callf,
(Kf , T, S, ∆) dKf ,
f , T, S, ∆) dK
f +
t
t
!ofPftQ,(Tas) follows:
""
! point
!
"
"
t, by
the
fair
value
of
the
time
at,
by arguments
arguments similar
similar to
to those
those leading
leading to
to Eq.
Eq. (3)
(3) in
in Section
Section 2,2,
the
fair
value
of
the
time
+
Put
(K
,
T,
S,
∆)
dK
+
f
f
0
f
(S,S+∆)
Q
t
T
T
bp
!
""
!
!
"
"
!
2
(t,T,S)
bp
ℓ̃
"F "ℓ̃ (t,
!
""
f
previous
section,atthe
fair value
of off
theat
time
rate-variance
(T!E)Q − !1e0ℓ̃ −
ST,+S,∆)
E !F eSP
Q T, S)
QEFtT
Q
bp bp
it rate-variance swap
generated
t, and
paying
T , deposit
is the Pfollowing:
(t,T,S)
ft (S,S+∆)
2 t∆)
f (t, T, S, ∆) = 2 ft P(S,
−e1ℓ̃f (t,T,S)
− Et
T,E
S)t F T ℓ̃bp (t, T,(18)
∆) = 2t ft2 (S,
2 f+
−ℓ̃bp1 (t,−
f P(t,
rate-variance
swap
generated
at
rate-variance
swapLIBOR
generated
at t,t, and
and paying
paying off
off
at TT,, isis the
the following:
following:
t (S, S t+ ∆) Et
efine
the basis point
integrated
rate-variance
as, at
#f$ (t, T, S, ∆) =
%S)
#
%
$
FT
FT
FT
FT
FT
FT
FT
FT
FT
FT
FT
FT
f
f
t
FT
FT
t
FT
FT
t
f
t
f
t
t
t
t
FT
FT
f
t
t
FT
swap generated at t, and paying off at T, is the following:
f,bp ! T
" T, S, ∆)"2, T ≤ S,
(T, S,2 ∆) −bp
Pbp
V
f,bp
f,bp
"T,fS,
" T≤
t (T,
f(t,(t,
−−PPbp
∆)
S,∆)
∆)
S,
Vtf,bp (T,VV
S,
≡ S,
f∆)
∆) T,
(S,
∆),," T
dτ, ≤S,
"vτ S,
tt ∆) (T,
f (t,
τ (S, S f+
+
f
FT
$ ∞
$ ft (S,S+∆)
#$
∞
t (S,S+∆)
$ ∞
2 where ℓ̃ f(t,
2 bp f
ft (S,S+∆)
T, S)+andPut
ℓ̃ (t,(K
T,
areS,
defined
inf ,Eqs.
and
(15).
2 fS), T,
f
,
Put
(K
T,
Call,fT,
(KS,
S,dK
∆) dK
∆)ft dK
+S,(17)
Callft (K
∆)
f +
f , T,
f∆) dK
f f f, (K
f
t
P
(T
)
+ time
Puttfis,
(K
∆) dKff+t
Call
0
ft (S,S+∆)
f , T, S,
f , T, S, ∆) d
Pt (T ) An index
t
of basist point
deposit
rate-volatility
0
(S,S+∆)
t
P (T )
& defined in Eqs. (17) (18)
(18) (15).
and
where ℓ̃f (t, T, S) and ℓ̃bp (t, T, S) are
Pbp
T, S, ∆)
bp
f (t,
bp
2×
is
T, S) are
defined
in
Eqs.
(17) and
(15).
where ℓ̃f (t, T, S) and ℓ̃ (t,
TD-VI
(t,
T,
S,
∆)
≡
100
f point time depositT rate-volatility
An
index
of basis
is,
−t
$where
#
An T,
index
ofare
basis
deposit
rate-volatility
is, in Eqs. (17) and (15).
, by arguments similar to those
leading to Eq. Q
(3)
value
ofT,the
timeℓ̃bpwhere
S) and
(t,
S)
and
(15).
ℓ̃f (t,
#in Section
T,defined
S)point
andtime
ℓ̃inbpEqs.
(t, T, (17)
S)
are
defined
ℓ̃f (t,
T
f,bp 2, the$$fair
F#
QQ
(t,
T,
S,
∆)
=
E
(T,
S,
∆)
.
(13)
V
Pbp
T
T
bp
f,bp
(13)
bp
f,bp
F
F
t
bp
&
deposit
rate-volatility
is,
T,
S,
==EEtoff
(T,
S,
(13)
VV,ttist the
PPff f(t,
ate-variance swap generated
at
paying
following:
(t,t,
T,and
S,∆)
∆)
(T,
S,∆)
∆) .. An index of basis point
(13)
An
of
point
timetime
deposit
rate-volatility
is,
where
Ptime
(t,index
T,
S, ∆)
as
in Eq.
(18).
f index
t at T
An
ofisbasis
basis
point
deposit
rate-volatility
is, &
Pbp
(t, T, S, ∆)
f portfolio
bp
Note that the above formula
an∆)
equally-weighted
of OTM options. ThisPisbp
in
TD-VIinvolves
≡ 1002 ×
f (t, T, S,&
f to (t, T, S, ∆)
−t &
bp of eachT option
e face two complications.
The first
is the same maturity mismatch arising in the
government
contrast
to the percentage volatility case where
the
is≡
inversely
bp weight(t,
f,bp The
bp
bp1002proportional
TD-VI
T,
S,
∆)
×
ace two
first
is
the
same
maturity
mismatch
arising
in
the
government
face
two complications.
complications.
The
first
is
the
same
maturity
mismatch
arising
in
the
government
P (t,
Pf (t, T, S, ∆)
V
(T, S, ∆) − P (t, T, S, ∆) , T ≤ S,
f T, S, ∆) 2
bpstrike. MO (2012; 2013d)
bp f additional
2 ×provide
the square
of its
TD-VI
T, S, ∆) ≡ 100
TD-VI
S, ∆) ≡intuition
100 ×about this feature by showing T − t
case. The second tcomplication isf that we are dealing with a notion of basis point
variance,
f (t,
f (t, T, T
where Pbp
∆) is as in Eq. (18).
f (t, T, S,
−differs
t
T − t index, and
se. The
variance,
ase.
The second
second complication
complication isis that
that we
we are
are dealing
dealing with
with aa notion
notion of
of basis
basis point
point
variance,
how the
hedging
portfolio of a basis point volatility index
from that of a percentage
Note that the above formula involves an equally-weighted portfolio of OTM options. This is in
mismatch
arising from
in thethe
government
bond
further analytical details on the behavior of the two indexes.
necessitates a different
treatment
percentage
case. case. The second provide
$
#
bp
contrast
to
the
percentage
volatility
case
where
the
weight
of
each
option
is
inversely
proportional to
ecessitates
treatment
from
the
percentage
case.
ecessitates aa different
different
treatment
from
the
percentage
case.
MO
contains
formulation
ofEq.
a variance
contract design cast in basis point terms
QFwe
complication
is=that
are dealing∆)
with
a notionwhere
of basis
where
P
(t,
S,first
∆)
isin as
(18).
T
where
Pofbp
(t,
T,T,
S,the
∆)
is as
Eq.in
(18).
(t,point
T, S, ∆)
as
in(2012)
Eq.
(18).
Pbp
ffits
f relates
the
square
strike. MO (2012; 2013d) provide additional intuition about this feature by showing
(t,is
T,by
S, ∆)
E
. its formulation
(13)
Vtf,bp (T, S, that
Pbp
the percentage case,
it
now
well-understood
to isathat
log-contract.
t
f
applies
to
fixed
income
markets–namely,
to
interest
rate
swaps.
In
earlier
work,
Carr
and Corso
e percentage
ititisisby
that
formulation
relates
to
aalog-contract.
he
percentagecase,
case,
bynow
nowwell-understood
well-understood
thatits
itstreatment
formulation
relates
toabove
log-contract.
Note
that
thethe
above
formula
involvesportfolio
an differs
equally-weighted
portfolio
of is
OTM
options. Th
variance,
which
necessitates a different
from
the
Note
that perthe
formula
involves
an
equally-weighted
offrom
OTM
options.
