Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Option Returns and Individual Stock
Volatility
Jie Cao, Chinese University of Hong Kong
Bing Han, University of Texas at Austin
Presented by Jie Cao
2010 NTU International Conference on Finance
December 10, 2010
1/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Motivation

Equity option market is big and grows very fast



Most option research focuses on pricing options relative to the
underlying stock, given the stock price
But not much is known about the returns investors can expect
to receive from holding various stock options
We examine the cross-sectional determinants of expected
equity option returns
 Focus on the role of volatility


Study delta-hedged options (control for the price movement of
underlying stocks), which are most sensitive to volatility risk
Use a large sample of individual stock options rather than index
options
2/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Related Literature

Index options



Coval and Shumway (2001) study index option returns and argue that
some other systematic factor, such as stochastic volatility, might be
priced by the market
Following stochastic volatility model, Bakshi and Kapadia (2003a)
examine delta-hedged returns of S&P 500 index options, and find a
negative market volatility risk premium (time-series tests)
Individual options



Bakshi and Kapadia (2003b) study 25 individual options, and argue
that 1) individual stock option prices also embed a negative market
volatility risk premium, 2) but idiosyncratic volatility is not priced
Duarte and Jones (2007) study individual options, and find that beta
to the market volatility risk conditionally matters in the cross-section
With a large sample, we study how total and idiosyncratic volatility
affect the cross-sectional delta-hedged individual option returns
3/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Daily Rebalanced Delta-Hedged Gains

Delta-hedged gain: following Bakshi and Kapadia (2003a)

Changes in the value of a portfolio of a long call position, and
hedged by a short position in the underlying stock, with the net
investment earning risk-free rate

Discrete version: daily rebalancing for empirical analysis

Normalized: the gain is scaled by stock (or option) price
4/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Under Stochastic Volatility Model


Under Black-Scholes model:
Under Stochastic Volatility model :


Volatility follows:
Bakshi and Kapadia (2003a) show:
is the market price of volatility risk


Assuming
total volatility
(Heston (1993)),
is linear in
5/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Data and Sample

Data and Sample





Options daily data from Option-Metrics
Underlying stock data obtained from CRSP, COMPUSTAT
Each month, construct a cross-section of at-the-money options
with a common short-term maturity (around 50 calendar days)
Apply several filters to ensure data quality: exclude options if
 Paying dividend, or violate no-arbitrage conditions
 Bid = 0, or (Bid + Ask)/2 < 1/8, or zero volume
 Moneyness (S/K) < 0.8 or > 1.2, or non-common maturity
Final sample: Jan 1996 – Dec 2006
 5,255 stocks, average 1,394 per month
 159,346 obs for call and 139,285 obs for put
6/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Summary Statistics
Variable
Mean
Median
Lower Upper
90 Pctl
Quartile Quartile
StDev
10 Pctl
Panel A: Call Options (159,346 Obs)
Gain till maturity / stock price (%)
-0.49
-0.65
3.66
Gain till maturity / option price (%)
-4.99
-10.17 58.30
Gain till month-end / stock price (%)
-0.38
-0.47
2.21
Gain till month-end / option price (%)
-4.32
-7.33
30.12
Days to maturity
50
50
2
Moneyness = S/K (%)
100.51 100.11
5.10
Vega
0.14
0.14
0.01
-3.58
-44.65
-2.51
-31.86
47
94.72
0.13
-1.90
-27.05
-1.36
-19.39
50
97.25
0.14
0.51
2.24
7.78
31.70
0.41
1.72
6.31
24.37
51
52
103.36 106.51
0.15
0.15
Panel B: Put Options (143,017 Obs)
Gain till maturity / stock price (%)
-0.54
-0.67
3.18
Gain till maturity / option price (%)
-6.58
-11.12 44.97
Gain till month-end / stock price (%)
-0.24
-0.42
2.38
Gain till month-end / option price (%)
-2.30
-6.92
35.26
Days to maturity
50
50
2
Moneyness = S/K (%)
99.84
99.72
4.86
Vega
0.14
0.14
0.01
-3.58
-46.71
-2.45
-31.95
47
94.23
0.13
-1.92
-28.87
-1.31
-19.66
50
96.83
0.14
0.55
2.41
8.51
34.72
0.52
2.00
8.25
29.06
51
52
102.75 105.63
0.15
0.15
Panel C:
Number of stocks with positive or negative average delta-hedged option return till maturity
Option Type
Total stocks
mean<0
t<-2
mean>0
t>2
Call
Put
5159
5073
3890
3975
1898
1928
1269
1098
62
68
7/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Delta-Hedged Gains and Volatility

