Modeling the Implied Volatility Surface
Jim GatheralGlobal Derivatives and Risk Management 2003BarcelonaMay 22, 2003
Jim Gatheral, Merrill Lynch, May-2003
This presentation represents only the personal opinions of the author andnot those of Merrill Lynch, its subsidiaries or affiliates.
Jim Gatheral, Merrill Lynch, May-2003
Outline of this talk
n A compound Poisson model of stock tradingn Empirical verification of modeling assumptionsn Stochastic volatilityn Empirical dynamics of SPX and VIXn Dynamics of the implied volatility skewn Which stochastic volatility model?n Do stochastic volatility models fit option prices?n Jumpsn The impact of large option trades
Jim Gatheral, Merrill Lynch, May-2003
Stock trading as a compound Poisson process
n Consider a random time change from conventional calendar time totrading time such that the rate of arrival of stock trades in a given(transformed) time interval is a constant .
n Intuitively, relative to real time, trading time flows faster when there ismore activity in the stock and more slowly when there is less activity.
n Suppose that the (random) size of a trade is independent of the level ofactivity in the stock.
n Assume further that each trade impacts the mid-log-price of the stock byan amount proportional to .
• This is a standard assumption in the market microstructure literature
n Then the change in log-mid-price over some time interval is given by
n Note that both the number of trades and the size of each trade in agiven time interval are random.
λ
n
n
1
sgn( )N
i ii
x n n=
∆ = ∑
N int∆
Jim Gatheral, Merrill Lynch, May-2003
Volatility and volume: a relationship
n The variance of this random sum of random variables is given by
n Rewriting this in terms of volatility, we obtain
n But is just the expectation of the volume over the timeinterval . Let where is the trading rate.
n The factor cancels and transforming back to real time, we see thatvariance is directly proportional to volume in this simple model.
n Moreover, the distribution of returns in trading time is approximatelyGaussian for large .
[ ] [ ] [ ][ ] [ ]
2
2
i i
i
Var x E N Var n Var N E n
E N E n
α α
α
∆ = + =
[ ] [ ] [ ]2iVar x t E N E nσ α2∆ = ∆ =
[ ] [ ]iE N E nt∆
t∆
t∆
[ ]E N tλ= ∆ λ
Jim Gatheral, Merrill Lynch, May-2003
The relationship
n A key assumption in our simple model is that market impact isproportional to the square root of the trade size . The followingargument shows why this is plausible:
• A market maker requires an excess return proportional to the risk of holdinginventory.
• Risk is proportional to where is the holding period.• The holding period should be proportional to the size of the position.• So the required excess return must be proportional to .
n
n
Tσ T
n
Jim Gatheral, Merrill Lynch, May-2003
Independence of trade size and trade frequency*
* Excluding trades over 10,000 shares
INTC: 30 minute buckets from 10/1/2002 to 4/25/2003
y = -0.0011x + 910.03R2 = 0.0003
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
Frequency
Ave
rage
Siz
e
Jim Gatheral, Merrill Lynch, May-2003
Independence of trade size and trade frequency*
* Excluding trades over 10,000 shares
MER: 30 minute buckets from 10/1/2002 to 4/25/2003
y = -0.0221x + 910.37R2 = 9E-05
0
500
1000
1500
2000
2500
0 100 200 300 400 500 600 700 800 900
Frequency
Ave
rag
e S
ize
Jim Gatheral, Merrill Lynch, May-2003
Empirical variance vs volume
INTC from 4/30/2001 to 4/25/2003
y = 5E-11xR2 = 0.2183
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0 40,000,000 80,000,000 120,000,000 160,000,000 200,000,000
Volume (Shares)
ln^2
(hi/l
o)
Jim Gatheral, Merrill Lynch, May-2003
Empirical variance vs volume
MER from 2/26/2001 to 4/25/2003
y = 3E-10xR2 = 0.2854
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0 5,000,000 10,000,000 15,000,000 20,000,000 25,000,000
Volume (Shares)
ln^2
(hi/l
o)
Jim Gatheral, Merrill Lynch, May-2003
Implications of our simple model
n Our simple but realistic model has the following modeling implications• Log returns are roughly Gaussian with constant variance in trading time
(sometimes call intrinsic time) defined in terms of transaction volume• Trading time is the inverse of variance• When we transform from trading time to real time, variance appears to be
random
n It is natural to model the log stock price as a diffusion processsubordinated to another random process which is really trading volumein our model - stochastic volatility
n What form should the volatility process take?
