Option data, missing tails, and the intraday
variation of implied moments
Anselm Ivanovas
Quantitative Methods Seminar
October 20, 2014
Motivation
Motivation
The moments of the risk-neutral distribution give interesting
insights into investor’s expectations.
They can be used to describe the density itself (Eriksson et al.,
2009): fit a Normal Inverse Gaussian density, (Polkovnichenko &
Zhao, 2013): probability weighting functions.
Or as explanatory variable (Bakshi et al., 2003): co-skewness, (Buss
& Vilkov, 2012): implied correlations and risk factors, (Conrad et al.,
2013): return predictions.
Dennis & Mayhew (2002) examine the effect of economic and
firm-specific factors on risk-neutral skewness.
Motivation
2
Contribution
Shared characteristic of the previous papers: they all use
model-free moments suggested by Bakshi et al. (2003) (BKM)
A detail often not discussed: how does this method hold up with
“finite samples”, i.e. issues with option data that exist in practice
Contribution
⊛ Examine finite sample properties
⊛ Propose a correction method for data truncation
⊛ Compute time series of moments for intraday data
Motivation
3
Model-free moments
The BKM formulas
Data issues
The BKM formulas I
BKM compute moments by constructing
⊛ The volatility contract with payoff R(t, T )2 and price
V(t, T ) = Et e−rτ R(t, T )2
⊛ The cubic contract with payoff R(t, T )3 and price
W(t, T ) = Et e−rτ R(t, T )3
⊛ The quartic contract with payoff R(t, T )4 and price
X(t, T ) = Et e−rτ R(t, T )4
from a sequence of options. R(t, T ) = log [S(T )] − log [S(t)] is the
underlying return between now and expiry date T .
Model-free moments
The BKM formulas
Data issues
4
The BKM formulas II
The prices of these contracts can be spanned algebraically from
option prices (Bakshi & Madan, 2000)
V(t, τ) =
Z∞
2 1 − log
K
S(t)
K2
S(t)
Z S(t) 2 1 + log
K
C(t, τ; K)dK
S(t)
+
W(t, τ) =
K2
0
Z∞
6 log
S(t)
P(t, τ; K)dK,
2
K
K
− 3 log
S(t)
S(t)
C(t, τ; K)dK,
K2
2
S(t)
S(t)
+ 3 log
K
K
P(t, τ; K)dK,
K2
0
2
3
K
K
Z∞
12 log
− 4 log
S(t)
S(t)
X(t, τ) =
C(t, τ; K)dK,
K2
S(t)
2
3
S(t)
S(t)
Z S(t) 12 log
+ 4 log
K
K
+
P(t, τ; K)dK,
2
K
0
−
Z S(t) 6 log
back
Model-free moments
The BKM formulas
Data issues
5
The BKM formulas III
By definition of skewness and kurtosis
h
3 i
Et R(t, τ) − Et [R(t, τ)]
SKEW(t, τ) ≡ 2 3/2
Et R(t, τ) − Et [R(t, τ)]
and
h
4 i
Et R(t, τ) − Et [R(t, τ)]
KURT(t, τ) ≡ 2 2
Et R(t, τ) − Et [R(t, τ)]
can now be stated in terms of prices of the aforementioned
contracts and consecutively current option prices.
Model-free moments
The BKM formulas
Data issues
6
The BKM formulas IV
κ3 (t, τ) =
erτ W(t, τ) − 3µ(t, τ)erτ V(t, τ) + 2µ(t, τ)3
[erτ V(t, τ) − µ(t, τ)2 ]
3/2
and
κ4 (t, τ) =
erτ X(t, τ) − 4µ(t, τ)erτ W(t, τ) + 6erτ µ(t, τ)2 V(t, τ) − 3µ(t, τ)4
[erτ V(t, τ) − µ(t, τ)2 ]
2
,
with
µ(t, τ) ≡ Et log
Model-free moments
S(t + τ)
erτ
erτ
erτ
= erτ − 1 −
V(t, τ) −
W(t, τ) −
X(t, τ)
S(t)
2
6
24
The BKM formulas
Data issues
7
Issues in practice I
The difference between assumption and option data
800
price
600
400
200
0
500
1000
1500
2000
K
Data is generally not available in (−∞, ∞)
Model-free moments
The BKM formulas
Data issues
8
Issues in practice II
The difference between assumption and option data
800
price
600
400
200
0
500
1500
1000
2000
K
Option data is discrete. Need a numerical integration scheme: Trapezoid rule
Model-free moments
The BKM formulas
Data issues
9
Issues in practice III
The difference between assumption and option data
800
price
600
400
200
0
500
1500
1000
2000
K
Data might be missing. Need to interpolate.
