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A parsimonious arbitrage-free implied volatility parameterization with

application to the valuation of volatility derivatives

Jim Gatheral

Global Derivatives & Risk Management 2004

Madrid

May 26, 2004

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Jim Gatheral, Merrill Lynch, May-2004

The opinions expressed in this presentation are those of the author alone,and do not necessarily reflect the views of of Merrill Lynch, its

subsidiaries or affiliates.

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Outline of this talk

n Roger Lees moment formulan A stochastic volatility inspired (SVI) pararameterization of the implied

volatility surface

n No-arbitrage conditions

n SVI fits to market data

n SVI fits to theoretical models

n Carr-Lee valuation of volatility derivatives under the zero correlation

assumption

n Valuation of volatility derivatives in the general case

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Roger Lees moment formula

{ }

{ }

2

* * BS

2*

* *

*

21* * BS

** *

*

Define : log( / ).

( , )Let : sup : and : limsup

k

1 1Then [0,2] and

2 2

( , )Also let : sup : and : limsup

k

1 1Then [0,2] and

2 2

q

Tk

p

Tk

k K F

k T Tq q S

q

k T Tp p S

p

+

+

=

= < =

=

= < =

=

E

E

2

n The derivation assumes only the existence of a Martingale measure: it

makes no assumptions on the distribution of .TS

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Implications of the moment formula

n Implied variance is always linear in kasn For many models, and may be computed explicitly in terms of the

model parameters.

n So, if we want a parameterization of the implied variance surface, it

needs to be linear in the wings!n and it needs to be curved in the middle - many conventional

parameterizations of the volatility surface are quadratic for example.

k *

p*q

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var( ; , , , , ) ( ) ( )k a b m a b k m k m = + + +2 2{ }

Left and right asymptotes are:var ( ; , , , , ) ( ) ( )

var ( ; , , , , ) ( ) ( )

L

R

k a b m a b k m

k a b m a b k m

= = + +

1

1

The SVI (stochastic volatility inspired)

parameterization

n Variance is always positive

n Variance increases linearly with |k| for k very positive or very negative

Intuition is that the more out-of-the-money an option is, the more volatilityconvexity it has.

n For each timeslice,

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a

b

m

a b m

gives the overall level of variance

gives the angle between the left and right asymptotes

determines how smooth the vertex is

determines the orientation of the graph

changing translates the graph

The next few slides show this graphically with base case

= = = = =0 04 0 4 01 0 4 0. , . , . , . ,

What the SVI parameters mean

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-1 -0.5 0.5 1k

0.1

0.2

0.3

0.4

0.5

0.6

Implied variance

a 0 08.

Changing a

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-1 -0.5 0.5 1k

0.2

0.4

0.6

0.8

1

Implied variance

b 0 8.

Changing b

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-1 -0.5 0.5 1k

0.1

0.2

0.3

0.4

0.5

0.6

Implied variance

0 2.

Changing

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-1 -0.5 0.5 1k

0.1

0.2

0.3

0.4

0.5

Implied variance

08.

Changing

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-1 -0.5 0.5 1k

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Implied variance

m 0 2.

Changing m

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Preventing arbitrage

n In practice, if SVI is fitted to actual option price data, negative verticalspreads never arise. In terms of the SVI (implied variance fit)

parameters, the condition is

n It turns out that if there are no negative vertical spreads, negative

butterflies are also excluded.

n In contrast, it is not obvious how to prevent negative time spreads. We

now show what conditions a parameterization needs to satisfy to exclude

arbitrage between expirations (calendar spread arbitrage).

( )4

1bT

+

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Necessary and sufficient condition for no calendar

spread arbitragen First we note that for any Martingale and it is easy to show

that

n Now consider the non-discounted values and of two options with

strikes and and expirations and with . Suppose the two

options have the same moneyness so that

( ) ( )2 1t t

X L X L+ +

E E

2 1t ttX

2 1t t>1C 2C

1K 2K 1t 2t

1 2

1 2

K Kk

F F=

n

Consider the martingale . Then, in the absence of arbitrage,we must have

/t t t

X S F

( ) ( )2 1

2 1

2 1

1 1t t

C CX k X k

K k k K

+ += == E E

n So, keeping the moneyness constant, option prices are non-decreasing in

time to expiration.