Jim Gatheral, Merrill Lynch, May-2003
Empirical observations
n In the underlying index return data we observe• Clustering: large moves follow large moves, small moves follow small
moves• Mean reversion of volatility• Negative correlation between implied volatility moves and index returns
n In the implied volatility data we observe• The volatility envelope: short-dated volatilities move more than long-dated
ones• A pronounced skew with downside implied volatilities significantly higher
than upside implied volatilities.• Implied volatilities move more when implied volatility is high.
Jim Gatheral, Merrill Lynch, May-2003
A generic stochastic volatility model
n We are now in a position to write down a generic stochastic volatilitymodel consistent with our observations. Let denote the log stock priceand denote its variance. Then
with .
n is a mean-reversion term, is the correlation between volatilitymoves and stock price moves and is called “volatility of volatility”.
n
n To estimate we analyze historical SPX and VIX data.
( )1
2
dx dt v dZ
dv v v dZβ
µ
α η
= +
= +
1 2,dZ dZ dtρ=
x
( )vα ρη
12
3 32 2
gives the Heston model
=1 gives Wiggins' lognormal model
gives Lewis's model
β
β
β
=
=
v
β
Jim Gatheral, Merrill Lynch, May-2003
SPX from 1/1/1990 to 4/25/2003
Correlation of vol changes with log returns
y = -1.0953x + 0.0003R2 = 0.6177
-15.00%
-10.00%
-5.00%
0.00%
5.00%
10.00%
15.00%
-8.00% -6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00% 8.00%
Log Return
Vo
lati
lity
Ch
ang
e
Jim Gatheral, Merrill Lynch, May-2003
n We regress vs for different values of .n Fit quality is plotted below:
Regressing VIX vs SPX
-1 -0.5 0.5 1 1.5 2a
0.52
0.54
0.56
0.58
0.62
R2
VIX VIXα ∆ ln(SPX)∆ α
2R 0α =
0
11
ααα
== −=
is lognormal
is 3/2 model
is Heston
0α =
n We note that is maximized at roughly which corresponds to alognormal process for volatility.
n At , so . Then, from the slope of theregression, with .
2 0.62R ≈ 0.79ρ ≈ −v v t Zη∆ ≈ ∆ 2.8η ≈
Jim Gatheral, Merrill Lynch, May-2003
Historical SPX implied volatility
VIX Index
0
10
20
30
40
50
60
Jan-90 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Jan-96 Dec-96 Dec-97 Dec-98 Dec-99 Dec-00
Jim Gatheral, Merrill Lynch, May-2003
n We regress the 20-day moving average std of vs fordifferent values of .
n Fit quality is plotted below:
Empirical dynamics of VIX
VIX∆ VIXα
2R
1
20
ααα
===
is lognormal
is 3/2 model
is Heston
n We note that is maximized at roughly . This would put thevolatility dynamics between lognormal and 3/2 with
n Imposing lognormality, the slope of the regression should give usvolatility of volatility directly.
α
0.5 1 1.5 2 2.5a
0.35
0.4
0.45
0.5
0.55
0.6
R2
1.4α =( 1) / 2 1.2β α= + ≈
Jim Gatheral, Merrill Lynch, May-2003
Regression of VIX volatility vs VIX level
y = 0.0635xR2 = 0.5629
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 45.0% 50.0%
20 day moving average
20 d
ay s
tand
ard
devi
atio
n
n The regression gives . Then,12 0.0635tσ η σ σ∆ ≈ ∆ ≈ 0.0635 16 2 2.0η ≈ × × ≈
Jim Gatheral, Merrill Lynch, May-2003
Unconditional distribution of VIX vs lognormal
VIX from 1/1/1990 to 4/25/2003
0
50
100
150
200
250
-3 -2.5 -2 -1.5 -1 -0.5 0
ln(VIX)
Freq
uenc
y
n Consistent with lognormal volatility dynamics!