This
in
how
the
hedging
portfolio
of aabove
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o
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the
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QUANT PERSPECTIVES
dRτ (T1 , · · · , Tn ) = Rτ (T1 , · · · , Tn ) στ (T1 , · · · , Tn ) · dWA (τ ) ,
τ ∈ [t, T ] ,
(19)
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the basis point realized variance of the forward swap rate arithmetic changes in the time interval
[t, T ],1
# T
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Rτ2 (T1 , · · · , Tn ) ∥στ (T1 , · · · , Tn )∥2 dτ.
t
How do we design a variance contract in this case that allows model-free valuation? Consider the
value of a payer swap withK:fixed rate, K:
SwapT (K; T1 , · · · , Tn ) ≡ PVBPT (T1 , · · · , Tn ) [RT (T1 , · · · , Tn ) − K] .
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of aand
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.),0}.information
about
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the replicating portfolios and the variance contract design.
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volatility andstead,
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areswaptions
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MO
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They
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inthe
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this
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The next
section illustrates
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isswaps
affected
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when
deriving
MO
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basis
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framework
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develop
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framework
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odel-free indexes
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basis point
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t
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Interest
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well-known
Mele,
2013,
Chapter
12)
iods
T1which
0 , ·is· ·also
n − Tformulation
n−1 is zeroused
R
(T
,T
a
martingale
unat
T,
where
(t,T)
is
the
fair
value
of
the
contract
such
that
tat
1,, where
n) is P
n
(t,
T
)
=
E
(t,
T
)
,
(20)
P
V
T
(t,
T
)
is
the
fair
value
of
the
contract
such
that
n T
t fixednsuch
, where
PnT(t,
TTany
), and
ismarket
the and
fairnuméraire
valuenof
the contract
, tenor
length
value of a forward starting swap (at T ≡ T0 , with reset dates T0 , · · · , Tat
of interest
in the
income that
space. Let us illustrate how this framework
n−
(T1a, der
·martingale
· · , Tthe
the
forward
swap rate
prevailing
at t, – n−1
i.e.,
fixed
rate
such
that the
n ) be
· ·Let
, TnR)t is
under
the
so-called
annuity
probability
QAthedefined
through
the
so-called
annuity
probability
Q
A
at
T
,
where
P
(t,
T
)
is
the
fair
value
of
the
contract
such
that
specializes
in the interest rate
! swap case. " !
n 12)
payment periods T1 − T0 , · · · , Tn − Tn−1 ) is zero at t. It is well-known (e.g., Mele, 2013, Chapter
"
length Tn −Consider
T , and
lue ofderivative,
a forward starting
swap (at T ≡ T0 , with reset dates T0 , · · · , Tn−1 , tenor
A starting
bp
dym
A , !the
bp
at
t)the
with
following"payoff:
(20)(20) by Eq. (19)
) = Eswap
(t,
TE
denotes
conditional
expectation
QA . Moreover,
whereQEA
Tn )follows:
is a martingale under the so-called annuity probability
through
thePn (t,aTvariance
that Rt (T1 , · · · , as
tPn V
(t,n Tunder
)=
) ,
(20)
Vannuity
tdefined
t
n (t, T probability
ayment periods T1 − T0!, · · · , Tn − Tn−1 ) is zero at t. It is well-known (e.g.,A Mele, 2013, Chapter 12)
! T ) = EA
" T) ,
Pn (t,
(20)
Vnbp (t,
t
Radon-Nikodym derivative, as follows:
"T
and Itô’sQlemma,
πn (t, T ) ≡ Vnbp (t, T ) − Pn (t, T ) × PVBPT (T1 , · · · , Tn )
PVBP
· · , Tn ) probability
T (T1 , · annuity
is Aa !!martingale
the
so-called
defined
through
the
at Rt (T1 , · · · , Tn )dQ
− t under
r
ds
s
A
A
A
!
,
=
e
"
A
where
E
denotes
conditional
expectation
under
the
annuity
t
expectation
under
the annuity
probability
Qprobability
by Eq. (19) by Eq. (19)
T
!
expectation
under
the
Ewhere
A . Moreover,
!
PVBP
Tn )whereconditional
dQ !GT asdQ
PVBP
(Twhere
, ·T ·(T
· ,1 ,TE· n·t·) ,denotes
t denotes
A . Moreover,
A!
adon-Nikodym derivative,
follows:
conditional
the annuity
annuity
probability
Q"A .QMoreover,
by Eq. (19)
EA
#conditional
at2T , where
Pn (t, T ) expectation
is$ the fair2 valueunder
of the contract
such that
t Adenotes
,
= e− t rs ds t 1
probability
Q
A − R (T1 , · · · , Tn ) = EA V bp (t, T ) = Pn (t, T ) .
R
(21)
(T
,
·
·
·
,
T
)
E
and
Itô’s
PVBP
dQ !GT
Tn )and Itô’sand
t (T
1 , · · · ,lemma,
1
n
t
T
t
lemma,
"n
!
Itô’s tlemma,
! ...
"
T
!
Pn (t, T ) = EA
(20)
Vnbp (t, T ) ,
PVBP
(T
· · , Tvalue
t
t (T
n− basis
1 , · the
n)
T
T1 , · · · , Tn ) is the “price dQ
value
the
point
–
”
i.e.,
at
t
of
annuity
paid
r
ds
A !1,of ,T
s
!
"
t
!!
" "
,value
e of
# at
$# 22
and PVBPt (T1 , · · · , Tn ) is the
“price=
value
the basis
point
– ,”· ·i.e.,
t of annuity
# paid
$ $ 2 2 A
A
bp
PVBP
· , where
Tthe
t (T
1 swap
n )E
i.e., the valuedQ
at !tGof
annuity paid
over
the
tenor.
,.by
Vprobability
(t,
T )TTQ)=
RR
·,··by
,· T· ,Eq.
)Tn−expectation
R=
, · ·V
· ,n
,bp
)) =
follows
RT2 second
(t,
TEE
)At A =
(21)
(T1 , · E
· ·Aequality
, Twhere
)2At (T
−
)(20).
E
p tenor.
(21)
T
n (t,
n (t,
nE
t (T
n
t· (T
t the
VPbp
R
(t,
).. PMoreover,
=
PTn )(t,
TEq.
) . (19) (21) (21)
·(T
·1 ,,·1Tconditional
(T
TTnnthe
=annuity
EA1T,denotes
1 ,1t· · ·under
n ) n− R
over the swap tenor.
...
Rta(T
diffusion
process,
as follows:
1, ,Tn) is
· · · ,RTn(T
) is, · a· · diffusion
process,
as afollows:
me that
and Itô’s lemma,
Similar
to our derivation
of Eq. (16), we now take the expectation under the annuity probability
t (T1 , that
, Tn ) is
diffusion
process,
as follows:
We R
assume
nd PVBP
value
of the
basis point
– ” i.e., the value at t of annuity paid
t (T1 , · · · , Tn )t is 1the “price
t
T
t
t
t
n
A
where follows
the second
follows by Eq. $(20).
!
"
where the second
equality
byequality
Eq.follows
Taylor’s
of(20).
RAt T2#R(T
···· ·· , ,T(20).
Tn )− with
secondexpansion
equality
Qwhere
2by
bp
1 ,1 ,Eq.
A of athe
er the swap tenor.
Vobtaining:
, ·remainder,
·· ,T
EA
= Pn the
(t, T )annuity
.
(21)
(T
Rt2 (T1take
E
n) =
t
n (t, T )under
Similar
to
our
derivation
ofT Eq.
(16),n )we
now
the
expectation
Tn1), =
(Tn1), ·σ·τ· (T
, Tn1), σ
, dW
Tn ) A
· dW
) ,τ ∈derivation
τ [t,
∈ [t,
(19)
dRτ (T1 , · · · ,dR
Tnτ )(T=1 , ·R· ·τ ,(T
· ·R
· τ, T
· ·τ ·(T, 1T, n· ·)· ·Similar
(τ to
)A,(τour
T ]T,] , (19)
(19)
-probability
of
Eq.
