8/3/2019 Mad Rid 2004
1/41
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
8/3/2019 Mad Rid 2004
2/41
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.
8/3/2019 Mad Rid 2004
3/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
4/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
5/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
6/41
Jim Gatheral, Merrill Lynch, May-2004
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,
8/3/2019 Mad Rid 2004
7/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
8/41
Jim Gatheral, Merrill Lynch, May-2004
-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
8/3/2019 Mad Rid 2004
9/41
Jim Gatheral, Merrill Lynch, May-2004
-1 -0.5 0.5 1k
0.2
0.4
0.6
0.8
1
Implied variance
b 0 8.
Changing b
8/3/2019 Mad Rid 2004
10/41
Jim Gatheral, Merrill Lynch, May-2004
-1 -0.5 0.5 1k
0.1
0.2
0.3
0.4
0.5
0.6
Implied variance
0 2.
Changing
8/3/2019 Mad Rid 2004
11/41
Jim Gatheral, Merrill Lynch, May-2004
-1 -0.5 0.5 1k
0.1
0.2
0.3
0.4
0.5
Implied variance
08.
Changing
8/3/2019 Mad Rid 2004
12/41
Jim Gatheral, Merrill Lynch, May-2004
-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
8/3/2019 Mad Rid 2004
13/41
Jim Gatheral, Merrill Lynch, May-2004
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
+
8/3/2019 Mad Rid 2004
14/41
Jim Gatheral, Merrill Lynch, May-2004
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.
8/3/2019 Mad Rid 2004
15/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
16/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
17/41
Jim Gatheral, Merrill Lynch, May-2004
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
Vols comparison for SPX Index expiring 07/16/04
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 0 1 ,3 00 .0 0Strike
Volatility
Bid Vol
Ask Vol
SVIVol
Vols comparison for SPX Index expiring 09/17/04
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
Vols comparison for SPX Index expiring 12/17/04
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
Vols comparison for SPX Index expiring 03/18/05
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
Vols comparison for SPX Index expiring 06/17/05
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
8/3/2019 Mad Rid 2004
18/41
Jim Gatheral, Merrill Lynch, May-2004
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!
8/3/2019 Mad Rid 2004
19/41
Jim Gatheral, Merrill Lynch, May-2004
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.
8/3/2019 Mad Rid 2004
20/41
Jim Gatheral, Merrill Lynch, May-2004
-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=
8/3/2019 Mad Rid 2004
21/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
22/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
23/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
24/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
25/41
Jim Gatheral, Merrill Lynch, May-2004
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=
8/3/2019 Mad Rid 2004
26/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
27/41
Jim Gatheral, Merrill Lynch, May-2004
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
= +
=
8/3/2019 Mad Rid 2004
28/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
29/41
Jim Gatheral, Merrill Lynch, May-2004
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.
8/3/2019 Mad Rid 2004
30/41
Jim Gatheral, Merrill Lynch, May-2004
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.
8/3/2019 Mad Rid 2004
31/41
Jim Gatheral, Merrill Lynch, May-2004
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.
8/3/2019 Mad Rid 2004
32/41
Jim Gatheral, Merrill Lynch, May-2004
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
=
8/3/2019 Mad Rid 2004
33/41
Jim Gatheral, Merrill Lynch, May-2004
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.
8/3/2019 Mad Rid 2004
34/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
35/41
Jim Gatheral, Merrill Lynch, May-2004
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 = = = = =
8/3/2019 Mad Rid 2004
36/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
37/41
Jim Gatheral, Merrill Lynch, May-2004
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
;
8/3/2019 Mad Rid 2004
38/41
Jim Gatheral, Merrill Lynch, May-2004
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
8/3/2019 Mad Rid 2004
39/41
Jim Gatheral, Merrill Lynch, May-2004
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!!
8/3/2019 Mad Rid 2004
40/41
Jim Gatheral, Merrill Lynch, May-2004
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.
8/3/2019 Mad Rid 2004
41/41
Jim Gatheral, Merrill Lynch, May-2004
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.
Top Related