Modeling the Implied Volatility Surface
Jim GatheralStanford Financial Mathematics SeminarFebruary 28, 2003
Jim Gatheral, Merrill Lynch, February-2003
This presentation represents only the personal opinions of the author andnot those of Merrill Lynch, its subsidiaries or affiliates.
Jim Gatheral, Merrill Lynch, February-2003
Outline of this talk
n A compound Poisson model of stock tradingn The relationship between volatility and volumen Clusteringn Correlation between volatility changes and log returnsn Stochastic volatilityn Dynamics of the volatility skewn Similarities between stochastic volatility modelsn Do stochastic volatility models fit option prices?n Jumpsn The impact of large option trades
Jim Gatheral, Merrill Lynch, February-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, February-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 .
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
t E n
α α
α λ
∆ = + = ∆
[ ] [ ]2iVar x t t E nσ α λ2∆ = ∆ = ∆
[ ]it E nλ∆t∆
t∆
t∆
Jim Gatheral, Merrill Lynch, February-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, February-2003
Average trade size is almost independent of activity
Jim Gatheral, Merrill Lynch, February-2003
Empirical variance vs volume
IBM from 4/30/2001 to 2/24/2003
y = 1E-10xR2 = 0.2613
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0 10,000,000 20,000,000 30,000,000 40,000,000 50,000,000
Volume (Shares)
ln^2
(hi/l
o)
Jim Gatheral, Merrill Lynch, February-2003
Empirical variance vs volume
MER from 2/26/2001 to 2/24/2003
y = 3E-10xR2 = 0.2895
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, February-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, February-2003
Mean reversion of volatility: an economic argument
n There is a simple economic argument which justifies the mean reversionof volatility (the same argument that is used to justify the mean reversionof interest rates):
• Consider the distribution of the volatility of IBM in one hundred years timesay. If volatility were not mean-reverting ( i.e. if the distribution of volatilitywere not stable), the probability of the volatility of IBM being between 1%and 100% would be rather low. Since we believe that it is overwhelminglylikely that the volatility of IBM would in fact lie in that range, we deducethat volatility must be mean-reverting.
Jim Gatheral, Merrill Lynch, February-2003
Empirical volatility term structure observations
n Short-dated implied volatilities move more than long-dated impliedvolatilities
n The term structure of implied volatility has the form of exponentialdecay to a long-term level
n The shape and dynamics of the volatility term structure imply thatvolatility must mean-revert i.e. that volatility changes are auto-correlated
n The following slides show that this is also true empirically.
Jim Gatheral, Merrill Lynch, February-2003
Volatility and Volume Clustering
IBM Log Returns vs Volume
0
10000000
20000000
30000000
40000000
50000000
60000000
70000000
8000000001
/03/
1990
01/0
3/19
91
01/0
3/19
92
01/0
3/19
93
01/0
3/19
94
01/0
3/19
95
01/0
3/19
96
01/0
3/19
97
01/0
3/19
98
01/0
3/19
99
01/0
3/20
00
01/0
3/20
01
01/0
3/20
02
01/0
3/20
03
Vo
lum
e
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Lo
g R
etu
rns
Jim Gatheral, Merrill Lynch, February-2003
Volatility and Volume Clustering
MER Log Returns vs Volume
0
5000000
10000000
15000000
20000000
25000000
3000000001
/03/
1990
01/0
3/19
91
01/0
3/19
92
01/0
3/19
93
01/0
3/19
94
01/0
3/19
95
01/0
3/19
96
01/0
3/19
97
01/0
3/19
98
01/0
3/19
99
01/0
3/20
00
01/0
3/20
01
01/0
3/20
02
01/0
3/20
03
Vo
lum
e
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Lo
g R
etu
rns
Jim Gatheral, Merrill Lynch, February-2003
Correlation between volatility changes and log returns
n The empirical fact that implied volatility is a decreasing function ofstrike price indicates that volatility changes must be negativelycorrelated with log returns.