(16),
we
now
take
the
expectation
under
the
annuity
probability
to
our
derivation
of
Eq.
(16),
we
now
take
the
expectation
under
the
annuity
probability
2
We assume that Rt (T1 , · · · , Tn ) is a diffusion process, as follows: QSimilar
, TEq.
of RT (T
#QA2 of a Taylor’s
$the second
1 , · · · by
n ) with
whereexpansion
follows
(20). remainder, obtaining:
T
2 equality
F
QA obtaining:
, · ·expansion
(T
, · · ·,annuity
expansion
R· T2, Ttation
(T
· under
·R
with
QA of a Taylor’s
n )1 , ·−
n )21the
T (T1of
aR
Taylor’s
of, tTR
(T
·,remainder,
·T·n,)T )probability
withobtaining:
remainder,
QEt of
WA (τ ) is a Brownian Motion under QA , and στ (T1 , · · · , Tn ) is adapted
WA (τ ),# and define
1 of Eq. n
Similar2 to $our derivation
(16), we now take the expectation under the annuity probability
A to to
T
) where
is a Brownian
Motion
under Q(T
, ,and
σ (Tσ1 , (T
· · · ,,· T· ·n ), Tis )adapted
Wτ(T
),Tand
define
Q...
%
'
...R
A∈1(τ
2 n) with 2remainder, obtaining:
F T ,T
& ∞
QA,Achanges
and
,[t,
,T
dRτ where
(T
,variance
TnA)(!= Rof
· · , Tτn ) swap
(τ ) , #!in
]Rn,)T2 interval
1,−
Ethe
(T
, of
· ·R
,T(19)
T&(T
, ·of
1, · · · W
τ Athe
1 ·forward
τ
1rate arithmetic
n · dW
obtaining:
a· Taylor’s
expansion
,T1n
) · ·R,TT(T
Q1Aof
n)R
n1), · · · , Tn ) with remainder,
the basis point
realized
t (T
t (T
1 ,···
t$ time
#
2
Q
p
#
$
%
'
T
r
int realized
variance
of
the
forward
swap
rate
arithmetic
changes
in
the
time
interval
2
2
Q
F
&
&
T
1
2
2
is adapted to WA (!
(K, T ) dK + ∞
Swpnn,t (K, T ) dK , (22)
Et
RT (T1=
−R
,2TR
[t, T ],
n)
n ) (T
n ) (TRt,(T·$1·,···
E, · ·F· , TR
, ·-t· (T
· ,21TQ, · T)· #· −
· ,T
,Swpn
2T ) n,t
p
t PVBPTt (T
r n)
· ·),· ,and
RT (T
ETt n )define
T
1, · · · , T
1 ,(τ
0 1 ,t· · · , Tn ) − Rt (TSwpn
Rt (T1 ,··· ,TSwpn
%
' T(22)
)'dK , (22)
=W
a Brownian
Motion #under
QA , andarithmetic
στ (T1 , · · · , Tn ) is 2adapted
to
here WA (τ ) isance
n)
A
'
n,t&(K, T ) dK +
n,t (K,
&
%&
of Vthe
swap
time
bp
# Tforward
&&t (T
Rt (T
)T&
∞
(T11,···
, · ·,T· n,%
T) ≡
Rτ2 (T1 ,rate
· · · , Tn ) ∥στ (T1 , · · · changes
, Tn )∥2 dτ. in the PVBP
n ) Rt (T
∞
0 1 ,··· ,Tn
R
)Rt (T1 ,··· ,Tn )
∞
1 ,··· ,Tn )
n (t,
2
p
p
r
2
r
e basis pointinterval
realized [t,T],
variance
forward swap rate
arithmetic
changes
in
the
time
interval
1 2 of the
2
bp
p
t
r
Swpn
(K,
T
)
dK
+
Swpn
(K,
T
)
dK
,
(22)
=
T ) dK + (K,
Swpn
) dK , (K,
(22)
n,t T ) dK +
n,t
Vn (t, T ) ≡
Rτ (T1 , · · · , Tn ) ∥στ (T1 ,=· · · , Tn )∥ dτ.=
T ) dK , at(22)
n,t (K, TSwpn
PVBPSwpn
·n,t
, Tn(K,
)
t (T1 , · · p
0 Swpn
Rt (T
n,t the
n,t
1 ,··· ,Tn )
(K,
T· ),(K,
Swpn
) denote
prices
receiver
and payer
swaptions
t,
(T1 , · valuation?
·PVBP
·Swpn
, where
Tn )trn,t
PVBP
p T
t where
Rt (T1 ,···
,Tn ) ofreceiver
T ],1 How do we design a variance
(T
Tand
n,t (K,
contract in this case that allows
model-free
Consider
the
t
10, · ·rn,t
n )T ) and
Swpn
0 Swpn
t (T1 ,··· ,Tand
n ) payer swaptions at t,
n,t (K, T ) denote the prices of R
# T
p
r
of aapayer
swap contract
with fixed rate,
K:case
T−and
strike
atT .T .and payer swaptions at t,
referencing
a Consider
swap awith
Tn T−)rTand
(K,
Swpn
(K,
T )K,
denote
theexpiring
prices of at
receiver
where
Swpn
2
2
n,t
wevalue
design
variance
that allows model-free
valuation?
thetenor
T and
strike
K,and
expiring
referencing
swap
withn,ttenor
p and
Vnbp (t, T ) in
≡ this R
τ (T1 , · · · , Tn ) ∥στ (T1 , · · · r, Tn )∥ dτ.
n.tn(K,T
n.t(K,T) denote the prices of rep
1
n
F
1
n
p tenor
− T and strike
K, and expiring
T.
with
(K, TaSwpn
)swap
denote
the
receiver
payer
swaptions
at t,
where Swpn (K, T ) andr Swpnreferencing
TTn)prices
denoteof the
pricesand
of atreceiver
n,t (K,swaptions
and payer
at t, referencing
a swap and
withpayer
tenor swaptions at t,
How do we design a variance contract in this casereferencing
that allows model-free
valuation?
the strike
− T tenor
and
K,and
andstrike
expiring
at T expiring
.
a referencing
swap with
tenor
nwith
K, and
at T .
a swapTConsider
Tn − T
n,t
) and
where Swpn (K, Tn,t
ceiver
SwapT (K; T1 , · · · , Tn ) ≡ PVBPT (T1 , · · · , Tn ) [RT (T1 , · · · , Tn ) − K] . n,t
yer swap with fixed rate, K:
t
Tn–T and strike K, and expiring at T.
lue
of apayoff
payer
swap
rate, K:
T1 , a· ·with
·payer
, Tnfixed
) ≡swaption
PVBP
(T1max
, · · ·{Swap
, Tn ) T[R(K;
, · · · and
, Tn )that
− K]of. a receiver is
Swap
The
·) 1, 0}
T is
T (T
T (K;of
max {−SwapT (K; ·) , 0}. Notice that swaption prices contain information about both interest rate
· , Tnmax
) ≡ {Swap
PVBPT (T
Swapswaption
1 , · · · , Tn ) [RT (T1 , · · · , Tn ) − K] .
T (K; T1 , · · is
of a payer
T (K; ·) , 0} and that of a receiver is
9
T (K; ·) , 0}. Notice that swaption prices contain information about both interest rate
he payoff of a payer swaption is max {SwapT (K; ·) , 0} and that of a receiver is
ax {−SwapT (K; ·) , 0}. Notice that swaption prices contain information about both interest rate
www.garp.org
9
9
10 10
10
10
10
M A R C H 2 0 1 4 RISK PROFESSIONAL
5
QUANT PERSPECTIVES
Matching Eq. (21) to Eq. (22) leaves:
Pn (t, T ) =
2
PVBPt (T1 , · · · , Tn )
!"
Rt (T1 ,··· ,Tn )
Swpnrn,t (K, T ) dK +
"
∞
Swpnpn,t (K, T ) dK
#
!
.
$ adapted to
Matching Eq. (21) to Eq. (22)
leaves: We assume that (1) loss-given-default
(23) (LGD) is constant; (2) the short-term rate r is a
origination.
