Robust Calculation of MFIV from Calibrated Surfaces Philip Stahl * and Philipp B. Rindler † Working Paper — Feb 2012 Abstract This paper proposes a new method to calculate model-free implied volatility from a calibrated option price surface. This circumvents com- mon interpolation/extrapolation problems found in established method- ologies, where prices enter calculation directly, and is numerically more stable. Areas outside the observable strikes are approximated better. Pre- dictive regressions over 180 months based on this new method show that the new method is indeed superior in most cases, even with only very few observable strikes. Keywords: Option pricing, model-free implied volatility, characteristic functions JEL-Classification: C61 C80 G13 1 Introduction For financial institutions and investors, reliable ways of forecasting risk of in- vestments is of utmost importance. Mostly reduced to the variation of returns, * [email protected]† [email protected]1
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Electronic copy available at: http://ssrn.com/abstract=2043503
Robust Calculation of MFIV from Calibrated
Surfaces
Philip Stahl∗and Philipp B. Rindler†
Working Paper — Feb 2012
Abstract
This paper proposes a new method to calculate model-free implied
volatility from a calibrated option price surface. This circumvents com-
mon interpolation/extrapolation problems found in established method-
ologies, where prices enter calculation directly, and is numerically more
stable. Areas outside the observable strikes are approximated better. Pre-
dictive regressions over 180 months based on this new method show that
the new method is indeed superior in most cases, even with only very few
which are the same as in Jiang and Tian (2005, parameter set 1). The strikes
are equally spaced with ∆K = USD5 and symmetrically distributed around S.
The difficulty of curve-fitting is evident.
Jiang and Tian (2005) simulate a price surface with stochastic volatility and
jumps in the price process. From this surface, they estimate the effectiveness
of their curve-fitting method to assess its appropriateness to overcome limited
information by reducing the strikes that enter the MFIV calculation, but only
consider isolated situations where either truncation or discretization occur and
strikes are equidistant. In earlier work, we have extended this analysis and
considered random drafts from a simulated option price surface of decreasing
7
size to replicate the patterns observed in real data (Stahl (2010)). We conclude
that the method of Jiang and Tian (2005) is appropriate if the observed volatility
surface is very smooth, but may be far out if option prices exhibit deviations
from a smooth surface or very few prices are available.
Carr and Wu (2009) perform two interpolations: the first over strikes, the
second over maturities. On the strike dimension, they follow Jiang and Tian
(2005). Over maturities, they interpolate linear between two available maturi-
ties T1 and T2 to gain the variance between t to T by
SWt,T =1
T − t[SWt,T1
(T1 − t)(T2 − T ) + SWt,T2(T2 − t)(T − T1)
T2 − T1] (3.1)
which they compare against the corresponding ex-post realized variance. This
is given by
RVt,T =365
M
M∑i=1
(Ft+i,T − Ft+i−1,T
Ft+i−1,T
)2
(3.2)
where M is the maturity in days such that t+M = T . To ensure comparability
of results, M = 30. We refer to (3.1) as discretized MFIV and to (3.2) as realized
variance (RV).
3.2 MFIV using calibrated models
Current implementations of MFIV are inaccurate and numerically instable.
Strangely priced options are dangerous to the stability of direct interpolation
because they enter the calculation of MFIV directly and influence the inter-
polated neighborhood on the surface, multiplying their effects. Strikes outside
[Kmin,Kmax] may be mispriced as well. We propose to step away from cal-
culating MFIV from prices directly, rather use them to calibrate a model and
generate a large and dense volatility surface, which is translated into option
8
prices at a density unobserved in practice that serve as a basis for MFIV. This
has several advantages:
• Academics have a thorough understanding of the behavior of parametric
models. Curve-fitting methods based on splines behave unpredictable and
can generate extreme outliers. Option prices can easily become negative.
Additional procedures are necessary to make sure that the interpolation
stays within reasonable boundaries. A calibrated model fulfills these con-
ditions by design.