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Derivation of no arbitrage condition continued

n Letn The Black-Scholes formula for the non-discounted value of an option

may be expressed in the form with strictly increasing in

the total implied variance .

n It follows that for fixed k, we must have the total implied variance

non-decreasing with respect to time to expiration.

( )2

,t BSw k t t

( , )BS t

C k wBS

C

tw

tw

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SVI fit to SPX with one week to expiration

Vols comparison for SPX Index expiring 05/21/04

10%

15%

20%

25%

30%

35%

40%

900.00 950.00 1,000.00 1,050.00 1,100.00 1,150.00 1,200.00 1,250.00Strike

Volatility

Bid Vol

Ask Vol

SVI Vol

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Other expirationsVols comparison for SPX Index expiring 06/18/04

10%

12%

14%

16%

18%

20%

22%

24%

26%

28%

30%

9 00 .0 0 9 50 .0 0 1 ,0 00 .0 0 1 ,0 50 .0 0 1 ,1 00 .0 0 1 ,1 50 .0 0 1 ,2 00 .0 0 1 ,2 50 .0 0Strike

Volatility

Bid Vol

Ask Vol

SVIVol


10%

12%

14%

16%

18%

20%

22%

24%

26%

28%

30%

9 00 .0 0 9 50 .0 0 1 ,0 00 .0 0 1 ,0 50 .0 0 1 ,1 00 .0 0 1 ,1 50 .0 0 1 ,2 00 .0 0 1 ,2 50 .0 0 1 ,3 00 .0 0Strike

Volatility

Bid Vol

Ask Vol

SVIVol


10%

12%

14%

16%

18%

20%

22%

24%

26%

28%

30%

9 00 .0 0 9 50 .0 0 1 ,0 00 .0 0 1 ,0 50 .0 0 1 ,1 00 .0 0 1 ,1 50 .0 0 1 ,2 00 .0 0 1 ,2 50 .0 0Strike

Volatility

Bid Vol

Ask Vol

SVIVol


10%

12%

14%

16%

18%

20%

22%

24%

26%

28%

30%

9 00 .0 0 9 50 .0 0 1 ,0 00 .0 0 1 ,0 50 .0 0 1 ,1 00 .0 0 1 ,1 50 .0 0 1 ,2 00 .0 0 1 ,2 50 .0 0Strike

Volatility

Bid Vol

Ask Vol

SVIVol


10%

12%

14%

16%

18%

20%

22%

24%

26%

28%

30%

9 0 0.0 0 9 5 0.0 0 1 ,00 0 .0 0 1 ,05 0 .0 0 1 ,10 0 .0 0 1 ,15 0 .0 0 1 ,20 0 .0 0 1 ,25 0 .00 1 ,30 0 .0 0 1 ,35 0 .0 0 1 ,4 00 .0 0Strike

Volatility

Bid Vol

Ask Vol

SVIVol


10%

12%

14%

16%

18%

20%

22%

24%

26%

28%

30%

90 0. 00 9 20 .00 9 40. 00 96 0. 00 9 80 .00 1 ,00 0. 00 1 ,0 20 .0 0 1 ,0 40 .0 0Strike

Volatility

Bid Vol

Ask Vol

SVIVol

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The total variance plot

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

-2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00

05/21/2004

06/18/200406/18/2004

09/17/2004

12/17/2004

03/18/2005

06/17/2005

12/16/2005

n Note no crossed lines so no arbitrage!

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Review of the work of Schoutens et al.

n In their recent Wilmott Magazine paper, Schoutens, Simons and Tistaertcarefully fitted 7 different models to the STOXX50 index option market

observed on 7-Oct-2003.

n The models fitted were

HEST, HESJ, BNS, VG-CIR, VG-OU, NIG-CIR, NIG-OU

n These models represent a range of quite different dynamics. For

example, HEST is a pure stochastic volatility model and VG-CIR is apure jump model with subordinated time.

