Rat

ional

Shap

es o

f th

e V

ola

tili

ty

Su

rfac

e

Jim

Gat

her

al

Glo

bal

Eq

uit

y-L

ink

ed P

rod

uct

s

Mer

rill

Ly

nch

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

2

Ref

eren

ces

•B

aksh

i, G

. ,C

aoC

., C

hen

Z., “

Em

pir

ical

Per

form

ance

of

Alt

ern

ativ

e

Op

tion

Pri

cing M

od

els”

Journ

al

of

Fin

an

ce, 52

, 20

03-2

04

9.

•J.

Gat

her

al,

Cou

ran

t In

stit

ute

of

Mat

hem

atic

al S

cien

ces

Lec

ture

No

tes,

htt

p:/

/ww

w.m

ath

.nyu.e

du

/fel

low

s_fi

n_

mat

h/g

ath

eral

/.

•H

ard

y M

.H

odg

es. “A

rbit

rag

e B

ound

s on

th

e Im

pli

ed V

ola

tili

ty S

trik

e

and T

erm

Str

uct

ure

s of

Euro

pea

n-S

tyle

Op

tion

s.”

Th

e Jo

urn

al

of

Der

iva

tive

s, S

um

mer

199

6.

•R

og

er L

ee, “L

oca

l vo

lati

liti

es i

n S

toch

asti

c V

ola

tili

ty M

odel

s”,

Work

ing

Pap

er,

Sta

nfo

rd U

niv

ersi

ty, 1

999

.

•R

. M

erto

n,

“Op

tio

n P

rici

ng W

hen

Und

erly

ing

Sto

ck R

eturn

s ar

e

Dis

con

tinu

ous,

”Jo

urn

al

of

Fin

an

cia

l E

con

om

ics,

3, Ja

nu

ary-F

ebru

ary

1976

.

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

3

Goal

s

•D

eriv

e ar

bit

rag

e b

oun

ds

on

th

e sl

op

e an

d

curv

atu

re o

f v

ola

tili

ty s

kew

s.

•In

ves

tig

ate

the

stri

ke

and

tim

e b

ehav

ior

of

thes

e

bo

und

s.

•S

pec

iali

ze t

o s

toch

asti

c v

ola

tili

ty a

nd

ju

mp

s.

•D

raw

im

pli

cati

on

s fo

r p

aram

eter

izat

ion

of

the

vo

lati

lity

su

rfac

e.

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

4

Slo

pe

Con

stra

ints

•N

o a

rbit

rag

e im

pli

es t

hat

cal

l sp

read

s an

d p

ut

spre

ads

mu

st b

e n

on-n

egat

ive.

i.e

.

•In

fac

t, w

e ca

n t

igh

ten

th

is t

o

∂ ∂≤

∂ ∂≥

C K

P K0

0 a

nd

∂ ∂≤

∂ ∂

F HGI KJ≥

C KK

P K0

0 a

nd

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

5

•T

ran

slat

e th

ese

equ

atio

ns

into

co

nd

itio

ns

on

th

e

imp

lied

to

tal

vo

lati

lity

a

s a

fun

ctio

n o

f

.

•In

co

nv

enti

on

al n

ota

tio

n, w

e g

et

()

FK

y/

ln=

σ[

]y

σπ

σπ

'[]

exp

'[]

exp

yd

Nd

yd

Nd

≤ ≥−

−

22

22

2

2

2

1

2

1

nsbg

nsbg

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

6

•A

ssu

min

g

w

e ca

n p

lot

thes

e

bo

und

s o

n t

he

slo

pe

as f

un

ctio

ns

of

.

yy

3.0

25

.0

][

−=

σ

y

-0.4

-0.2

0.2

0.4

-3

-2

-1123

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

7

•N

ote

th

at w

e h

ave

plo

tted

bo

und

s o

n t

he

slo

pe

of

tota

l im

pli

ed v

ola

tili

ty a

s a

fun

ctio

n o

f y

. T

his

mea

ns

that

th

e b

ou

nds

on

th

e sl

op

e o

f B

S i

mp

lied

vo

lati

lity

get

tig

hte

r as

ti

me

to e

xp

irat

ion

incr

ease

s b

y

.

1/

T

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

8

Con

vex

ity C

onst

rain

ts

•N

o a

rbit

rag

e im

pli

es t

hat

cal

l an

d p

uts

mu

st h

ave

po

siti

ve

con

vex

ity

. i.

e.