Jim Gatheral, Merrill Lynch, May-2003
Dynamics of the volatility skew
n Our empirical investigations so far indicate that the appropriatespecification of stochastic volatility is lognormal with volatility ofvolatility somewhere between 2.0 and 3.0.
n This choice of specification has implications for the dynamics of theimplied volatility skew. We can check to see whether or not thevolatility skew behaves consistently with this specification.
η
Jim Gatheral, Merrill Lynch, May-2003
Vol term structure and skew under stochastic volatility
n All stochastic volatility models generate volatility surfaces withapproximately the same shape
n The Heston model has an impliedvolatility term structure that looks to leading order like
It’s easy to see that this shape should not depend very much on theparticular choice of model.
n Also, Gatheral (2002) shows that the term structure of the at-the-moneyvolatility skew has the following approximate behavior for all stochasticvolatility models of the form
n So we can estimate by regressing volatility skew against volatilitylevel.
( )1 / 2
2
0
1/2
(1 ), 1
with /2
T
BSx
v ex T
x T T
v
β λ
β
ρησ
λ λ
λ λ ρη
′− −
=
−
∂ −≈ − ′ ′∂
′ = −
( )dv v v dt v dZβλ η= − − +
( )dv v v dt v dZλ η= − − +
( ) ( )2 (1 ),
T
BSe
x T v v vT
λ
σλ
−−≈ + −
β
Jim Gatheral, Merrill Lynch, May-2003
SPX 3-month ATM volatility skew vs ATM 3m volatility
0.25 0.3 0.35ATM vol
-0.13
-0.12
-0.11
-0.09
-0.08
ATM vol skew
Jim Gatheral, Merrill Lynch, May-2003
Interpreting the regression of skew vs volatility
n Recall that if the variance satisfies the SDE
at-the-money variance skew should satisfy
and at-the-money volatility skew should satisfy
n The graph shows volatility skew to be roughly independent of volatilitylevel so again consistent with lognormal volatility dynamics.
~dv v dZβ
1/2
0k
vv
kβ −
=
∂∝
∂
2 2
0
BSBS
kkβσ
σ −
=
∂∝
∂
1β ≈
Jim Gatheral, Merrill Lynch, May-2003
Implied vs empirical stochastic volatility parameters
n It is often claimed that stochastic volatility parameters obtained by fittingstochastic volatility models to option prices are inconsistent withhistorical parameters. Specifically, the implied volatility of volatility isoften thought to be extreme.
n The typical volatility of volatility returned from a fit of the Heston modelto option prices is .
n The relationship between lognormal vol of vol and Heston vol of volshould be . With an implied volatility level of around 30%,would give which should be compared with fromour empirical analysis.
1Hη ≈
H BSη η σ≈ × 1Hη ≈3.3η ≈ (2.0,3.0)η ∈
Jim Gatheral, Merrill Lynch, May-2003
Do stochastic volatility models fit option prices?
n Once again, we note that the shape of the implied volatility surfacegenerated by a stochastic volatility model does not strongly depend onthe particular choice of model.
n Given this observation, do stochastic volatility models fit the impliedvolatility surface? The answer is “more or less”. Moreover, fittedparameters are reasonably stable over time.
n For very short expirations however, stochastic volatility models certainlydon’t fit as the next slide will demonstrate.
Jim Gatheral, Merrill Lynch, May-2003
Short Expirations
n Here’s a graph of the SPX volatility skew on 17-Sep-02 (just beforeexpiration) and various possible fits of the volatility skew formula:
n We see that the form of the fitting function is too rigid to fit the observedskews.
n Jumps could explain the short-dated skew!