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-0.6 -0.4 -0.2 0.2 0.4k

0.05

0.15

0.2

0.25

0.3

0.35

0.4

Implied Vol .

-0.3 -0.2 -0.1 0.1 0.2 0.3k

0.05

0.15

0.2

0.25

0.3

0.35

0.4

Implied Vol .

-0.15 -0.1 -0.05 0.05 0.1 0.15k

0.05

0.15

0.2

0.25

0.3

0.35

0.4

Implied Vol .

-1.25 -1 -0.75 -0.5 -0.25 0.25 0.5 0.75k

0.05

0.15

0.2

0.25

0.3

0.35

0.4

Implied Vol .

-1.5 -1 -0.5 0.5k

0.05

0.15

0.2

0.25

0.3

0.35

0.4

Implied Vol .

-1.5 -1 -0.5 0.5 1k

0.05

0.15

0.2

0.25

0.3

0.35

0.4

Implied Vol .

Plots of implied vol. vs log-strike k for 6 of the

Schoutens et al. models

0.0361t= 0.2000t= 1.1944t=

2.1916t= 5.1639t=4.2056t=

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SVI fit to VG-OUmodel with Schoutens et al.

parameters

n The above graph shows the SVI fit to the first timeslice (t=0.0361 years).

This slice is invariably the hardest to fit in practice; SVI fits almost

perfectly.

-0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2k

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

Implied variance

c 6.1610, g 9.6443, m 16.026, l 1.679, a 0.3484, b 0.7664, y0 1

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Total variance plot for VG-OUwith Schoutens et al.parameters

n Note that the surface has to be arbitrage-free because it is generated from

explicit arbitrage-free dynamics.

n Note also that SVI fits every slice extremely well!

-1 -0.5 0.5 1

k

0.1

0.2

0.3

0.4

0.5

Implied total variance

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SVI fit to the Heston model with Schoutens et al.

parameters

n The above graph shows the SVI fit to the first timeslice (t=0.0361 years).

Again the fit is very good but fitting this slice on its own is not asking

very much...

l 0.6067, r -0.7571, h 0.2928, vbar 0.0707, v .0654

-0.2 -0.1 0.1 0.2k

0.02

0.04

0.06

0.08

Implied variance

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Total variance plot for Heston with Schoutens et al.

parameters

n SVI fits somewhat less well this time.

-1.5 -1 -0.5 0.5 1 1.5k

0.1

0.2

0.3

0.4

0.5

Implied total variance

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The Heston Model

n The model is given by

with

1

2

1 2

2

( )

,

vdx dt v dZ

dv v v dt v dZ

dZ dZ dt

= +

= +

=

( )0log / x F S=

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The Heston model as

n From (e.g.)Lewis, we can derive the the level, slope and curvature of at-the-money forward (ATMF) implied volatility from the parameters of

the model. In the case of the Heston model as , these are

T

with

n From Roger Lee, we can compute the slope of the asymptotes of the

implied variance by finding the poles of the Heston characteristic function. The

resulting slopes of the asymptotes again as are given by

k

{ }

{ }{ }

2 2 2

2 2

2 2 2

2

2 2 2

2

4(0) 4 (1 ) 2

(1 )

4(0) 4 (1 ) 2

(1 )2

(0) 4 (1 ) 2

T

T

T

vv

v

T

vv T

+

+

+

:

:

:

{ }

{ }

* 2 2 2

* 2 2 2

2( ) ( ) 4 (1 ) 2

(1 )

2( ) ( ) 4 (1 ) 2

(1 )

T

T

v T T

v T T

+ = +

= + +

/ 2; .v v = =

T

T

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Consistency with SVI

n It turns out that these formulae are totally consistent with SVI if we havethe following correspondence between SVI parameters and Heston

parameters (as ):T 2

2

(1 )2

2

1

a

b

m

=

=

=

=

=

with { }2 2 22 24

4 (1 ) 2(1 )

v

v T

= +

=

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Review of Carr-Lee valuation of volatility derivatives

n Assumes zero correlation between quadratic variation and returns.n Carr and Lee show how under this assumption, the prices of volatility

derivatives are completely determined by the prices of European options

(which we assume to be known). Formally

wheneverf(y) has a Laplace transform for on the right of all

singularities ofF

n Friz and Gatheral show that more often than not, the weights are ill-

defined and the problem of inverting to find the weights is ill-posed.

n However, they also show how to compute the value of volatility

derivatives from the values of European options to any desired accuracy.