τ
(23)
Eq. (23) provides the expression for the value of the variance swap in a model-free fashion. It
r. Let n #
be the
initial
inshort-term
the index decided
origination.
We
assume
that of
(1)names
loss-given-default
(LGD)
isataconstant; (
origination.
We arrives
assume
that
(1) loss-given-default
(LGD)
isnumber
constant;
the
rτ is
!of"equally weighted
is a portfolio
swaptions,
as in (3)
the case
for time deposits
18);
however,process with intensity
diffusion
process;
and
default
as
a Cox
λ adapted
to r. (2)
Let
n be
the (2) therate
" ∞(see Eq.
Rt (T1 ,···
,T )
origination.
We
assume
that
(1)
loss-given-default
(LGD)
is
constant;
short-term
rate rτ
time
t"T
each
constituent
have
a
notional
value
1/n,
0, and letand
by the ninverse of the
annuity
factor
(the
numéraire
in
the
interest
rate
2 this portfolio is rescaled
diffusion
process;
(3)
default
arrives
as
a
Cox
process
with
p
r
diffusion
and(the
(3)
default
arrives
as),dK
aand
Coxlet
process
with intensity
λ aadapted
to r. Let n be the intensity
T) =
(K,
) dK
+
Swpn
T
.
each
constituent
have
notional
initial
of
names
inbyTprocess;
the
index
at
≡
T
swap market),
whereasnumber
the portfolio Swpn
in (18)
isn,t
rescaled
the
price
of a zerodecided
numéraire
in time
the n,tt(K,
0
diffusion
process;
and (3) initial
defaultnumber
arrives as a Cox process
with decided
intensity
λtime
adapted
Letletn eac
be
$.
, · · government
· , Tn ) bond
PVBP
2
0market).
Rt (T
)
in let
theeach
index
≡ Tto0 ,r.and
1 ,···
and the
valuet (T
of1an
annuity.
This
another
fundamental
difference
between
equity at time tof≡names
1 reveals yetinitial
constituentathave
a tnotional
number
ofsame
names
in,Tnthe
index
T0 , and
, the swap
same
and
the
intensity,
λ. decided
value
Intuitively,
tilting n
a variance
by theLGD,
market numéraire
at
T (which
is one
in the cases
dealt
The
number
of
names
having
survived
up
to
T
i is
,
and
let
each
constituent
have
a
notio
initial
number
of
names
in
the
index
decided
at
time
t
≡
T
(23)
1
0
1
!
, loss-given-default
the
the same
intensity,
λ.thethe
value
d income markets. Options
ononequities
to rate
a single
source
of risk:
the
stock
price.
In-(1)
nsame LGD, and
with in sections
time depositsrelate
and interest
swaps)
causes
its
fair
value
to be and
defined
under
the
same
LGD,
the
same
intensity,
λ.
value
origination.
We
assume
that
(LGD)
is is
constant;
(2)(2)
short-term
rate
rτ
n (1)
origination.
We
assume
that
loss-given-default
(LGD)
constant;
short-term
rate
n ,by
1 of the
however,
this
portfolio
isinformation
rescaled
the
inverse
annuity
S
(T
(1 − I{τjλ.
The
number
of of
names
having
survived
up
to
T
!
i is and
i ) ≡same
≤Ti!
} ) where τ!jj is the time at which
j=1
,
the
same
LGD,
the
intensity,
value
q. (23) provides the
expression
the
value
the
variance
in
a
model-free
fashion.
It
a market
space wherefor
all the
relevant
is given
by the
price swap
of available
derivatives—an
n
n
nnames
is
S
(T
)
≡
The
number
of
names
having
survived
up
to
T
(·),
and
the
annuity
waps and swaptionsfactor
are
affected
by
two
sources
of
risk:
the
swap
rate,
R
3
i
i
T
!
1
is
S
(T
)
≡
(1
−
I
)
where
τ
is
the
The
number
of
having
survived
up
to
T
diffusion
process;
and
(3)
default
arrives
as
a
Cox
process
with
intensity
λ
adapted
to
r.
Let
n
origination.
We
assume
that
(1)
loss-given-default
(LGD)
is
constant;
(2)
the
short-term
rate
r(
j=1b
expectation
under
the
probability
of interest.
j
i (τ ) and
diffusion
process;
and notional
(3) default
arrives
as iaS
Cox
process
intensity
λN(!)=1/n
adapted
to r. Let
n
(the
numéraire
in name
the
interest
rate
swap
market),
whereas
≤T
}
ni1.
j=1 Nwith
j≡
j defaults,
the
(τ
(t){τnotional
time
atnuméraire
which
defaults,
and
the
outstanding
is
N
=
portfolio of equally weighted
swaptions,
asswap
involatility
the jcase
for time
deposits
(seeofEq.
18);name
however,
iswith
S (Tand
(1 −isI{τ
) whereis τN
The
number
names
having
survived
up njto
Ti)outstanding
j is
i ) ≡the
Finally, an index
of interest rate
is
j=1
1
j ≤T
i}
VBPT (·).
time
at
which
name
defaults,
outstanding
notional
(τ
1S(!)
STlet
(τ!let
with
N
(t)
≡to1.have
time
at
name
defaults,
and
the
outstanding
notional
is
Nt ≡
(τ
)loss
a not
initial
ofisof
names
indefault
the
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time
T
diffusion
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(3)
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Cox
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λconstituent
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r.have
Let
n
b
the portfolio
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the obligor
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n I{t≤τjrate
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2012) show how toméraire
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ow that the annuity factor
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1 index
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n Ithe
nprotection
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)
S
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Finally,
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t
i
CDX
)
×
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minus
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−
I
)
where
τj τij
The
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names
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to
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,
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λ.
value
2
b
n
t
i
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S
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)
≡
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−
I
)
where
The
number
of
names
having
survived
up
to
T
i
i
where Pn (t, T ) is as in Eq. (23).
{τj{τ
≤T
i
i
j=1
i} i}
b
1n
1
≤T
n
nment bond market).
j=1
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Spercentage
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at Tfor(which
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dealt with
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b contract
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atat
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jCDX
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−
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τ(t)
i
The
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up
to Ti is notional
) where
withN N
time
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name
jdefaults,
defaults,
andthetheoutstanding
outstanding
N
(τ
) =
j (t
ntuitively, tilting adeposits
variance
swapinterest
by the rate
market
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at
(which
istoone
the
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dealt
DSX
LGD
· vbe
−
) ·and
vsurvived
t =
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1t , 1
1 S i(τ
Credit
and
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dent variance terms. More
generally,
MO (2013d,
Chapter
2) develop
avalue
framework
that
handles
1 1 t = LGD · v0t − 1CDXt (M ) · v1t ,
bτjshould
DSX
obligor
j
default
is
LGD
I
,
whereas
the
premium
at
The
index
loss
at
τ
S
(τ
)
with
N
(t)
time
at
which
name
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and
the
outstanding
notional
is
N
(τ
)
=
DSX
=
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·
v
−
CDX
(M
)
·
v
,
1
should
obligor
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I
,
whereas
the
premium
a
The
index
loss
at
j
tj
t
1t n n
{t≤τ
} }
in sections on time deposits
andbeinterest
rate
swaps)
causes
its the
fair
value
toriskbe defined
under0t b
j ≤T
bMbM
{t≤τ
j ≤T
bn
volatility
priced
through
credit default
swaptions,
nature ofhow
credit this
ket and numéraire ofCredit
interest
incanthe
fixed
income
space.
Let although
us
framework
DSXt = LGD · v0t − CDX
1 1illustrate
1credit
1 t (M ) · v1t ,
1 (T
calls
for relevant
a where
number of new
features
to take
into
account.
For
example,
we
need
to(M
consider
S
).
Finally,
the
value
of
protection
leg
minus
premium
leg
is is premium at
CDX
(M
)
×
should
obligor
j
default
is
LGD
I
,
whereas
the
The
index
loss
at
τ
b
v
is
the
value
at
t
of
$1
paid
off
at
the
time
of
default
of
a
representative
firm,
provided
CDX
)
×
S
(T
).