Fama-MacBeth Regressions


Stock volatility




Dependent variable: delta-hedged gain till maturity / stock price
Total volatility: VOL -- s.t.d of previous month daily returns
Idiosyncratic volatility: IVOL -- AHXZ (2006): FF-3 factors
Systematic volatility measures
 SysVOL = sqrt (VOL2 - IVOL2)
 Betas to MKTRF, SMB, HML, and change in VIX
Key results



The delta-hedged gains decrease with total volatility
The result is driven entirely by idiosyncratic volatility
Consistent with stochastic volatility model prediction
8/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Determinants of Delta-Hedged Option Returns: Volatility
VOL
Model 1
-0.0113
(-7.38)
Model 2
-0.0124
(-8.62)
Model 3
-0.0271
(-15.13)
IVOL
SysVOL
ΔVIX Beta
Model 4
Model 5
-0.0279
(-18.60)
-0.0058
(-1.36)
-0.0291
(-19.52)
MKTRF Beta
SMB Beta
HML Beta
Ln (VOLt-1 / IVt-1)
Ln (IVt / IVt-1)
Vega
Contemporaneous
stock return
-0.1825
(-8.38)
0.0326
(11.46)
0.0223
(20.71)
0.0340
(23.55)
-0.0469
(-2.30)
0.0300
(11.10)
0.0219
(20.63)
0.0341
(23.68)
-0.0485
(-2.43)
0.0302
(11.36)
0.0200
(1.53)
-0.0002
(-0.91)
-0.0001
(-1.32)
-0.0003
(-1.64)
0.0217
(20.41)
0.0341
(24.16)
-0.0485
(-2.50)
0.0306
(11.91)
9/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Robustness