n The following slide shows that volatility changes really are anti-correlated with stock price changes
Implied Volatility vs StrikeJune 2002 options as of 4/24/2002
10.00%
15.00%
20.00%
25.00%
30.00%
35.00%
40.00%
45.00%
50.00%
55.00%
700 800 900 1,000 1,100 1,200 1,300 1,400
Jim Gatheral, Merrill Lynch, February-2003
SPX from 1/1/1990 to 2/24/2003
Correlation of vol changes with log returns
y = -1.1066x + 0.0003R2 = 0.6218
-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
latil
ity C
han
ge
Jim Gatheral, Merrill Lynch, February-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
( )1
2
dx dt v dZ
dv v v v dZβ
µ
α η
= +
= +
1 2,dZ dZ dtρ=
x
( )vα ρη
12
0 gives the Heston model
= gives Wiggins' lognormal model
β
β
=
v
Jim Gatheral, Merrill Lynch, February-2003
Dynamics of the volatility skew
n So far, we have ascertained that the volatility process must be mean-reverting and that volatility moves are anti-correlated with log returns.
n Can we say anything about the diffusion coefficient?n The following four slides give a sense of the dynamics of the volatility
surfacen We see that as volatility increases
• so does volatility of volatility• and so does the volatility skew
Jim Gatheral, Merrill Lynch, February-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, February-2003
Regression of VIX volatility vs VIX level
y = 0.0954x1.339
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
0.0% 10.0% 20.0% 30.0% 40.0% 50.0%
20 day moving average
20 d
ay s
tand
ard
devi
atio
n
Jim Gatheral, Merrill Lynch, February-2003
SPX implied volatility skew vs implied volatility level
0%
1%
2%
3%
4%
5%
6%
7%
8%Ja
n-98
Mar
-98
May
-98
Jun-
98
Aug-
98
Oct
-98
Nov-
98
Jan-
99
Mar
-99
Apr-9
9
Jun-
99
Aug-
99
Sep-
99
Nov-
99
Dec-
99
Feb-
00
Apr-0
0
May
-00
Jul-0
0
Aug-
00
Oct
-00
Dec-
00
Jan-
01
Mar
-01
May
-01
Jun-
01
Aug-
01
Oct
-01
Thre
e M
onth
Impl
ied
Put S
kew
0%
5%
10%
15%
20%
25%
30%
35%
40%Three M
onth ATM Call + Put Im
plied Volatility
Put Skew
ATM Put + Call Vol
Note that skew is defined to be the difference in volatility for ± 0.25 delta
Jim Gatheral, Merrill Lynch, February-2003
Regression of skew vs volatility level
y = 0.3774x1.6316
R2 = 0.5476
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% 35.00% 40.00%
3m ATM Volatility
Note that skew is defined to be the difference in volatility for ± 0.25 delta
Jim Gatheral, Merrill Lynch, February-2003
Similarities between stochastic volatility models
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 volatility skewhas the following approximate behavior for all stochastic volatilitymodels of the generic form :
( ) ( )
( )
2 (1 ), 1
with / 2
T
BS
v ex T
x T T
v
λρη βσ
λ λ
λ λ ρη β
′− ∂ −≈ − ′ ′∂
′ = −
( ) ( )dv v dt v v dZα η β= +
( )dv v v dt v dZλ η= − − +
( ) ( )2 (1 ),
T
BSe
x T v v vT
λ
σλ
−−≈ + −
Jim Gatheral, Merrill Lynch, February-2003
Implications for the volatility process
n In our generic stochastic volatility model parameterization
n Transforming to volatility gives
n gives the lognormal model of Wiggins (1987) and gives the3/2 model studied in detail by Lewis (2000).
n All four graphs conclusively reject the Heston model which predicts thatvolatility of volatility is constant, independent of volatility level.