0.5 1 1.5 2t
-1
-0.8
-0.6
-0.4
-0.2
ATM skew
Jim Gatheral, Merrill Lynch, May-2003
More reasons to add jumps
n The statistical (historical) distribution of stock returns and the optionimplied distribution have quite different shapes.
n The size of the volatility of volatility parameter estimated from fits ofstochastic volatility models to option prices is too high to be consistentwith empirical observations
Jim Gatheral, Merrill Lynch, May-2003
J
instantaneous variancelong-term mean variancerate of mean reversion of variancevolatility of volatilitycorrelation between stock and volatility changesrate of arrival of jumps
mean log-jump
vvbarληρλ
α sizestandard deviation of log-jump sizeδ
Fitting STOXX50 volatilities using Heston and SVJ
n Heston and SVJ models were fitted to SX5E implied volatility data from18-Oct-2002.
n Heston and SVJ parameter definitions are as follows:
Jim Gatheral, Merrill Lynch, May-2003
Heston FitsThe orange line is a parametric fit to the implied volatility data; the dashed blue line is the Heston fit.
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.5
0.6
0.7
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.45
0.5
0.55
0.6
0.65
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.35
0.4
0.45
0.5
0.55
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.35
0.4
0.45
0.5
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.35
0.4
0.45
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.25
0.35
0.4
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.25
0.275
0.325
0.35
0.375
0.4
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.26
0.28
0.32
0.34
0.36
0.38
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.26
0.28
0.32
0.34
0.36
l ® 3.9, r ® -.7, h ® 1.15, v® 0.542, vbar ® .312
Jim Gatheral, Merrill Lynch, May-2003
SVJ FitsThe orange line is a parametric fit to the implied volatility data; the dashed blue line is the SVJ fit.
l ® 3.5, r ® -.75, h ® 1.15, v® 0.552, vbar ® .252, lJ ® 1, a ® -.1, d ® .1
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.5
0.6
0.7
-0.6 -0.4 -0.2 0.2 0.4 0.60.35
0.45
0.5
0.55
0.6
0.65
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.35
0.4
0.45
0.5
0.55
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.35
0.4
0.45
0.5
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.25
0.35
0.4
0.45
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.25
0.35
0.4
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.25
0.275
0.325
0.35
0.375
0.4
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.24
0.26
0.28
0.32
0.34
0.36
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.26
0.28
0.32
0.34
Jim Gatheral, Merrill Lynch, May-2003
Observations on the fits
n We note that the Heston model cannot fit volatility levels at allexpirations simultaneously nor can it fit the skew at the front.
n On the other hand, the SVJ model allows us to fit the short term skewand also more or less all of the at-the-money volatility levels.
n Both fits return volatility-of-volatility parameters that are larger thanseem reasonable.
Jim Gatheral, Merrill Lynch, May-2003
Volatility skew from the characteristic function
n We can compute the volatility skew directly if we know thecharacteristic function .
n The volatility skew is given by the formula (Gatheral (2002)):
n The Heston characteristic function is given by
where and are the familiar Heston coefficients withparameters .