• Calibrated models replicate the behavior of option prices far out-of-the-
money better than constant extrapolation. MFIV considers a cross-section
of prices that are weighted by K2, thus strikes far from the forward price
are less important for the calculation, but in situations where all observ-
able strikes are truncated close to the forward price, truncation errors may
be considerable.
• If only one strike is observable (not uncommon – see table 1), the indi-
rect approach is still able to approximate the dynamics of the volatility
surface. Constant extrapolation would not only be as informationally inef-
ficient as common at-the-money implied volatility, but also introduce the
approximation as a new source of error.
• Parametric models follow from economic reasoning and empirical analy-
sis. Standard curve-fitting methods are not specifically designed for op-
tion markets. Economic reasoning may improve accuracy where educated
guesses are required by the real world.
Choosing the underlying model, we follow Gatheral (2006), who suggests using
a stochastic volatility model with jumps in the stock price process (SVJ model).
His reasoning is that while jumps are empirically necessary to explain volatility
9
surface behavior, estimation of simultaneously jumping prices and volatilities
(as with the SVJJ model) is computationally expensive and inaccurate, because
volatility cannot be observed directly and thus estimation of jumps makes fitting
the surface hard. Furthermore, Gatheral (2006) demonstrates that while the
SVJ model increases the fit compared to the Heston (1993) model, improvements
of the SVJJ model over the SVJ model are small2.
For every day and maturity, the specified model needs to be recalibrated.
Even though one would expect that it is sufficient to calibrate the model per
stock globally every once in a while and continue for the next dates with lo-
cal optimizations, errors grow fast and are cumulative. Every calibration is
therefore performed globally in a two step process: First, adaptive simulated
annealing (ASA) is performed on the SVJ model’s characteristic function to
find a set of parameters v0, κ, θ, η, ρ, α, δ, and λ that is likely to be close
to the global optimum. Second, pattern search is performed to find the local
optimal set, starting from the last set from ASA. By employing the second step,
the very slow transition from a good state to the best state of ASA can be
avoided and the resulting set is likely to fit better. Details and boundaries of
the optimization can be found in the appendix.
With a parameter set for a given t and T an indefinitely dense option price
surface can be calculated. Here, strikes are ranged from $1 to $5000 and spaced
by $2, so the interpolated option price surface contains 2500 prices. As the in-
terpolating SVJ model’s stochastic process is known by assumption, this can be
done computationally efficient through fourier inversion3. On this surface, op-
tion prices are sorted over strikes and the integral of weighted implied variances
2As characteristic functions are used for calibration, extension of results to the SVJJ orVariance-Gamma model is straightforward. These can be found in Gatheral (2006).
3See Gatheral (2006) or the appendix for details.
10
is approximated by trapezoidal numerical integration.
The weighting factor K2 pronounces strikes at-the-money, thus the most im-
portant part the model has is the interpolation. Extrapolation of far out-of-the-
money strikes has a relatively low impact. The underlying model is known, so
the behavior of the surface is stable. The calibration may correct false prices
and filters the effects of outliers on the estimation. However, only prices for a
specific date and maturity are considered, so in cases where only one option is
observable, the model may be vastly mis-specified.
4 Empirical Results
4.1 Data
The data to test the proposed methodology are daily historical option prices
from OptionMetrics, starting January 2, 2000, and ending December 31, 2010,
stocks where quotes that start later or end earlier are not considered. Maturity
of options is limited to be between 8 and 365 days to avoid anomalies of very long
term options or shortly before exercise date. All options are American options,
however OptionMetrics corrects these by subtracting an approximation of the
early exercise premium following a binomial tree approach. Even though this
might contain other biases, these are consistent over all analyzed stocks and do
not affect the main results. Furthermore, positive bid-ask spreads are required.
Stock prices, Black-Scholes implied volatilities, and the interest rate curve are
from WRDS and prices are dividend adjusted, but stock splits are not corrected.