Model-free moments
The BKM formulas
Data issues
10
Issues in practice IV
The difference between assumption and option data
800
price
600
400
200
0
500
1500
1000
2000
K
Data might contain errors: Smoothing in IV with splines
Model-free moments
The BKM formulas
Data issues
11
Numerical experiment
Experimental setup
Densities
Discretization error
Truncation error
Data in practice
Experimental setup
How I analyze the performance of the BKM equations
1. Compute moments and option prices from a known density
2. Take different subsets/variations of the full set of prices
3. Compute BKM moments from the reduced data set
4. Report errors between true and computed moments
Option prices and moments
⊛ Black & Scholes (1973) model
◮ Moments (of log-returns) of a normal
◮ Prices via closed-form pricing equation
⊛ Bates (1996) model (or SVJ) more
◮
◮
Numerical experiment
Moments via the characteristic function
Option prices via Monte Carlo
Experimental setup
Densities
12
Specific densities
What the model generates
σ
κ3
κ4
τ
0.06
0.14
0.19
-0.12
-0.08
-0.11
0.28
0.56
0.79
30
180
360
set 2
0.12
0.31
0.44
-0.74
-1.22
-1.17
1.24
3.18
2.93
30
180
360
set 3
0.06
0.16
0.22
-1.69
-2.59
-2.44
5.76
14.10
12.72
30
180
360
12.5
10.0
7.5
5.0
set 1
2.5
12.5
10.0
7.5
density
5.0
2.5
12.5
10.0
7.5
5.0
2.5
-1.0
-0.5
0.0
log-return
Numerical experiment Experimental setup
0.5
Densities
Discretization error
13
Discretization error
Black Scholes option prices with volatility of 0.22 and r = 0.02.
6e-04
4e-04
σ
2e-04
0e+00
0.00075
ε
κ3
0.00050
0.00025
0.00000
0.00
-0.05
κ4
-0.10
0
10
20
30
∆K
τ
Numerical experiment Densities
1
12
0.5
1
Discretization error
Truncation error
14
Truncation error
SVJ parameter set 2. Results only shown for quantiles in [0.1%, 20%] and [80%, 99.9%]
right truncation
left truncation
0.00
-0.02
σ
-0.04
-0.06
-0.08
0.5
κ3
ε
1.0
0.0
-0.5
0
-1
κ4
-2
-3
-4
-2.0
-1.5
-1.0
-0.5
τ
Numerical experiment Discretization error
0.0
log-moneyness
1
12
0.5
Truncation error
0.25
0.50
0.75
1.00
1
Data in practice
15
Data availability
Median (empirical) truncation limits, depending on the sampling interval
put
call
1.00
τ
0.75
0.50
0.25
0.00
-1.00
-0.75
-0.50
-0.25
0.1
0.2
0.3
log-moneyness
sampling interval (seconds)
100
Numerical experiment Truncation error
200
Data in practice
300
16
Tail correction
Parametric tails
Correction term
Distributional assumptions
Robustness
Parametric tails
Illustration
price
OTM puts
OTM calls
K
density
Kl
Tail correction
Kp,i
Parametric tails
Kc,i
Ku
ST
Correction term
17
Correction term
Example for the call side
We have several terms of the form
Z∞
f (S(t), K) C(t, τ; K)dK.
S(t)
cf. BKM equations
Let qT (r) be a non-negative function that represents
h the risk-neutral
i
Ku
distribution of log-normal returns on the domain log S(t)
,∞
Then the correction term for the integral in the tail is
Z∞ K
X
1−
qT log
dXdK,
X
S(t)
K
Ku
Z∞
Z∞
1
K
X
f (S(t), K) 1 − Q log
=
−K
dX dK.
qT log
S(t)
S(t)
Ku
K X
R (Ku , S(t)) =
Tail correction
Z∞
f (S(t), K)
Parametric tails
Correction term
Distributional assumptions
18
Distributional assumptions
I use two types of distributional assumptions for the tails
⊛ Generalized extreme value distribution
⊛ Generalized Pareto distribution
Fitted by price matching (Ivanovas & Meier, 2014). Correction term:
numerical integration
Further, as benchmark (popular in recent papers): Constant implied
volatility extrapolation.