•T

ran

slat

ing

th

ese

into

ou

r v

aria

ble

s g

ives

∂ ∂≥

∂ ∂≥

2

2

2

20

0C K

P

K a

nd

∂ ∂≥

∂ ∂

2

2C y

C y

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

9

•W

e g

et a

co

mp

lica

ted

ex

pre

ssio

n w

hic

h i

s

nev

erth

eles

s ea

sy t

o e

val

uat

e fo

r an

y p

arti

cula

r

fun

ctio

n

.

•T

his

ex

pre

ssio

n i

s eq

uiv

alen

t to

dem

and

ing

th

at

bu

tter

flie

s h

ave

no

n-n

egat

ive

val

ue.

][y

σ

σ�@y

D≥

1

4σ@y

D3

I−4σ@y

D2+8yσ@y

Dσ′@y

D−4y2σ′@y

D2+σ@y

D4σ′@y

D2M

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

10

•A

gai

n, as

sum

ing

an

d

we

can

plo

t th

is l

ow

er b

ou

nd

on t

he

con

vex

ity

as a

fu

nct

ion

of

.

25

.0

][

=y

σ

y

3.0

]['

−=

yσ

-1

-0.5

0.5

1

-10

-7.5

-5

-2.5

2.55

7.5

10

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

11

Imp

lica

tion

fo

r V

aria

nce

Sk

ew

•P

utt

ing

tog

eth

er t

he

ver

tica

l sp

read

an

d c

on

vex

ity

con

dit

ion

s, i

t m

ay b

e sh

ow

n t

hat

im

pli

ed v

aria

nce

may

no

t g

row

fas

ter

than

lin

earl

y w

ith

th

e lo

g-

stri

ke.

•F

orm

ally

,

2[

][

] s

om

e co

nst

ant

as

yB

Sy

vy

Ay

y

σ≡

→→

∞

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

12

Lo

cal

Vo

lati

lity

•L

oca

l v

ola

tili

ty

is

giv

en b

y

•L

oca

l v

aria

nce

s ar

e n

on

-neg

ativ

e if

f ar

bit

rag

e

con

stra

ints

are

sat

isfi

ed.

()

TK

,σ

()

2

22

2

,

2

CK

TT

CK

K

σ∂ ∂

=∂ ∂

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

13

Tim

e B

ehav

ior

of

the

Skew

•S

ince

in

pra

ctic

e, w

e ar

e in

tere

sted

in

th

e lo

wer

bo

und

on

th

e sl

op

e fo

r m

ost

sto

cks,

let

’s

inv

esti

gat

e th

e ti

me

beh

avio

r o

f th

is l

ow

er b

ou

nd

.

•R

ecal

l th

at t

he

low

er b

ou

nd

on

th

e sl

op

e ca

n b

e

exp

ress

ed a

s

−−

−2

21

2

1π

exp

dN

dnsbg

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

14

•F

or

smal

l ti

mes

,

so Rei

nst

atin

g e

xp

lici

t d

epen

den

ce o

n T

, w

e g

et

Th

at i

s,

fo

r sm

all

T.

σπ

'[]

02

≥−

dN

d1

11 2

0≈

−≈

an

d bg

σπ

BS

T'[

]0

2≥

−

T

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

15

•A

lso

,

•T

hen

, th

e lo

wer

bo

un

d o

n t

he

slo

pe

•M

akin

g t

he

tim

e-d

epen

den

ce o

f

ex

pli

cit,

σπ

σ

'[]

exp

[]

02

2

12 0

1

2

1

1

≥−

−

≈−

=−

dN

d

d

nsbg

dt

1

0 2=

→∞

→∞

σ[

] a

s

σ[

]0

σσ

BS

BS

TT

'[]

[]

01

2

0≥

−→

∞ a

s

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

16

•In

par

ticu

lar,

th

e ti

me

dep

end

ence

of

the

at-t

he-

mo

ney

sk

ew c

ann

ot

be

bec

ause

fo

r an

y c

ho

ice

of

po

siti

ve

con

stan

ts a

, b

σB

ST

'[]

01

≈−

∃−

<−

Ta T

b T l

arg

e en

ou

gh s

.t.

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

17

•A

ssu

min

g

, w

e ca

n p

lot

the

var

ian

ce

slo

pe

low

er b

ou

nd

as

a fu

nct

ion o

f ti

me.

25

.0

]0[

=B

Sσ 50

100

150

200

tHyearsL

-1.2

-1

-0.8

-0.6

-0.4

-0.2

Slope

T/4

T

v 02π

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

18

A P

ract

ical

Ex

amp

le o

f A

rbit

rage

•W

e su

pp

ose

th

at t

he

AT

MF

1 y

ear

vo

lati

lity

an

d

skew

are

25

% a

nd

11

% p

er 1

0%

res

pec

tiv

ely

.