From this data set we selected 15 stocks4 randomly. While other authors (for
example Carr and Wu (2009)) choose stocks according to availability of option
4The low number of stocks is mainly due to the time intensive computation. Tests will beextended in future versions of this paper.
11
Table 1: List of stocks in the sampleNo. Security ID Ticker Starting Date Sample size Min. Strikes Max. Strikes Name
Table 1 summarizes the sample of stocks used in this paper. Quotes for calculations start 01-Jan-2000 and end 31-Dec-2010. Samplesize refers to the number of observations per stock, minimum and maximum strikes refer to the number of strikes observable for achosen maturity.
quotes, this shortage of data is what the proposed methodology is designed for
to overcome. A summary of the used data can be found in table 1.
4.2 Predictive regressions
To assess the forecast ability of both methods we follow Shu and Zhang (2002)
and run regressions on both the method as in Carr and Wu (2009) and the
indirect method on the future realized variance. The regression models are
RVt = α+ βMFIV CWt + εt (4.1)
RVt = α+ βMFIV CSt + εt (4.2)
where superscript CW denotes the Carr and Wu (2009)-Method, and CS de-
notes MFIV from calibrated surfaces. Table 2 summarizes the rsults of both
regressions on MFIV calculations for the first trading day of every month on 30
days in the future per security. In most cases, the new method’s R2 is higher
than before, and F -statistics as well as p-value are in favor of the MFIV from
calibrated surfaces. In absolute terms, R2 seems to be low, but there are two
possible explanations: First, the data is not adjusted for stock splits. This may
introduce an error in the calculation of the variance, depending on the exact
12
adjustment procedure of options. Second, the stocks are selected randomly from
all available stocks in the WRDS database, and may be illiquid or inactively
traded. Both apply to every stock in the data set, and do not affect the results
of the predictive regressions (4.1) or (4.2).e
5 Conclusion
In this paper, we develop a new method to calculate MFIV which does not
directly incorporates prices and is numerically more robust than prior methods.
It specifically addresses the interpolation/extrapolation method by Jiang and
Tian (2005), which relies on polynomial splines between observed option prices,
which are hard to control and may easily result in negative prices or volatilities.
Furthermore, the assumption of constant strike prices outside the observable
range is unrealistic. By calibrating a SVJ model to option prices, the price
surface can be approximated better, and will stay within boundaries that are
relatively well understood. Instead of taking the volatility from the model itself,
we calculate MFIV from the cross section of option prices, using all available
information. Predictive regressions confirm that the proposed method is indeed
better suited to extract future realized variance from option markets.
There are still issues with the calculation of MFIV from incomplete observa-
tions. First, for every considered date, the price surface has to be calibrated
globally. This is not only expensive in terms of computation, but it still requires
some data to calibrate on. In cases where only one or no option prices are ob-
servable, the model might be miscalibrated. One way to circumvent this could
be to analyze the stability of model parameters over time, and use them not only
as a starting set for the next fitting, but also to interpolate over time. Second,
empirical evidence points into the direction that volatility follows a continuous
13
Tab
le2:
Reg
ress
ion
stati
stic
sS
ecu
rity
IDC
arr
and
Wu
(200
9)-
Met
hod
olo
gy
MF
IVfr
om
cali
bra
ted
surf
ace
Coeffi
cien
tIn
terc
ept
R2
F-s
tati
stic
p-v
alu
eσε
Coeffi
cien
tIn
terc
ept
R2
F-s
tati
stic
p-v
alu
eσε
1009
430.
0408
0.21
420.
0023
0.4
154
0.5
201
0.2
153
1.2
974
0.0
101
0.0
918
17.9
925
0.0
000
0.1
960
1011
720.
2343
0.58
430.
0444
7.8
553
0.0
057
0.9
144
0.6
584
0.3
745
0.0
933
17.3
999
0.0
000
0.8
676
1013
280.