Correction via additional option prices
Tail correction
Correction term
Distributional assumptions
Robustness
19
Truncation error with tail correction
right truncation
left truncation
0.00
-0.02
σ
-0.04
-0.06
1.0
ε
κ3
0.5
0.0
0
-1
κ4
-2
-3
-4
-1.6
-1.2
-0.8
-0.4
correction
Tail correction
Correction term
NO
0.3
0.2
log-moneyness
GEV
GPD
Distributional assumptions
0.4
0.5
IV
Robustness
20
Robustness
Pricing errors can lead to heavy tail fits
0.006
0.004
0.002
σ
0.000
-0.002
-0.004
0.1
κ3
ε
0.0
-0.1
-0.2
-0.3
4
3
2
κ4
1
0
-1
0.05
Tail correction
Distributional assumptions
0.10
0.15
m
Robustness
0.20
0.25
21
Time series
Data
Tail extension intraday
Interpolation
Kurtosis time series
Descriptive statistics
Tail Shape
Reaction to events
Intraday patterns
Intraday data
Sampling frequency
S&P 500 option data from market data express. All trading days
2008 - 2011
⊛ Using tick data, specifically level I quotes
⊛ Sampling cross section every 2 minutes
⊛ Rolling window over 10 minutes
Time series
Data
Tail extension intraday
22
Example in 2011
Comparison to the VIX index
2011-05-24
2011-05-23
2011-05-25
2011-05-27
2011-05-26
0.20
σ
0.18
0.16
-1.0
correction
-1.5
GEV
κ3
GPD
IV
-2.0
NO
-2.5
15
10
κ4
5
09:00 10:00 11:00 12:00 13:00 14:00 15:00
Time series
Data
09:00 10:00 11:00 12:00 13:00 14:00 15:00
09:00 10:00 11:00 12:00 13:00 14:00 15:00
time
Tail extension intraday
09:00 10:00 11:00 12:00 13:00 14:00 15:00
09:00 10:00 11:00 12:00 13:00 14:00 15:00
Interpolation
23
Interpolation to fixed maturities
σ
E1
E2
E3
E4
κ3
κ4
E1
E2
E3
E4
E1
E2
E3
E4
E1
E2
E3
E4
1
-
0.97
1
-
0.79
0.89
1
-
0.79
0.90
0.99
1
1
-
0.85
1
-
0.69
0.90
1
-
0.64
0.85
0.94
1
1
-
0.80
1
-
0.62
0.86
1
-
0.57
0.80
0.90
1
Linear correlation between intraday BKM moments of different expiries.
⊛ Issue: moving expiry dates, moments always for different time
horizons
⊛ High correlation between moments of similar expiries →
interpolate to two fixed maturities
⊛ Using interpolation in total implied variance, with subsequent
GEV tails
Time series
Tail extension intraday
Interpolation
Kurtosis time series
24
Kurtosis over four years
dark blue: 30 days, light blue: 0.5 years
20
15
2008
10
5
0
20
15
2009
10
excess kurtosis
5
0
20
15
2010
10
5
0
20
15
2011
10
5
0
January
Time series
April
Interpolation
July
Kurtosis time series
October
Descriptive statistics
January
25
Descriptive statistics
σ
τ
κ3
κ4
30 days
0.5 years
30 days
0.5 years
30 days
0.5 years
1.03
0.53
0.28
0.24
0.16
0.14
0.72
0.49
0.31
0.28
0.22
0.20
0.61
-1.04
-1.72
-1.67
-2.68
-4.12
-0.93
-1.31
-1.87
-1.87
-2.47
-3.58
29.99
14.31
6.59
5.88
1.97
0.48
23.07
10.42
6.23
6.16
2.79
0.19
Maximum
95% quantile
Mean
Median
5% quantile
Minimum
Descriptive statistics for the intraday BKM moments over the years 2008-2011.
30 days
σ
κ3
κ4
0.5 years
σ
κ3
κ4
σ
κ3
κ4
1
-
0.13
1
-
-0.19
-0.96
1
1
-
0.12
1
-
-0.14
-0.99
1
Correlation between the different intraday BKM moments for the
years 2008-2011.