Su

pp

ose

th

at w

e ex

trap

ola

te t

he

vo

l sk

ew u

sin

g a

rule

.

•N

ow

, b

uy

99

pu

ts s

tru

ck a

t 1

01 a

nd

sel

l 10

1 p

uts

stru

ck a

t 99

. W

hat

is

the

val

ue

of

this

po

rtfo

lio

as

a fu

nct

ion

of

tim

e to

ex

pir

atio

n?

T/

1

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

19

Arb

itra

ge!

Cu

rren

t M

ark

et1

00

.00

10

0.0

01

00

.00

100

.00

Div

iden

ds

(cts

. yie

ld o

r sc

hed

ule

)0.0

0%

0.0

0%

0.0

0%

0.0

0%

Str

ike

101

.00

99.0

01

01

.00

99

.00

Sta

rt D

ate

(dat

e o

n w

hic

h s

trik

e is

set

)0

3-A

pr-

98

03-A

pr-

98

03-A

pr-

98

03-A

pr-

98

Shar

es =

s,

No

tio

nal

= n

ss

ss

Exp

irat

ion D

ate

03-A

pr-

99

03-A

pr-

99

03-A

pr-

02

03-A

pr-

02

Sto

ck R

ate

(sa/

36

5 r

ate

or

curv

e)0

.00

0%

0.0

00

%0

.00

0%

0.0

00

%

Pay

Rat

e (s

a/3

65 r

ate

or

curv

e)0

.00

0%

0.0

00

%0

.00

0%

0.0

00

%

Vo

lati

lity

(nu

mber

or

curv

e)2

3.9

0%

26

.10

%2

4.4

5%

25.5

5%

Cal

l =

c, P

ut=

pp

pp

p

Op

tion

Pric

e10.0

79.8

419.9

219.5

8

Delt

a-0

.4690

-0.4

329

-0.4

113

-0.3

916

Gam

ma (

per

1%

)0.0

166

0.0

151

0.0

080

0.0

075

Vega p

er 1

% v

ol

0.3

976

0.3

932

0.7

774

0.7

675

Th

eta

per

day

-0.0

130

-0.0

141

-0.0

065

-0.0

067

Posi

tion

99

-101

99

-101

Valu

e996.7

2

(993.7

0)

1,9

72.3

4

(1,9

77.1

8)

Port

foli

o V

alu

e3.0

2

(4.8

3)

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

20

Wit

h m

ore

rea

sonab

le p

aram

eter

s, i

t ta

kes

a l

ong

tim

e to

gen

erat

e an

arb

itra

ge

tho

ugh

….

Curr

ent

Mar

ket

100.0

0100.0

0100.0

0100.0

0

Div

iden

ds

(cts

. yie

ld o

r sc

hed

ule

)0.0

0%

0.0

0%

0.0

0%

0.0

0%

Str

ike

101.0

099.0

0101.0

099.0

0

Sta

rt D

ate

(dat

e o

n w

hic

h s

trik

e is

set

)03-A

pr-

98

03-A

pr-

98

03-A

pr-

98

03-A

pr-

98

Shar

es =

s,

No

tio

nal

= n

ss

ss

Expir

atio

n D

ate

03-A

pr-

99

03-A

pr-

99

03-A

pr-

48

03-A

pr-

48

Sto

ck R

ate

(sa/

365 r

ate

or

curv

e)0.0

00%

0.0

00%

0.0

00%

0.0

00%

Pay

Rat

e (s

a/365 r

ate

or

curv

e)0.0

00%

0.0

00%

0.0

00%

0.0

00%

Vo

lati

lity

(num

ber

or

curv

e)24.7

0%

25.3

0%

24.9

6%

25.0

4%

Cal

l =

c, P

ut=

pp

pp

p

Op

tion

Pric

e10.3

99.5

263.0

761.6

1

Delt

a-0

.4668

-0.4

340

-0.1

902

-0.1

864

Gam

ma (

per

1%

)0.0

161

0.0

156

0.0

015

0.0

015

Vega p

er 1

% v

ol

0.3

975

0.3

934

1.8

909

1.8

670

Th

eta

per d

ay

-0.0

135

-0.0

136

-0.0

013

-0.0

013

Posi

tion

99

-101

99

-101

Valu

e1,0

28.2

1

(961.9

2)

6,2

44.1

4

(6,2

22.6

8)

Port

foli

o V

alu

e66.3

0

21.4

6

50

Yea

rs!