5246
-0.1
359
0.14
4330.0
086
0.0
000
4.3
426
1.0
715
0.2
668
0.0
355
6.5
586
0.0
113
4.8
944
1014
750.
0593
0.27
430.
0047
0.6
909
0.4
072
0.3
513
1.1
173
0.0
423
0.1
144
18.9
874
0.0
000
0.3
126
1023
220.
5154
-0.0
952
0.48
4844.2
200
0.0
000
0.7
490
2.7
777
-0.2
953
0.6
640
92.8
977
0.0
000
0.4
884
1049
390.
1539
0.06
880.
2499
59.3
042
0.0
000
0.2
082
1.2
623
-0.0
795
0.5
625
228.8
609
0.0
000
0.1
214
1059
640.
5398
0.06
630.
0403
7.1
446
0.0
083
0.0
120
0.9
855
0.0
001
0.4
252
125.7
466
0.0
000
0.0
072
1066
520.
4378
2.80
230.
0002
0.0
222
0.8
818
319.6
534
-6.5
125
5.9
541
0.0
098
1.0
569
0.3
063
316.5
926
1067
760.
0168
0.58
190.
0011
0.1
943
0.6
599
0.3
132
1.1
663
0.0
484
0.3
842
111.0
680
0.0
000
0.1
931
1068
920.
4006
0.26
510.
0871
15.5
441
0.0
001
0.5
542
1.7
661
-0.0
652
0.4
680
143.4
065
0.0
000
0.3
230
1070
150.
1125
0.15
900.
0178
3.2
237
0.0
743
0.3
573
2.0
152
-0.0
435
0.1
862
40.7
369
0.0
000
0.2
961
1096
951.
5990
0.54
140.
0010
0.0
231
0.8
805
2.5
954
-3.6
681
0.9
539
0.0
288
0.7
117
0.4
072
2.5
231
1106
810.
3788
0.61
650.
0546
8.3
179
0.0
045
1.9
033
1.0
260
0.3
964
0.0
570
8.6
969
0.0
037
1.8
986
1107
400.
2029
0.08
210.
0490
9.0
156
0.0
031
0.0
478
1.7
047
-0.0
276
0.5
489
212.9
644
0.0
000
0.0
227
1122
190.
1621
0.25
380.
0479
8.9
002
0.0
033
0.0
842
1.3
592
0.0
197
0.2
787
68.4
032
0.0
000
0.0
638
Tab
le2
stat
esre
gres
sion
resu
lts
ofb
oth
met
hod
olo
gie
son
reali
zed
vari
an
ce.
All
vari
an
ces
hav
eb
een
calc
ula
ted
at
the
firs
ttr
ad
ing
day
ofea
chm
onth
for
am
atu
rity
of30
day
s.D
ata
start
sJanu
ary
2,
2000
an
den
ds
Dec
emb
er31,
2010.
14
process. High frequency data, which has established to be standard for realized
variance estimations, should be analyzed in the context of of MFIV. This may
also improve ones ability to interpolate over time, as parameters are likely to
be more stable over 5 minutes than over 30 days.
It seems promising to apply the proposed methodology to estimate the vari-
ance risk premium. With a more accurate estimation, a part of what is un-
derstood as premium might be identified as measurement error. Further tests
could check wether classic risk factors influence the variance risk premium, or
wether current risk factors are brought into existence by variance risk premia.
A Calibration of SVJ model
A.1 Derivation of the Valuation Equation
In general, the price of a European call option can be computed as
C(Ft, τ) = e−rτEQ [max(FT −K, 0)] (A.1)
It turns out that it is quite straightforward to get option prices by inverting the
characteristic function of a given stochastic process if it is known in closed form:
C(Ft, τ) = Ft −√FtK
1
π
∫ ∞0
Re[eiuxtφτ (u− i
2 )]
u2 + 14
du (A.2)
where φT (u) is the characteristic function of S and xt = log(FtK
). In the fol-
lowing, we’ll follow the derivation of Carr and Madan (1999) and Lewis (2000)
to derive (A.2).