( ) ( ) ( , )ftf x dk w k c k T = E

fw

with / 2

( ) 4 ( ) cosh 1/4 22

a ik

f

a i

ew k F k d

i

+

= + ( )F a R

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Extension to non-zero correlation

n Suppose that (e.g.) Heston dynamics really drive the market.n We note that the values of volatility derivatives are independent of the

correlation between stock returns and volatility moves.

n To compute the fair value of a volatility derivative, all we would need to

do is to set to zero and apply the inversion method described in Friz

and Gatheral.

n This suggests the following general recipe:

n Start with some parameterization of the volatility surface.

n Noting that when is zero, the implied variance smile is symmetric

around k=0, rotate the curve keeping its overall level the same.

Although this step is intuitively clear, its meaning is far from precise and its

implementation is far from obvious

n Apply the Friz algorithm to compute the value of the volatility

derivative.

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The Friz inversion algorithm

n We recall from Hull and White that in a zero-correlation world, we maywrite

( ) ( ) ( , )BSc k dy g y c k y= where is the total variance. In words, we can compute an

option price by averaging over Black-Scholes option prices conditioned

on the BS total implied variance.

n We discretize the above to obtain

2:

BSy T=

1

( 1,.., )m

i ij j

j

c A g i n=

= =with .

n We may then compute the probability vector g using the Moore-Penrose

pseudo-inverse as

m n?

( )1

:T T= =g A A A c M c

n This is effectively a method for deducing the law of implied variance

from option prices. Then, the expectation of any function of variance

may be computed.

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Example: SVI

n

Although SVI has explicitly as a parameter, its not enough to setand apply the Friz method.

This can be seen easily by computing the value of a variance swap before

and after setting .

n Other of the SVI parameters may and do in fact depend on . To see

how they should transform, we refer back to the mapping between SVI

and Heston parameters in the limit. Specifically,

0 =

0 =

T

2

0

0

1

m

Instead ofspecifying exactly how a and b transform (which is pretty

complicated) we note that they transform as

2(1 )

a a

b b

We find by imposing that the expected variance be conserved under

the transformation.

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A formula for computing expected total variance

n

It may be shown that for diffusion processes

2

0

( ) ( )BSTx dz N z z

= E

with

2

( , )( ) 2( , )

BS

BS

k T Tkz d k

k T T

=

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Recipe for valuation of volatility derivatives

n

Fit SVI parameterization to the implied volatility surfacen Transform the surface so it is symmetric around the forward price as

follows:

2

2

(1 )

0

0

1

a a

b b

m

n Adjust using the total variance formula to keep the expected total

variance invariant.

n Apply the Friz method or Carr-Lee directly to compute the value of the

volatility derivative directly in this new zero-correlation world.

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Example: One year Heston with BCC parameters

n

We compute one year European option prices in the Heston model usingparameters from Bakshi, Cao and Chen. Specifically

0.04; 0.39; 1.15; 0.64v v = = = = =

n We fit SVI and obtain the following SVI parameters:

-1 -0.5 0.5 1

0.02

0.04

0.06

0.08

0.12

.0159479; 0.0577371; 0.568899; 0.127445; 0.165476a b m = = = = =

n Applying the total variance formula gives which

is not too far off the exact value of 0.04.