No

arb

itra

ge!

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

21

So

Far

….

•W

e h

ave

der

ived

arb

itra

ge

con

stra

ints

on

th

e sl

op

e

and

co

nv

exit

y o

f th

e v

ola

tili

ty s

kew

.

•W

e h

ave

dem

on

stra

ted

th

at t

he

ru

le f

or

extr

apo

lati

ng

th

e sk

ew i

s in

consi

sten

t w

ith

no

arb

itra

ge.

T

ime

dep

end

ence

mu

st b

e at

mo

st

for

larg

e T

T/

1

T/1

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

22

Sto

chas

tic

Vo

lati

lity

•C

on

sid

er t

he

foll

ow

ing

sp

ecia

l ca

se o

f th

e H

esto

n

mo

del

:

•In

th

is m

od

el, it

can

be

sho

wn

th

at

()

dx

dt

vd

Z

dv

vv

dt

vd

Z

µ

λη

=+

=−

−−

0

11

1T

BS

y

ve

yT

T

λ

ηλ

λ

−

=

∂−

≈−

−

∂

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

23

•F

or

a g

ener

al s

toch

asti

c v

ola

tili

ty t

heo

ry o

f th

e

form

:

wit

h

we

clai

m t

hat

(v

ery

ro

ug

hly

)

()

()

1

2

dx

dt

vdZ

dv

vv

dt

vv

dZ

µ

λη

β

=+

=−

−−

12

,d

Zd

Zd

tρ

= ()

0

11

1T

BS

y

ve

vy

TT

λ

ρη

βλ

λ

−

=

∂−

≈−

∂

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

24

•T

hen

, fo

r v

ery

sh

ort

ex

pir

atio

ns,

we

get

-a

resu

lt o

rig

inal

ly d

eriv

ed b

y R

og

er L

ee a

nd

fo

r

ver

y l

ong

ex

pir

atio

ns,

we

get

•B

oth

of

thes

e re

sult

s ar

e co

nsi

sten

t w

ith

the

arb

itra

ge

bo

un

ds.

()

02

BS

y

vv y

ρη

β

=

∂≈

∂

()

0

BS

y

vv y

T

ρη

β λ=

∂≈

∂

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

25

Does

n’t

Th

is C

on

trad

ict

?

•M

ark

et p

ract

itio

ner

s’ru

le o

f th

um

b i

s th

at t

he

skew

dec

ays

as

.

•U

sin

g

(fro

m B

aksh

i, C

ao a

nd

Ch

en),

we

get

th

e fo

llo

win

g g

rap

h f

or

the

rela

tiv

e si

ze o

f

the

at-t

he-

mo

ney

var

ian

ce s

kew

:

T

1T

1.1

5λ

=

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

26

0.5

11.5

2T

-1.8

-1.6

-1.4

-1.2-1

-0.8

Relative

Skew

AT

M S

kew

as

a F

unct

ion

of

Sto

chas

tic

Vol.

(

)

T

1.1

5λ

= 1T

Act

ual

SP

X s

kew

(5/3

1/0

0)

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

27

-1

-0.5

0.5

1

0.04

0.06

0.08

0.12

0.14

Hes

ton

Im

pli

ed V

aria

nce

Imp

lied

Var

ian

ce

y=

ln(K

/F)

Par

amet

ers:

fro

m B

aksh

i, C

ao a

nd C

hen

.

vv

=0

.=

0.

,=

.,

=-0

.,

=0

.0

4,

04

115

39

64

λρ

η

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

28

A S

imp

le R

egim

e S

wit

chin

g M

od

el

•T

o g

et i

ntu

itio

n f

or

the

imp

act

of

vo

lati

lity

con

vex

ity

, w

e su

pp

ose

th

at r

eali

sed

vola

tili

ty o

ver

the

life

of

a o

ne

yea

r o

pti

on

can

tak

e o

ne

of

two

val

ues

eac

h w

ith

pro

bab

ilit

y 1

/2. T

he

aver

age

of

thes

e v

ola

tili

ties

is

20%

.

•T

he

pri

ce o

f an

op

tion

is

just

the

aver

age

op

tio

n

pri

ce o

ver

th

e tw

o s

cen

ario

s.

•W

e g

rap

h t

he

imp

lied

vo

lati

liti

eso

f th

e re

sult

ing

op

tio

n p

rice

s.