A covered call G(xt, τ) is a long position in S and a short position in a call
option C(S, τ) written on S with payoff G(k, 0) = min(ST ,K) = min(FT ,K).
15
To derive (A.2), consider the Fourier transform payoff of G(k, τ) with respect
to the log-strike k = log(KFt
)= −x defined by
G(u, τ) =
∫ ∞−∞
eiukG(k, τ)dk (A.3)
Then
1
FG(u, T − t) =
∫ ∞−∞
eiukE[min(exT , ek)|xt = 0
]dk
= E[∫ ∞−∞
eiuk min(exT , ek)dk∣∣∣xt = 0
]= E
[∫ xT
−∞eiukekdk +
∫ ∞xT
eiukexT dk∣∣∣xt = 0
]= E
[e(1+iu)xT
1 + iu− e(1+iu)xT
iu
∣∣∣xt = 0
]only if 0 < Im[u] < 1!
=1
u(u− i)E[e(1+iu)xT
∣∣xt = 0]
=1
u(u− i)φT (u− i)
To get the call option price in terms of the characteristic function, we express
the option in terms of the covered call position and invert the Fourier transform,
integrating along the line Im[u] = 12 .
C(F, τ) = F −G(F, τ)
= F − F 1
2π
∫ ∞+ i2
−∞+ i2
e−iuk
u(u− i)φT (u− i)du
= F − F 1
2π
∫ ∞−∞
e−i(u+i2 )k
(u+ i2 )(u− i
2 )φT (u− i
2)du
= F −√FK
1
π
∫ ∞0
Re[eiuxtφT (u− i
2 )]
u2 + 14
du
16
A.2 Used characteristic functions
As in Gatheral (2006, Chapter 5).
A.2.1 Black-Scholes GBM
In the Black-Scholes model, the dynamics are given by
dS = µSdt+ σSdW (A.4)
The characteristic function is given by
φT (u) = exp
iu(µ− 1
2σ2)T − 1
2u2σ2T
(A.5)
A.2.2 Merton Jump Diffusion
In the jump-diffusion model, the dynamics are assumed to be
dS = µSdt+ σSdW + (J − 1)Sdq (A.6)
where
dq =
0 with probability 1− λdt
1 with probability λdt
is a homogeneous Poisson process with arrival intensity λ and the jump size
J assumed to be log-normally distributed with mean log-jump α and standard
deviation δ. The characteristic function is given by
φT (u) = exp
(−1
2u(u+ i)σ2 − λ
[−(eiuα−
u2δ2
2 − 1)
+ iu(eα+
δ2
2 − 1)])
T
(A.7)
17
A.2.3 Heston Stochastic Volatility
The derivation of the characteristic function in the Heston model is somewhat
involved. It is given by
φT (u) = expC(u, τ)θ +D(u, τ)v0 (A.8)
where
C(u, τ) = κ
[r−τ −
2
η2log
(1− ge−dτ
1− g
)]D(u, τ) = r−
1− e−dτ
1− ge−dτ
g =r−r+
r± =b± dη2
d = d =√b2 − 4ac
c =η2
2
b = κ− ρηiu
a = − u2
2− iu
2
A.2.4 Stochastic Volatility with Jumps
To add jumps to the stochastic volatility Heston model, it is combined with the
jump part of the Merton model by multiplying the characteristic functions such
that
φT (u) = exp
C(u, τ)θ +D(u, τ)v0−(
λ[−(eiuα−
u2δ2
2 − 1)
+ iu(eα+
δ2
2 − 1)])
τ
(A.9)
18
Derivations of the characteristic functions above as well as for the SVJJ and
the Variance-Gamma model can be found in Gatheral (2006).
A.3 Adaptive simulated annealing
For the first step of the calibration, the last best set is used after either the
root mean squared error change over 500 iterations is smaller than 1e− 6 or the
algorithm has evaluated 24000 sets.
In the calibration of the SVJ model, the the parameters are limited to