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the
value
of
protection
leg
minus
premium
leg
t
i
j
rket space where all
the
information
is
given
by
the
price
of
available
derivatives—an
{t≤τ
≤T
}
0t
t
i
j
bM
b
n
n
is variance
given
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the price of available derivatives — an expectation
n
es in the interest rate
swap
case.
swaps
on loss-adjusted forward position in a CDS index, and web must deal with the survival
vthe
istime
the of
value
at t of
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1 value
1 $1 paid where
0t
where
is
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at
t
of
off
at
default
representative
firm,
provided
3v0t numéraire
3
CDX
(M
)
×
S
(T
).
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the
value
of
protection
leg
minus
leg
isof default
contingent
probability
and
the
defaultable
annuity
market
to
account
for
default
risk.
ctation under the numéraire
probability
of
interest.
the
value
t value
of
an
annuity
of
$1 off
paid
at
default
occurs
before
theofpayoff:
index
where
t and
under
the
numéraire
probability
interest.
bexpiry;
nvaluevi 1t
where
v0t isatthe
at
t of 1$1
paid
at
the
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where
v0t
is the
atis
t
of
$1
paid
off
at
the
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of
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representative
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ider a variance swap
starting
at t with
the
following
We only present the percentage variance contract formulation here. The risk
we are dealing with
1
is the
valueprovi
at t
occursv1tbefore
the
index
and where
v1t paid
isLGD
the
value
at
texpiry;
of
an
annuity
of $1
at
default
occurs
before
the
index expiry;default
and
where
inally, an index of interest
rate
swap
volatility
is periodic
DSX
=
LGD
·
v
−
CDX
(M
)
·
v
,
an
index
of
rate
swap
volatility
is
,
·
·
·
,
T
,
until
either
a
default
of
the
representative
firm
or
the
expiry
of
the
index
(whichever
DSX
=
·
v
−
CDX
(M
)
·
v
,
t
0t
t
1t
isFinally,
that of aT
CDS
index,
forbM
which
ainterest
buyer pays
premium
(the
CDS
index
spread)
and
the
1
t and where 0t
t
isb the ofvalue
at 1tt of an annuity
of
$1
paid
default occurs before the Tindex
expiry;
v1tdefault
!
"
b
1
,
·
·
·
,
T
,
until
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the
representative
firm
or
the
ex
seller insuresbp
losses from defaults by any T
of the
index’s
constituents
during
the
term
of
the
contract.
1
bM = LGD
· · ·PVBP
, (T
until
a default of the representative
firm
expiry
(whichever
DSX
· v1tor
−the
·ofv1texpiry
,index
theCDX
value
atof)the
t the
an
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of index
$1 (whiche
value
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defaultable
Voccurs
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)−
)1 , ×
,· ·· ··either
·and
πn (t, T )If ≡
tthe representative
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t (M
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n )index
TT
n (t,
a constituent
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the P
defaulted
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is $
removed
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the
the
continues
,,TTannuity.
, until
either occurs
a default
of
firm
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of
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11,index,
bM
b
first)–the
ofeither
atime
defaultable
annuity.
1a CDS where
occurs
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value
defaultable
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v of0tisa is
the
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off
atatthe
default
to be traded with abp
notional amount.
index
are European-style,
to
paid
TT$1
,...
,TabM
,value
until
aofdefault
of ofthe
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the
value
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t at
of$1
off
the
time
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ofa arepresentative
representativefirm,
firm,prov
pr
2 ×Options
1paid
Anprorated
CDS
payer
ison an
option
abuy
CDS
index
at
aspaid
protection
buyer
with
strike
Poccurs
IRS-VI
(t,protection
T )index
≡ at100
T ) to0tventer
n (t,
(payers) or sell (receivers)
the strike spread upon option
expiry.first)–the value of a defaultable annuity.
T
−
t
A
CDS
index
payer
is
an
option
to
enter
a
CDS
index
at
T
aspro
a
ere Pn (t, T ) is the fairWevalue
the
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payer
isis
an
option
to
enter
CDS
index
at
Ttime
buyer
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strike
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value
trepresentative
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anan
annuity
of of
$1
pa
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index
and
vas
where
the
valuethe
at
t index
of $1aexpiry;
paid
off
atwhere
the
ofprotection
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ofat
aat
firm,
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the
value
t of
annuity
$1
default
before
the
expiry;
and
where
1tv a
0t, Toccurs
) with
length
assume of
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events
may occur over
aA
sequence
of regular
intervals
(Tvoccurs
0
Rt (T1 ,··· ,Tn )
i−1 buyer
i
1t arising from
spread K. Upon exercise, the protection
would
a front-end
A CDS
index
payer also
isspread
anreceive
option
to enter
aprotection
CDS
index
at T buyer
as a protection
buyer
with
st
K.
Upon
exercise,
the
protection
receive
a$1(wh
fron
of
a
defaultable
annuity.
!
"
spread
K.
Upon
exercise,
the
protection
buyer
would
also
receive
a
front-end
protection
from
,
·
·
·
,
T
,
until
either
a
default
of
the
representative
or
the
ofalso
the
index
T
the
value
atexpiry
twould
ofarising
an
annuity
of (which
pa
default
occurs
before
the
index
expiry;
and
where
v
TbM
, until either
a default
of the representative
firm
or
the
expiry
of
the
index
1Tmatures.
1t is firm
1 , · · ,bM
losses occurring
the option
Accordingly,
consider
a loss-adjusted
forward
position
at
A before
bp
e Pn (t, T ) is as in Eq.
(23).
spread
K.
Upon
exercise,
the
protection
buyer
would
also
receive
a
front-end
protection
arising
where Pnn (t,T
(t, T ) = Et Vn (t,
T) ,
T
asindex
occurring
before
the option
matures.
Accordingly,
consider
a(whic
lossfr
11 occurring
matures.
Accordingly,
consider
a loss-adjusted
forward
at
occurs
value
oflosses
aa(20)
defaultable
annuity.
, which
· · · ,first)–the
Tthe
,option
until
either
offollowing:
the
representative
firm
or the
expiryposition
of the
TT1before
first)–the
value
of
adefault
defaultable
annuity.
bM
t in a CDS indexlosses
that starts
atlosses
,occurs
can
be
shown
to
equal
the
occurring before theta option
matures.
Accordingly,
consider
a loss-adjusted
position
protection
buyerto
with
strike
spread
K. Upon
exercise,
theforward
proaan
CDS
index
that
starts
at
Tindex
, which
shown
to
equal
the
folls
t in a CDS index
that
starts
at value
Tpayer
, payer
which
can
beoption
shown
equal
the
following:
A ACDS
an
option
toannuity.
a aCDS
index
atatT can
buyer
with
occurs
first)–the
ofisin
ais
defaultable
CDSindex
index
toenter
enter
CDS
Tasasabeaprotection
protection
buyer
with
A denotes conditional
Credit
expectation
under
the
annuity
probability
Q
.
Moreover,
by
Eq.
(19)
t
in
a
CDS
index
that
starts
at
T
,
which
can
be
shown
to
equal
the
following:
tection
buyer
would
also
receive
a
front-end
protection
arising
"
#
edit
A
1
spread
K.K.Upon
exercise,
protection
also
receive
protection
A CDS
index
payer
isthe
an
option
to buyer
enter
awould
CDS
index
at1 T aasfront-end
protection
buyerarising
with
spread
Upon
exercise,
protection
would
also
a afront-end
protection
arisins
) losses
−
CDX
, before
DSXL
"
τ (M the
t (M ) buyer
" occurring
#receive
t,T (τ ) ≡ N (τ ) v1τ CDX1
the
matures.
Accordingly,
lemma,
L option
b DSXL (τ ) ≡ from
CDX
(τ
)
≡
N
(τ
)
v
(M
)
−
CDX
DSX
"
#
1τ
τ
t (M
CDX
N
(τ
)
v
(M
)
−
CDX
(M
)
,
t,T
1
losses
occurring
before
the
option
matures.
Accordingly,
consider
a
loss-adjusted
forward
positi
spread
K.