n Adding a jump in the stock price (SVJ) gives the characteristic function
with
( ) E[ ]Ti u xT u eφ =
2 / 820
0
2 1 Im[ ( / 2)]1 / 4
BS TBS T
k
u u ie du
k uTσσ φ
π
∞−
=
∂ −= −
∂ +∫
{ }( ) exp ( , ) ( , )SVT u C u T v D u T vφ = +
( , )C u T ( , )D u T, ,λ η ρ
( ) ( ) ( )SVJ SV JT T Tu u uφ φ φ=
( ) ( ){ }2 2 2/ 2 / 2( ) exp 1 1J i u uT J Ju iu T e T eα δ α δφ λ λ+ −= − − + −
Jim Gatheral, Merrill Lynch, May-2003
SVJJ
n Adding a simultaneous jump in the volatility gives the characteristicfunction (Matytsin (2000)):
with
where
with (usual Heston notation)
( ) ( ) ( ) ( )VSVJ SV J JT T T Tu u u uφ φ φ φ=
( ){ }2 2 / 2( ) exp ( ) 1VJ i u uT Ju v T e I uα δφ λ −= −
( )( )( , )( , )
0 0
21( )
1 / 1 /V
V
zT D u TD u T V e dzI u d t e
T p p z p z p
γγ γ −−
+ − + −
= = −+ +∫ ∫
{ }2Vp b iu d
γρη
η± = − ±
Jim Gatheral, Merrill Lynch, May-2003
2 4 6 8 10
-0.15
-0.125
-0.075
-0.05
-0.025
Comparing ATM skews from different models
n With parameters
we get the following plots of ATM variance skew vs time to expirationl ® 2.03, r ® -0.57, h ® 0.38, v® 0.1, vè ® 0.04, lJ ® 0.59, a ® -0.05, d ® 0.07, gv® 0.1
SVJJ
SVJ
SV
Jim Gatheral, Merrill Lynch, May-2003
0.05 0.1 0.15 0.2 0.25
-0.15
-0.125
-0.1
-0.075
-0.05
-0.025
Short expiration detail
n SV and SVJ skews essentially differ only for very short expirations
SVJJ
SVJSV
Jim Gatheral, Merrill Lynch, May-2003
Other problems for diffusion models
n If the underlying stochastic process for the stock is a diffusion, weshould be able to get from the statistical measure to the risk neutralmeasure using Girsanov’s Theorem
• This change of measure preserves volatility of volatility.
n However, historical volatility of volatility is significantly lower than theSV fitted parameter.
n Finally, in the few days prior to SPX expirations, out-of-the moneyoption prices are completely inconsistent with the diffusion assumption
• For example a 5 cent bid for a contract 10 standard deviations out-of-the-money.
Jim Gatheral, Merrill Lynch, May-2003
Simple jump diffusion models don’t work either
n Although jumps may be necessary to explain very short dated volatilityskews, introducing jumps introduces more parameters and this is notnecessarily a good thing. For example, Bakshi, Cao, and Chen (1997and 2000) find that adding jumps to the Heston model has little effect onpricing or hedging longer-dated options and actually worsens hedgingperformance for short expirations (probably through overfitting).
n Different authors estimate wildly different jump parameters for simplejump diffusion models.
Jim Gatheral, Merrill Lynch, May-2003
SPX large moves from 1/1/1990 to 4/25/2003
Log returns over 4%
-8.00%
-6.00%
-4.00%
-2.00%
0.00%
2.00%
4.00%
6.00%
8.00%
-15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00%
Volatility Change
Log
Ret
urn
Jim Gatheral, Merrill Lynch, May-2003
SPX large moves from 1/1/1990 to 4/25/2003
Vol changes over 6%
-8.00%
-6.00%
-4.00%
-2.00%
0.00%
2.00%
4.00%
6.00%
-15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00%
Volatility Change
Log
Ret
urn
Jim Gatheral, Merrill Lynch, May-2003
Empirical jump observations
n A large move in the SPX index is invariably accompanied by a largemove in volatility
• Volatility changes and log returns have opposite sign
n This is consistent with clustering• If there is a large move, more large moves follow i.e. volatility must jump
n We conclude that any jumps must be double jumps!
Jim Gatheral, Merrill Lynch, May-2003
What about pure jump models?n Dilip Madan and co-authors have written extensively on pure jump
models with stable increments• The latest versions of these models involve subordinating a pure jump
process to the integral of a CIR process – trading time again.
n Pure jump models are more aesthetically pleasing than SVJJ• The split between jumps and and diffusion is somewhat ad hoc in SVJJ
n However, large jumps in the stock price don’t force an increase inimplied volatilities.