⊛ 198,562 observations for 30 days to expiry, and 198,934 observations
for 0.5 years.
⊛ Occurrences with kurtosis > 30 removed
Time series
Kurtosis time series
Descriptive statistics
Tail Shape
26
Put tail heaviness
Weekly box plots
-0.1
2008
-0.2
-0.3
-0.4
-0.1
2009
-0.2
-0.3
ξ
-0.4
-0.1
2010
-0.2
-0.3
-0.4
-0.1
2011
-0.2
-0.3
-0.4
January
Time series
April
Descriptive statistics
July
Tail Shape
October
Reaction to events
January
27
Lehman Brothers
September 2008
2008-09-08
2008-09-09
2008-09-10
2008-09-11
2008-09-12
2008-09-15
2008-09-16
2008-09-17
2008-09-18
2008-09-19
2008-09-22
2008-09-23
2008-09-24
2008-09-25
2008-09-26
0.40
0.35
σ
0.30
0.25
-0.8
-1.2
κ3
-1.6
-2.0
8
6
κ4
4
2
Time series
Tail Shape
Reaction to events
Intraday patterns
28
Fukushima
March 2011
2011-03-07
2011-03-08
2011-03-09
2011-03-10
2011-03-11
2011-03-14
2011-03-15
2011-03-16
2011-03-17
2011-03-18
2011-03-21
2011-03-22
2011-03-23
2011-03-24
2011-03-25
0.30
0.25
σ
0.20
-1.5
κ3
-2.0
-2.5
16
12
κ4
8
4
Time series
Tail Shape
Reaction to events
Intraday patterns
29
Intraday patterns
σ
κ3
κ4
103
102
2008
101
100
99
106
104
2009
normalized moment
102
100
98
103
102
2010
101
100
99
103
102
2011
101
100
99
98
09:00
10:00
11:00
12:00
13:00
14:00
15:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
time
Time series
Reaction to events
Intraday patterns
30
Conclusion
Conclusion
⊛ BKM formulas are sensitive to amount of data used in the tail
⊛ Assuming excess kurtosis and negative skewness, the typical
data range is not enough
⊛ Missing data con be corrected with parametric tails
⊛ Including the tail correction, BKM can be applied to intraday
data
⊛ Resulting moment time series are driven by the put tail
⊛ Moments are not constant intraday, and react to financial and
non-financial news
Conclusion
31
References
References (1)
Bakshi, G., Kapadia, N., & Madan, D. (2003). Stock return characteristics, skew laws, and the differential pricing of individual equity
options. The Review of Financial Studies, 16, 101–143.
Bakshi, G. & Madan, D. (2000). Spanning and derivative-security valuation. Journal of Financial Economics, 55, 205–238.
Bates, S. S. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. The Review of Financial
Studies, 9, 69–107.
Black, F. & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654.
Buss, A. & Vilkov, G. (2012). Measuring equity risk with option-implied correlations. Review of Financial Studies, 25, 3113–3140.
Conrad, J., Dittmar, R. F., & Ghysels, E. (2013). Ex ante skewness and expected stock returns. Journal of Finance, 68(1), 85–124.
Dennis, P. & Mayhew, S. (2002). Risk-neutral skewness: Evidence from stock options. Journal of Financial and Quantitative Analysis, 37,
471–493.
Eriksson, A., Ghysels, E., & Wang, F. (2009). The normal inverse gaussian distribution and the pricing of derivatives. The Journal of
Derivatives, 16, 23–37.
Ivanovas, A. & Meier, P. (2014). Estimating risk neutral density tails: a comparison. Working paper.
Polkovnichenko, V. & Zhao, F. (2013). Probability weighting functions implied in options prices. Journal of Financial Economics, 107,
580–609.
References
32
Appendix
The Bates model
Generating known theoretical densities
The Bates (or SVJ) model:
√
dSt
(1)
= (r − d − λk̄)dt + vt dWt + dZt
St
√
(2)
dvt = κ(θ − vt )dt + σ vt dWt ,
(1)
(2)
where Cov(dWt , dWt ) = ρdt, Zt is compound Poisson with intensity λ
and jumps J with
ln(1 + J) ∼ N(ln(1 + k̄) −
1 2 2
δ , δ ).
2
The characteristic function of this model is known analytically.
Appendix
33