Upon
exercise,
the
protection
buyer
would
also
receive
a
front-end
protection
arising
1τ
τ
t
t,T
losses
occurring
before
theL option
a loss-adjusted forward posi
the nature
of default
credit risk
new
b
it volatility can bealthough
priced through
credit
swaptions,
although
the of
nature
ofDSX
risk matures. Accordingly, consider
bcredit
! calls for
"a number
t,T (τ ) ≡ N (τ ) v1τ CDXτ (M ) − CDXt (M ) ,
#
$
bp
2
2 into account. A
bwhich
tlosses
awe
starts
atat
T ,that
which
can
bebe
shown
toconsider
equal
following:
the
option
matures.
Accordingly,
athe
loss-adjusted
to− take
example,
need
cont) in
aoccurring
CDS
index
that
starts
T
,T
can
shown
to
equal
following: forward positi
the
value
of
CDX
(M
),
set
such
a forward
position
at
τ inthe
the
that
starts
at
,features
· · · ,features
Tn )where
Rto
(T
·(M
, Tn)account.
) is
= defined
EtForVFor
(t,
Tin
=CDS
Pwe
(t,index
T
)to. before
(21)
EA
for a number
of1 new
take
example,
need
to
consider
credit
τthat
n
1 , · ·τinto
n as
t RT (T
t CDX
(M
)
is
defined
as
the
value
of
CDX
setthe
such that a
where
CDX
τ
τ (M
Lτ (M
)
is
defined
as
the
value
of
CDX
(M
),
set
such
that
a
forward
position
at τ), in
where
CDX
sider
credit
variance
swaps
on
loss-adjusted
forward
position
τ
index
that the
starts
at T , which can be shown to equal the following:
(τtand
)in=awe
0,CDS
is worthless,
DSX
nce swaps on loss-adjustedindex
forward
position in viz
a CDS
index,
must(M
deal with
survival
τ,T where
# # position at τ in
asLthe
value1 of
CDX
(M"L), (τ
set) =
such
that a forward
CDX
τ L ) is defined
τ"
1
0,
index
is
worthless,
viz
DSX
L
e second
equality and
follows
Eq. (20). annuity
(τ )default
= 0, DSX
index
is worthless,
viz
DSXτ,T for
τ,T
(τ (τ
)≡
(τ (τ
) v)1τv1τCDX
) −) −
CDX
) ), ,
DSX
) ≡NN
CDX
CDX
τ (M
t (M
t,Tt,T
ngent
probability
theby
defaultable
market
numéraire
toworthless,
account
risk.
τ (M
t (M
"
#
(τ )F= 0, b1 b
index
is
viztoDSXL
τ,T L
probability
and
the
defaultable
annuity
market
numéraire
ar
to
our
derivation
of
Eq.
(16),
we
now
take
the
expectation
under
the
annuity
probability
1
v
v
CDX
(M
)
−
CDX
(M
) ,
(τ
)
≡
N
(τ
)
v
DSX
0τ
1τ
τ
t
We only present the percentage variance contract formulation here.
The
risk
we
are
dealing
with
τ
t,T
CDXτ (M ) = LGD
+
. vb (M
v0τ set vτF
2
account
for default
risk. remainder, obtaining:
vτF (M ), 1set
!0τ of CDX
! (M),
, Tn ) with
Taylor’s
expansion
b where
v1τdefined
N)as(τ
)
v
(M
)1is
defined
the
value
such
that
a
forward
position
atat
where
F
1τ
T (T1 , ·a· ·buyer
CDX
(M
)
=
LGD
+
.τ in
CDX
)CDX
is
as
the
value
of
CDX
(M
),
set
such
that
a
forward
position
τ
τ index
τ
at
of a CDS
index, of
forRwhich
pays periodic premium
(the CDX
CDS
spread)
and
the
τ
τ (M
τ
(M
=
LGD
+
.
1
v0τ b
vτ
v1τ viz
N (τ ) v1τ
- τsuchL that
b isviz
N
(τ ) v(M
a=forward
! set
in the
index
CDX
) =position
LGD
+
. isaworthless,
1τ of
1τat),
L(τthe
τ v(M
)
0,
index
is
worthless,
DSX
(M
)
defined
as
value
CDX
such
that
forward
position
at
τ
i
where
CDX
$
insures
losses
from
defaults
by
any
of
the
index’s
constituents
during
the
term
of
the
contract.
) = 0,
index is worthless,
viz DSX
τ
τ
τ,TLτ,T
b (τIndeed,
(τ ) v1τ
CDXt (Mv1τ
) is aNmartingale
RT2 (T1 , · · · , Tn ) − Rt2 (T1 , · · · Note
, Tn ) that N (τ ) v1τ is the natural numéraire in -this market.
L !,T(!)=0,
constituent defaults,
fromN
the
and the
index
continues
=
index
worthless,
viz'DSX
natural
numéraire
this market. Indee
Note
N (τmarket.
) v1τ is the
%& the defaulted obligor is removed
τ,T (τ )that
natural
numéraire
in0,this
Indeed,
CDX
(M
ainmartingale
Note&that
(τ index,
) vis
1τ is the
tF
, )defined
through
Radon-Nikodym
derivative,
as ) is CDX
under
probability”
Qsc(τ
Rt (T1 ,···
,Tn ) the “survival contingent
∞
1 1thenuméraire
v0τvmarket.
vτFvIndeed,
the
natural
in
this
) is a marting
Note
that
N
v1τ isunder
2 with a prorated
t (Mthrough
p European-style,
traded
notional
amount.
Options
on
a
CDS
index
are
to
buy
0τ
r
τ
defined
the R
“survival
probability”
) contingent
=
LGD
++ F
.Q.sc , derivative,
Swpn
Tunder
) dK losses
+the “survival
Swpn
(K,any
Tprobability”
) dK
defined
through
the Radon-Nikodym
as
contingent
spread) follows:
and the
seller
insures
from defaults
of , (22)Qthe
CDX
)=
LGD
τ (M
sc ,CDX
n,t (K,
n,tby
τ (M
$
(T1sell
, · · ·(receivers)
, Tn )
BPt or
b1 b
v1τ
NN
(τvτ(τ
) the
v)1τv1τ
% T contingent probability”
through
Radon-Nikodym derivative
“survival
Qsc , defined
v1τ
0 protection at the strike spread upon
Rt (T1under
,···
,Tn ) the
0τ
ers)
option
expiry.
$
follows:
N
(T
)
v
dQ
+ $
. %T
sc $
1T
follows:
− τ r (u) $du
τ (M ) = LGD
%b TCDX
N
(T ) v1T
$ Nlength
follows:
sc $$(τ ) v1τ −
,the
Ti )sc with
We assume
credit events
may occur
overthea defaulted
sequence of
regular
intervals
(Ti−1
$,N(!)
$ r = efrom
N
(T
) natural
v1T v1τ dQ
dQ
(u) du N
%the
−
r
(u)
du
constituent
defaults,
obligor
is
removed
dQ
(τ
)
v
p
Note
that
v
is
numéraire
in
this
market.
Inτ r
1!
r
$
T
1τ
=
e
,
τ
$
Note
that
N
(τ
)
v
is
the
natural
numéraire
in
this
market.
Indeed,
CDX
) is
a marti
F
= the
e dQ
is
natural
numéraire
in,du
this
market.
CDX
(M
Note
that swaptions
N (τ )1τv$1τ at
t (M
NdQ
(T
)$ vr1T Indeed,
t,
wpnn,t (K, T ) and Swpnn,t (K, T ) denote the prices of receiver and
tN
T payer
sc $
−
r
(u)
(τ) )isv1τa mar
r i = 1, · · · , bM , where
is the
of years the
index
runs,
T0 is
the time
τ ) v1τ
dQ ofFrthe index t $(M)=
(τ
eaN
,
FT “survival
index,Mand
the number
index
continues
to
be
traded
with
a
prorated
is
martingale
under
the
contingent
T the natural
r denotes
through
the
Radon-Nikodym
thethe
“survival
probability”
Qsc
is
numéraire
in
this N
market.
Indeed,
CDXt (M ) is derivativ
a deriva
marti
Note
that
N, (τ
) contingent
v1τcontingent
dQ
(τ
) v1τonly.