• This is very intuitive; vols should move around more if the volatility level is100% than if it is 10%
n The regression of VIX volatility vs VIX level gives for the SPXn To interpret the result of the regression of skew vs volatility level, we
need to do some more work...
v v v t Zβη∆ ≈ ∆
2
2t Zβη
σ σ∆ ≈ ∆
vσ =
0.67β ≈
12β = 1β =
Jim Gatheral, Merrill Lynch, February-2003
Interpreting the regression of skew vs volatility
n Recall that if the variance satisfies the SDE
at-the-money variance skew should satisfy
n Delta is of the form with
n Then, if the regression of volatility skew vs volatility is of the form
we must also have
n Referring back to the regression result, we get . The twoestimates are not so far apart! Both are between lognormal and 3/2.
/ 2 0.82β γ= ≈
~dv v v dZβ
0k
vv
kβ
=
∂∝
∂1( )N d
0 0
2 BSBS
k k
vk k
γσ δσ σ
δ= =
∂∂ ∂= ∝
∂ ∂ ∂
1 2k T
dT
σσ
= − +
0
BS
k
γσσ
δ =
∂∝
∂
Jim Gatheral, Merrill Lynch, February-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, February-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, February-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, February-2003
SV versus SVJJ
n Duffie, Pan and Singleton fitted a stochastic volatility (SV) model and adouble-jump stochastic volatility model (SVJJ) to November 2, 1993SPX options data. Their results were:
n Note the unreasonable size of the volatility of volatility parameter inSV!
SV SVJJρ -0.70 -0.82v 0.019 0.008λ 6.21 3.46η 0.61 0.14Jλ 0.47
Jµ -0.10
Jσ 0.0001
vµ 0.05
Jρ -0.38
0v 10.1% 8.7%
Vol of vol is 4.5 times greater in the SV model!
η
Jim Gatheral, Merrill Lynch, February-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, February-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, February-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, February-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, February-2003
Estimating Volatility of Volatility
n Recall that variance in the Heston model follows the SDE (dropping thedrift term)
n Converting this to volatility terms gives
n So
n implies that annualized SPX volatility moves aroundper day.
n A more typical fit of Heston to SPX implied volatilities would giveimplying a daily move of around 2.5 annualized volatility points per day.
n Historically, the daily move in short-dated implied volatility is around1.5 volatility points as shown in the following graph.
dv v dZη≈
d dZσ σ ησ2 ≈
2d dZ
ησ ≈
0.61η ≈ 2%2
tη
∆ ≈
0.80η ≈
Jim Gatheral, Merrill Lynch, February-2003
Empirical volatility of volatility
20 Day Standard Deviation of VIX Changes
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
05/90 09/91 01/93 06/94 10/95 03/97 07/98 12/99 04/01 09/02
Jim Gatheral, Merrill Lynch, February-2003
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, February-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, February-2003
SPX large moves from 1/1/1990 to 2/24/2003
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, February-2003
SPX large moves from 1/1/1990 to 2/24/2003
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, February-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, February-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, February-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, February-2003
The implied volatility impact of an option trade
n To get implied volatility from instantaneous volatilities, integrate localvariance along the most probable path (see Gatheral (2002))
with .
2 2
0
1( , ) ( , )
T
BS tK T S t dtT
σ σ≈ ∫ %
/
00
t T
tK
S SS
≈
%
Jim Gatheral, Merrill Lynch, February-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, February-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, February-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, February-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, February-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 SVJJ
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, February-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
for affine jump-diffusions. Econometrica, 68, 1343-1376.n Geman, Hélyette, and Thierry Ané, 2000, Order flow, transaction clock, and
normality of asset returns, The Journal of Finance 55, 2259-2284.n Gatheral, J. (2002). 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 FinancialStudies 6, 327-343.
n Leland, Hayne (1985). Option Pricing and Replication with Transactions Costs.Journal of Finance 40, 1283-1301.
n Lewis, Alan R. (2000), Option Valuation under Stochastic Volatility : withMathematica Code, Finance Press.
n Matytsin, Andrew (2000). Modeling volatility and volatility derivatives, CiranoFinance Day, Montreal.http://www.cirano.qc.ca/groupefinance/activites/fichiersfinance/matytsin.pdf
n Wiggins J.B. (1987). Option values under stochastic volatility: theory andempirical estimates. Journal of Financial Economics 19, 351-372.