0.0400846T

x = E

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Example: continued

n

We rotate the SVI parameters according to our recipe to obtain the newSVI parameters

n Graph is as expected and total variance is 0.0400846 by construction

-1 -0.5 0.5 1

0.05

0.06

0.07

0.08

-1 -0.5 0.5 1

0.02

0.04

0.06

0.08

0.12

n As a check, regenerate Heston option prices with and refit SVI.We obtain

0 =

0.0233521; 0.06087352; 0; 0.1914214; 0;a b m = = = = =

n To recap, we guessed a SVI rotation transformation from the

limit. This transformation seems to work pretty well even for T=1!

T

0.024973; 0.0611502; 0; 0.154966; 0;a b m = = = = =

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True zero-correlation Heston and rotated SVI smiles

n The closeness of agreement of the two curves is all we need to show that

the values of all volatility derivatives are correctly obtained through therotation recipe (at least in this case).

n We now compute the value of the volatility swap to verify this.

-1 -0.5 0.5 1

0.05

0.06

0.07

0.08

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A formula for computing expected volatility

n

It may be shown that for diffusion processes under the zero-correlationassumption

with

/ 2 1( ) 2 ( ) 2 2

k

f

kw k k e I

= +

( ) ( , )ftx dk w k c k T = E

where is a modified Bessel function of the first kind.

n To motivate this formula, recall that

( )1 .I

(0)

(0, ) 2

BST

c t

;

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Heston formula for expected volatility

n

By a well-known formula

n Then, taking expectations

3 / 2

0

1 1

2

ye

y d

=

n We know the Laplace transform from the CIR bond formula.

n So we can also compute expected volatility explicitly in terms of the

Heston parameters.

3 / 20

11

2

Tx

T

e

x d

=

E

E

Tx

e

E

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Results of volatility swap computation

n

Applying the CIR bond formula and computing Heston one yearexpected volatility exactly gives

0.187429T

x = E

n Applying the SVI rotation recipe and computing expected volatility

using the Carr-Lee weights gives

0.185871T

x = E

n Error is only 0.16 vol. points!!

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Summary

n

Motivated by the asymptotic behavior of the implied volatility smile atextreme strikes, we introduced a parameterization (SVI).

n We demonstrated that this parameterization not only fits SPX option

prices extremely well but that it fits option prices generated by many

theoretical models. These models include stochastic volatility and pure

jump models.

n

By identifying the form of the implied volatility smile as SVI for theHeston model in the limit of infinite time to expiration, we were able to

map the SVI parameters to parameters of the Heston model (in this

limit).

n Noting that volatility derivatives (under a stochastic volatility

assumption) cannot depend on the correlation between underlying

returns and volatility changes, we developed a recipe transforming a

given set of SVI parameters into their zero-correlation world equivalents.

n We then applied Carr-Lee to the option prices thus generated to value

volatility derivatives.

n We demonstrated that in the particular case of Heston one year European

options with BCC parameters, agreement was very close.

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References

n

Bakshi, G., Cao C. and Chen Z. (1997). Empirical performance of alternativeoption pricing models. Journal of Finance, 52, 2003-2049.

n Carr, Peter and Lee, Roger (2003), Robust Replication of Volatility Derivatives,

Courant Institute, NYU and Stanford University.

n Friz, Peter and Gatheral, J. (2004), Valuation of Volatility Derivatives as an

Inverse Problem,Merrill Lynch and Courant Institute, NYU.

n Gatheral, J. (2003). Case studies in financial modeling lecture notes.

http://www.math.nyu.edu/fellows_fin_math/gatheral/case_studies.htmln Heston, Steven (1993). A closed-form solution for options with stochastic

volatility with applications to bond and currency options. The Review of Financial

Studies6, 327-343.

n Hull, John and White Alan (1987), The pricing of options with stochastic

volatilities, The Journal of Finance 19, 281-300.

n Lee, Roger (2002). The Moment Formula for Implied Volatility at Extreme

Strikes.Forthcoming inMathematical Finance.

n Lewis, Alan R. (2000), Option Valuation under Stochastic Volatility : with

Mathematica Code, Finance Press.

n Schoutens W., Simons E. and Tistaert J. (2004). A Perfect Calibration! Now

What?, Wilmott Magazine, March, pages 66-78.

Mad Rid 2004

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