Jim Gatheral, Merrill Lynch, May-2003
The instantaneous volatility impact of option trades
n Recall our simple market price impact model
where is the number of days’ volume represented by the trade.n We can always either buy an option or replicate it by delta hedging until
expiration: in equilibrium, implied volatilities should reflect thisn If we delta hedge, each rebalancing trade will move the price against us
by
n In the spirit of Leland (1985), we obtain the shifted expectedinstantaneous volatility
n What does this mean for the implied volatility?
x σ ξ∆ =
ξ
n SS
ADV ADVδ
σ σ σ ξ∆ Γ ∆
= = Γ ∆
2ˆ 1 2S
tσ ξ
σ σπ δ
Γ ≈ ±
Jim Gatheral, Merrill Lynch, May-2003
The implied volatility impact of an option trade
n To get implied volatility from instantaneous volatilities, integrate localvariance along the most probable path from the current stock price todayto the strike price at expiration (see Gatheral (2002))
with .
2 2
0
1( , ) ( , )
T
BS tK T S t dtT
σ σ≈ ∫ %
/
00
t T
t
KS S
S
≈
%
Jim Gatheral, Merrill Lynch, May-2003
20
70
120
170
3m6m
9m 12m 15
m 18m 21
m 24m 27
m 30m 33
m 36m 39
m 42m 45
m 48m 51
m 54m 57
m 60m
25.00%
26.00%
27.00%
28.00%
29.00%
30.00%
31.00%
32.00%
33.00%
32.00%-33.00%
31.00%-32.00%
30.00%-31.00%
29.00%-30.00%
28.00%-29.00%
27.00%-28.00%
26.00%-27.00%
25.00%-26.00%
The volatility impact of a 5 year 120 strike call
5 year 120 call, 10 days volume
(original surface flat 40% volatility)
Jim Gatheral, Merrill Lynch, May-2003
20
70
120
170
3m6m
9m 12m 15
m 18m 21
m 24m 27
m 30m 33
m 36m 39
m 42m 45
m 48m 51
m 54m 57
m 60m
35.00%
36.00%
37.00%
38.00%
39.00%
40.00%
41.00%
42.00%
41.00%-42.00%
40.00%-41.00%
39.00%-40.00%
38.00%-39.00%
37.00%-38.00%
36.00%-37.00%
35.00%-36.00%
The volatility impact of a 5 year 100/120 collar trade
5 years, 100/120 collar, 3 1/2 days volume
(original surface flat 40% volatility)
Jim Gatheral, Merrill Lynch, May-2003
Liquidity and the volatility surface
n We see that the shape of the implied volatility surface should reflect thestructure of open delta-hedged option positions.
n In particular, if delta hedgers are structurally short puts and long calls,the skew will increase relative to a hypothetical market with no frictions.
• Part of what we interpret as volatility of volatility when we fit stochasticvolatility models to the market can be ascribed to liquidity effects.
Jim Gatheral, Merrill Lynch, May-2003
How option prices reflect the behavior of stock pricesn Short-dated implied vol. more
volatile than long-dated implied vol.n Significant at-the-money skew
n Skew depends on volatility level
n Extreme short-dated impliedvolatility skews
n High implied volatility of volatility
n Clustering - mean reversion ofvolatility
n Anti-correlation of volatility movesand log returns
n Volatility of volatility increases withvolatility level
n Jumps
n Stock volatility depends on thestrikes and expirations of open delta-hedged options positions.
Jim Gatheral, Merrill Lynch, May-2003
Conclusions
n Far from being ad hoc, stochastic volatility models are naturalcontinuous time extensions of simple but realistic discrete-time modelsof stock trading.
n Stylized features of log returns can be related to empirically observedfeatures of implied volatility surfaces.
n By carefully examining the various stylized features of option prices andlog returns, we are led to reject all models except stochastic volatilitywith simultaneous index and volatility jumps.
n Investor risk preferences and liquidity effects also affect the observedvolatility skew so SV-type models may be misspecified and fittedparameters unreasonable.
Jim Gatheral, Merrill Lynch, May-2003
Referencesn Clark, Peter K., 1973, A subordinated stochastic process model with finite
variance for speculative prices, Econometrica 41, 135-156.n Duffie D., Pan J., and Singleton K. (2000). Transform analysis and asset pricing
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