, defined
through
the
Radon-Nikodym
T and Fstrike
K, and
at under
T .under
“survival
probability”
Q,short-term
ng a swap with tenor Tn −where
scdefined
theexpiring
information
set
at
time
T
which
includes
the
path
of the
rate
FrT
T
r denotes
- the pat
sc
r denotes the information set at
where
F
the
information
set
at
time
T
,
which
includes
p
where
F
time
T
,
which
includes
the
path
of
the
short-term
rate
only.
follows:
,
defined
through
the
Radon-Nikodym
derivativ
under
the
“survival
contingent
probability”
Q
T
$ $ τ ∈ [t,sc
follows:
Treceiver with
rstrike
T ;the
M ) path
≡ of the short-term rate o
The prices
of a(receivers)
payer andprotection
K spread
expiring tive,
at T ,asare,
forat
any
T
τ (K,
T%],T SW
to buy (payers)
or sell
at Fthe
strike
where
set
time
T−
,%which
includes
follows:
$
11
N
(T(T
) vT
dQ
T denotes the information
p
N
)1Tv],1TSW
dQ
rfor
(u)
dudu
The prices
ofsc a$$$scpayer
and
with
strike
at T , are, for any
−τ receiver
r
(u)
The
prices
of
a
payer
and
receiver
with
strike
K
expiring
at
T
,
are,
any
τ
∈
[t,
$
follows:
=
e
, K, expiring
τ
τ (K, T ; M ) ≡p
= expiring
e% T
upon option expiry.
$$ r$ K
The prices of a payer and receiver with
strike
at
T
,
are,
for
any
τ
∈
[t,
T
],
SWτ (K, T ; M
dQdQ
(τ
)
v
N
(T
)
v
dQ
N
(τ
)
v
1τ
r
sc $FT F
1T1τ
−
r (u) du
,
$ r T= e τ
dQ
N
(τ
)
v
1τ
r denotes
FT time T , which includes the
r where12
...F,bM,
the
information
setset
at
path
of the short-term rate
lar intervals (Ti-1,Ti) with length 1/b, forwhere
i=1,
Mthe
where
information
at time T , which includes the
path
ra
TFT denotes
12of the short-term
12
10
p p
r time
rset at
is the number of years the index runs, The
T0The
isprices
the
of
the
(K,
T
of
a
payer
and
receiver
with
strike
K
expiring
at
T
,
are,
for
any
τ
∈
[t,
T
],
SW
where
F
denotes
the
information
time
T
,
which
includes
the
path
of
the
short-term
rate
F Twith
denotes
theK12
information
at time
T, τwhich
prices
of a payer andwhere
receiver
strike
expiring at Tset
, are,
for any
∈ [t, Tin], SW
τ τ (K,;TM
T
thewith
pathstrike
of the
short-term
only.
a pτ (K, T ; M
The prices of a payer andcludes
receiver
K expiring
at Trate
, are,
for The
any τprices
∈ [t, Tof], SW
1
b,
for i = 1, · · · , bM , where M is the number of years the index runs, T0 is the time of the index
1212
12
6
RISK PROFESSIONAL M A R C H 2 0 1 4
www.garp.org
QUANT PERSPECTIVES
payer and receiver with strike K expiring at T, are, for any indexes viable for serving as the underlying of tradable prodp
+
r
ucts"such as volatility futures and options.
M)N(!) v1! . E!sc
! (K,T
T (M)-K)!
!
! !%[t,T
+"
+
r
sc
sc
!
"
!
" ."
. Er!scSW
v1τ · Eτ (CDX
)−
K) v1!and
≡ N+].(τ ) v1τ sc
· Eτ! (K − CDXT (M ))
T (M
!sc
"
+
τ -(K, T ; MT)(M))
(K,T
M)N(!)
[K
! T (M ) − K)+ and" SWr (K, T ; M ) ≡ N (τ ) v1τ · E sc (K − !CDXT (M ))+ +
"
(CDX
.
τ
+ SWττ (K, rT ; M ) ≡ N (τ ) v1τ · E
and
− CDXT (M )) . +
K)
sc (CDX
T (M
τ ·(K
eτ·(CDX
assume
that)T −
and SWτ (K, T ; M ) ≡ N (τ ) v1τ
E
(M
) − K)
Esc
τ (K − CDXT (M )) .
#
$
$
#
sume that dCDXτ (M )
#= − #Esc ej(τ ;M ) $− 1 η $(τ ) dτ
dCDXCDX
$
$
τ (M ) (M ) # sc# τj(τ ;M )
$ dτ $ $
# ) − 1 η (τ )#
dCDXdCDX
Eτ # ej(τ ;M
τ (M ) τ = −
= −) =Esc
−;M
1 ) η (τ ) dτ
CDXτ (Mτ) (M
τ− eEsc ej(τ
sc − 1 ηj(τ
;M
) ) −$
dτ1 dJ sc (τ ) ,
τ ) · dW (τ )#+ e (τ
CDXτCDX
(M ) (M ) + σ (τ ; M
τthat
me
that
-
τ
sc
;M )
income asset classes.
+ σ (τ ; M )" · dW sc
(τ ) + #ej(τ
# )− 1 $dJ! scsc(τ
$ ),
!
"
j(τ ;M
+) · dW
r(τ ) sc
sc (CDX
+
σ
(τ
;
M
+
e
1· E)scτ −
dJ(K
) sc
, T(τ
j(τ
sc
sc ))+ .
and
SW
−(τ
CDX
N
(τ
)
v
·
E
(M
)
−
K)
(K,
T
;
M
)
≡
N
(τ
) v−
(M
1τwhere
1τ;M
T (! + σ (τ ; M ) · dW
sc
sc
τ
τ
a methodological research perspective, the logical next
(τ
)
+
e
1
dJ
)
,
W
(!)
W (τ ) is a multidimensional Brownian motion and J (τ ) is a Cox process (with From
intensity
We assume that
sc jump
.
.
step
in
supporting
the creation of a market for standardized
&(
)
and
size
j(
))
under
the
τnd
) is
a
multidimensional
Brownian
motion
and
J
(τ
)
is
a
Cox
process
(with
intensity
jump size j (·)) under the survival
contingent
probability.
#
$ sc$
#
dCDX
Brownian
motion
(τ
)
is
a
Cox
process
(with
intensity
τ (M )
scis(τa) multidimensional
sc
sc
j(τ ;M ) and J
survival
contingent
probability.
aprice
multidimensional
and
(τ ) is
=
− Eoriginated
− 1atηt,
(τ and
) dτJpaying
e motion
mp
sizeis
j (·))
under
the
survivalBrownian
contingent
probability.
e wish
to
a credit
variance
swap
off aatCox
T , asprocess
follows: (with intensity
τ
CDXτ (M )
#
$
sizeprice
jsize
(·))
under
the survival
contingent
probability.
t, follows:
and options on these indexes in a way that is consistent with the unto
a jcredit
variance
originated
at
t,
and
off
jump
(·)) under
theswap
survival
contingent
probability.
sc
j(τpaying
;M )
sc at T , as
+ σ (τ ; M ) · dW (τ ) + e
− 1 dJ (τ ) ,
) − Pvar,M (t,
)) ×
) v1T , off at T , as follows:
oish
price
a credit
variance
swap
originated
atT t,
and
paying
paying
offvariance
at(VT,
as Tfollows:
M (t,
to price
a credit
swap
originated
atNt,(T
and
paying off at T , as derlying
follows: yield or credit curves, the indexes themselves and the
where W sc (τ ) is a(V
multidimensional
Brownian motion and J sc (τ ) is a Cox process (with intensity
term
M (t, T ) − Pvar,M (t, T )) × N (T ) v1T ,
M
structure of volatility in order to facilitate risk managetrading strategies by end users.
η (·)(t,and
sizefair
underof
thethe
contingent
Pvar,M
T )jump
is the
value
contract,
we(T
have
the percentage variance
as, formulation of
ment and
(Vj (·))(t,
T) −
Psurvival
(t,
T ))and
× probability.