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

29

Hig

h V

ol:

21

%;

Lo

w V

ol:

19

%

19

.98

0%

20

.00

0%

20

.02

0%

20

.04

0%

20

.06

0%

20

.08

0%

20

.10

0%

20

.12

0%

-0.5

-0.4

-0.3

-0.2

-0.1

00

.10

.20

.30

.40

.5

y

Implied Volatility

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

30

Hig

h V

ol:

39

%;

Lo

w V

ol:

1%

15

%

17

%

19

%

21

%

23

%

25

%

27

%

29

%

31

%

33

%

-0.5

-0.4

-0.3

-0.2

-0.1

00

.10

.20

.30

.40

.5

y

Implied Volatility

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

31

Intu

itio

n•

As

, im

pli

ed v

ola

tili

ty t

end

s to

th

e h

igh

est

vo

lati

lity

.

•If

vo

lati

lity

is

un

bo

un

ded

, im

pli

ed v

ola

tili

ty m

ust

also

be

unb

ou

nd

ed.

•F

rom

a t

rad

er’s

per

spec

tiv

e, t

he

mo

re o

ut-

of-

the-

mo

ney

(O

TM

) an

op

tion

is,

th

e m

ore

vo

l

con

vex

ity

it

has

. P

rov

ided

vo

lati

lity

is

un

bou

nd

ed,

mo

re O

TM

op

tio

ns

mu

st c

om

man

d

hig

her

im

pli

ed v

ola

tili

ty.

y→

∞

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

32

-1

-0.5

0.5

1

0.02

0.04

0.06

0.08

0.12

0.14

Asy

mm

etri

c V

aria

nce

Gam

ma

Imp

lied

Var

ian

ce

Imp

lied

Var

ian

ce

y=

ln(K

/F)

Par

amet

ers:

w=

0.0

4,

=0

.1,

=-1

.ν

θρ

50

4,

.=

−

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

33

Jum

p D

iffu

sio

n

•C

on

sid

er t

he

sim

ple

st f

orm

of

Mer

ton

’s j

um

p-

dif

fusi

on

mo

del

wit

h a

co

nst

ant

pro

bab

ilit

y o

f a

jum

p t

o r

uin

.

•C

all

op

tion

s ar

e v

alued

in

th

is m

od

el u

sin

g t

he

Bla

ck-S

cho

les

form

ula

wit

h a

sh

ifte

d f

orw

ard

pri

ce.

•W

e g

rap

h 1

yea

r im

pli

ed v

aria

nce

as

a fu

nct

ion

of

log

-str

ike

wit

h

:

λ

0.0

4,

0.0

5v

λ=

=

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

34

-1

-0.5

0.5

1

0.1

0.2

0.3

0.4

0.5

Jum

p-t

o-R

uin

Mo

del

Imp

lied

Var

ian

ce

y=

ln(K

/F)

Par

amet

ers:

v=

0.0

4,

=0

.05

λ

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

35

•S

o, ev

en i

n j

um

p-d

iffu

sion

, i

s li

nea

r in

as

.

•In

fac

t, w

e ca

n s

ho

w t

hat

fo

r m

any

eco

nom

ical

ly

reas

on

able

sto

chas

tic-

vo

lati

lity

-plu

s-ju

mp

mo

del

s,

imp

lied

BS

var

ian

ce m

ust

be

asy

mp

toti

call

y l

inea

r

in t

he

log

-str

ike

a

s .

•T

his

mea

ns

that

it

does

no

t m

ake

sen

se t

o p

lot

imp

lied

BS

var

ian

ce a

gai

nst

del

ta. A

s an

ex

amp

le,

con

sid

er t

he

foll

ow

ing

gra

ph

of

v

s. d in

th

e

Hes

ton

mo

del

:

yy

→∞

v

yy

→∞

v

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

36

Var

iance

vs d in

th

e H

esto

n

Mo

del

0.2

0.4

0.6

0.8

1

0.05

0.06

0.07

0.08

Var

iance

d

Ris

k 2

000

, T

ues

day

13

Jun

e 20

00

37

Imp

lica

tion

s fo

r P

aram

eter

izat

ion

of

the

Vo

lati

lity

Su

rfac

e

•Im

pli

ed B

S v

aria

nce

mu

st b

e p

aram

eter

ized

in

term

s o

f th

e lo

g-s

trik

e (

vsd

elta

do

esn

’t w

ork

).

•is

asy

mp

toti

call

y l

inea

r in

as

•d

ecay

s as

as

•te

nd

s to

a c

onst

ant

as

y→

∞

vy

vy

0y

v y=

∂ ∂

1 TT

→∞

0y

v y=

∂ ∂0

T→