N
) v defined
,
M (V (t, T var,M
1T
)−
Pvar,M
(t, Tat))t, ×
(T ) voff1Tat, T , as follows:
M variance
We wish to price a credit
swap
originated
andNpaying
% Tcontract, and we %have
(t, T ) is the fair value of the
defined the percentage variance as,
T
) ≡(t,T)
∥σ
(τ
MP)∥
dτ(t,we
+
M, ) dJthe
(τpercentage
) .we have variance as,
Vvalue
where
fair
value
ofhave
contract,
and
(t,the
T );−
T ))
×the
Nj (T(τ
) v;1T
(Vcontract,
var,M
M (t, T
t, T ) is the fair
and
defined
Mis
var,M
%of Tthe
of
ar,M (t, T ) is the fair value
T t we have defined the percentage variance as,
t the contract,% and
FOOTNOTES
2
2
sc
≡%
(τ ;ofMthe
)∥ contract,
dτ + %andT jwe (τ
; Mdefined
) dJ the
(τpercentage
).
M (t,
where PVvar,M
(t,T
T)
) is
theTfair∥σ
value
variance as,
%have
MO (2013d, Chapter 5, Appendix
D), 2we showt that
T
t % T
sc
%M
≡ T ) ≡∥σ (τ ;∥σ
)∥; M
dτ)∥+2 dτ +%j 2T (τ ;jM
VM (t,VT )(t,
2 ) dJ (τ ) sc
T (τ
(τ ; M ) dJ . (τ ) .
2
M
&
2
2
2
sc
'
sc
(t, T )(M
≡) we
∥σr show
(τ ; M )∥that
dτt+
j (τ ; M ) dJ (τ ) . p
%VtMCDX
2013d, Chapter 5, 2Appendix
t % ∞
tt D),
SW
SWt (K, T ; M )
t
t (K, T ; M ) t
Pvar,M (t, T ) =
dK
+
dK showed the remarkable property that the fair value of a
2
2
&
' variance swap remains the same in this case.
K
K
v
13d,
Chapter
5,
Appendix
D),
we
show
that
1t
0
CDX
(M
)
% CDX
% ∞
In MO
(2013d,
Chapter
5, Appendix
D),we
we show
show that
t
r D),
O (2013d,
Chapter
5, Appendix
that
t (M )
2
SWpt *
(K, T ; M +)
(%& TSW
) t (K, T ; M )
'
dK + % ∞1 2
dK '
ar,M (t, T ) = &%
p
) 2) r
sc 2 % CDXtj(τ
;M
sc )
'(24)
SW−
T2; M
v
pK
−
2E
er (MK
1 −T ;jM(τ) ;dK
M%+
)CDX
−
; Mt )(K,
dJ
(τ ) .
CDX
)
∞ t (M
t (K,
0&
)(τSW
%tt(M
%j ∞
pTdK
2P1t
T
;
M
)
SW
(K,
;
M
) ; M )2. Merener
var,M (t, T ) =CDXSW
t (M t) (K, r
(2012) considers the replication of variance
2
2
2
t
t
v1t
K TdK
K SWt (K,
2
SW
(K,
;
M
)
T
(%
)
*
+
0
CDX
(M
)
t
t
(t,
T
)
=
+
dK
M
T
2
dK1CDX
+2 (M ) *
dK
Pvar,M (t, T
(%
+2
K
K
v1t)−=2E
j(τ
;M
sc
sc
T) )
2
2
0
t ; M)
v1t
e is −ej(τ1;M
−K
j1 (τ
dJ (τ ) K
.
(24)
) j (τ ; M ) −
CDX
t
dingly, a credit volatility
−index
2Esc
− 1 − j (τ ; M ) 2− j 2 (τ
; M )t (M
dJ)sc (τ ) . +
(24)
t
(% Tt0 (%
)
*
2
)t
*
+ approximation, and does not rescale for the relevant no-
1 2 1
T ;M )
j(τ
sc
,
− 2Easc
−
1 )−−j 1(τ−
;M
)−
j (τ ; M
j(τ
;M
sc e
2 ) dJ (τ ) sc.
t−
2E
e
j
(τ
;
M
Accordingly,
credit
volatility
index
is
1 2) − j (τ ; M ) dJ (τ )
a credit volatility index
is
tt C-VI
2 T)
) ≡ 100 × index isPvar,M (t,
M (t, T volatility
Accordingly, atcredit
,T − t
1
,
credit
volatility
indexindex
is C-VIisM (t, T ) ≡ 100 ×1 T − t Pvar,M (t, T )
gly,
a credit
volatility
Pvar,M (t, T )
(t, T ) ≡ 100 ×
M (24).
Pvar,M (t, T ) is as C-VI
in Eq.
market
. tion of(24)
(24)numéraire in the manner we suggest in this
section, thereby providing neither the index formulae and
pricing in this section nor the hedging details described in
, T −,t
where
Pvar,M
(t, T )RHS
is as in
ers
variance
swaps
priced
and
hedged
based
on
parametric
he first
term
on the
of Eq.
Eq.(24).
(24) is model-free,
once
we
estimate
the
CDX
default
intensity
1
1 we (t,
The firstwhere
term
on M
the(t,
RHS
model-free,Ponce
estimate
default intensity
C-VI
T
)of(t,
≡Eq.
T )(t,the
var,M
assumptions.
(t,T
var,Mterm
second
isT100
small
for
all
intents
and
purposes,
and should
not materially
porated by
v . The
Pvar,M
C-VI
)(24)
≡×is100
T )CDX
T× −
t
M (t, T ) is as1tin Eq. (24). M
incorporated by v1t . The second term is small for all intents
T −and
t purposes, and should not materially
the
value
of the
index
approximated
by by
only
retaing
thefirst
firstterm.
term.
term
on the
RHS
is model-free,
once we
the CDX default intensity
affect
the value
of of
theEq.
index(24)
approximated
only retaing
the estimate
. The
1t
3. Interestingly, our model-free expression for the variance
ment of the forward swap rate, as it turns out by compar-
t,by
T )(t,
asThe
Eq.in(24).
second
term
is small for all intents and purposes, and should not materially
dar,M
vis
1t .T
) in
is as
Eq.
(24).
second
term
is small
for all intents
and
purposes, the
and should
erm
on
the
RHS
of
Eq.
(24)
model-free,
once
we
default
intensity
lue
of
the
index
approximated
by
only
retaing
theonce
firstestimate
term.
rst term on the RHS of Eq. is
(24)
is model-free,
we estimateCDX
the CDX
default
intensity
not materially affect the value of the index approximated by
. The
term term
is small
for allfor
intents
and purposes,
and should
not materially
y
v1tby
The second
is small
all intents
and purposes,
and should
not materially
ted
v1t . second
evalue
of theofindex
approximated
by
only
retaing
the
first
term.
13
the index approximated by only retaing the first term.
13
Closing Thoughts
The viability of volatility indexing using the methodologies
13
13
that complements the asset pricing foundations laid down
describe the behavior of variance risk-premiums under
both interpretations of n (t,T
13
options used in the various index formulas render the resulting
www.garp.org
M A R C H 2 0 1 4 RISK PROFESSIONAL
7
QUANT PERSPECTIVES
REFERENCES
Journal of Financial Economics
able from www.garp.org/risk-news-and-resources/2013/
december/
Interest Rate
Finance Institute.
Journal of Finance
Energy & Power Risk Management 4,
Quantitative Finance 12, 249-261.
Journal of Portfolio Management 20, 7480.
-
Quantitative Finance 1, 19-37.
script.
-
Institute.
Journal of Financial Economics
Mele, Antonio, 2013. Lectures on Financial Economics
manuscript. Available fromhttp://www.antoniomele.org.
Fixed Income Securities
Index. Available from http://www.cboe.com/micro/srvx/
default.aspx.
Antonio Mele is a Professor of Finance with the Swiss Finance Institute in Lugano.
as a tenured faculty at the London School of Economics. His academic expertise covers
-
from its think tank of academic researchers. Previously, he managed U.S. and Asian
Ibid, 2013d. The Price of Fixed Income Market Volatility.
manuscript.
8
RISK PROFESSIONAL M A R C H 2 0 1 4
www.garp.org