The results hold after controlling
 for vega (moneyness)
 for contemporaneous stock returns
 for jump risk – option implied Skewness & Kurtosis
 for volatility-related mispricing (Goyal and Saretto (2009))
 for past stock returns over different horizons
 for stock liquidity and transaction costs
 for option demand pressure and transaction costs
The results hold using alternative volatility measures:
 Expected idiosyncratic volatility from EGARCH (1,1)
 Implied total volatility
The results hold for delta-hedged gain till month-end, or scaled by
the option price
The results hold for both call options and put options
10/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
More Controls
Dependent Variables
VOL
Ret (-1,0)
Ret (-12,-1)
Ret (-36, -13)
(Option open interest /
stock volume) *103
Option bid-ask spread (%)
Ln (Illiquidity)
Stock price
Ln (ME)
Ln (VOLt-1 / IVt-1)
Ln (IVt / IVt-1)
Vega
Contemporaneous
stock return
Gain till maturity
stock price
Gain till month-end
stock price
Gain till maturity
option price
-0.0264
(-18.81)
0.0016
(1.20)
0.0017
(7.50)
0.0005
(3.87)
-0.0162
(-8.54)
0.0070
(5.34)
-0.0022
(-11.05)
0.0001
(10.11)
-0.0021
(-9.97)
0.0197
(20.09)
0.0340
(27.56)
-0.0184
(-1.06)
0.0336
(13.85)
-0.0174
(-17.98)
-0.0003
(-0.38)
0.0010
(8.13)
0.0002
(4.00)
-0.0142
(-10.24)
0.0007
(0.92)
-0.0010
(-9.42)
0.0000
(0.93)
-0.0008
(-7.52)
0.0129
(24.92)
0.0476
(28.63)
0.0452
(4.51)
0.0034
(4.50)
-0.2106
(-10.11)
0.0130
(0.68)
0.0204
(6.07)
0.0044
(3.23)
-0.2009
(-10.57)
-0.0395
(-2.09)
-0.0162
(-5.90)
0.0024
(8.85)
-0.0232
(-6.93)
0.1956
(12.69)
0.5337
(38.46)
-0.6067
(-2.14)
0.4569
(11.05)
11/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Put Options
Dependent Variables
VOL
Ret (-1,0)
Ret (-12,-1)
Ret (-36, -13)
(Option open interest /
stock volume) *103
Option bid-ask spread (%)
Ln (Illiquidity)
Stock price
Ln (ME)
Ln (VOLt-1 / IVt-1)
Ln (IVt / IVt-1)
Vega
Contemporaneous
stock return
Gain till maturity
stock price
Gain till month-end
stock price
Gain till maturity
option price
-0.0266
(-18.40)
-0.0025
(-1.89)
0.0015
(8.20)
0.0004
(5.21)
-0.0241
(-9.54)
0.0133
(9.03)
-0.0028
(-11.69)
0.0000
(1.41)
-0.0020
(-10.40)
0.0211
(23.57)
0.0367
(26.77)
0.1172
(7.17)
-0.0242
(-7.64)
-0.0159
(-13.96)
-0.0019
(-1.90)
0.0010
(8.50)
0.0002
(4.99)
-0.0191
(-12.10)
0.0024
(2.59)
-0.0010
(-8.86)
0.0000
(0.34)
-0.0008
(-7.84)
0.0126
(23.59)
0.0507
(29.67)
0.0633
(5.53)
-0.0021
(-2.70)
-0.0039
(-8.07)
-0.0019
(-3.76)
0.0002
(1.86)
0.0001
(3.64)
-0.0034
(-5.04)
-0.0007
(-1.32)
-0.0004
(-6.66)
-0.0000
(-4.35)
-0.0003
(-6.18)
0.0044
(15.21)
0.0101
(23.09)
-0.0054
(-1.40)
0.0001
(0.39)
12/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Monthly Rebalanced Delta-Hedged Returns



Daily rebalancing is very costly in practice
Delta-hedged gain scaled by stock or option price is not a proper
measure for return
Construct a return measure from covered call writing




At the beginning of each month,
sell one call and buy delta-unit stocks: cost Vt = (∆t*S t - C t)>0
At the end of each month
buy the call and sell the stocks: gain Vt+1 =(∆t*S t+1 - C t+1)
Rt = (Vt+1 - Vt )/ Vt
R should increase with total or idiosyncratic volatility in the crosssection
13/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Returns of Covered Call Writing Sorted on Volatility
Quintile
1-Low
2
3
4
5-High
5-1
CAPM FF-3 CarhartAlpha Alpha 4 Alpha
Panel A: Portfolio Return (%) Sorted on Total Volatility (VOL)
Equal-weighted
Stock-value-weighted
Option-value-weighted
2.32
2.33
3.95
2.98
2.56
2.05
1.62
(13.18) (13.24) (14.01) (12.32) (15.05) (10.37) (9.69)
1.61
1.66
3.18
2.41
2.02
1.65
1.52
(11.43) (10.94) (12.81) (9.21) (11.52) (6.77) (5.83)
2.25
2.31
3.76
2.55
2.27
1.71
1.45
(9.72) (9.61) (11.86) (7.66) (12.16) (7.87) (7.03)
2.32
(9.23)
1.70
(5.77)
2.31
(6.47)
2.31
(9.64)
1.77
(6.32)
2.31
(6.90)
Panel B: Portfolio Return (%) Sorted on Idiosyncratic Volatility (IVOL)
Equal-weighted
Stock-value-weighted
Option-value-weighted
2.27
2.30
2.32
2.32
3.94
3.05
2.52
2.03
1.62
(13.55) (12.73) (12.87) (13.65) (15.43) (11.37) (10.72) (9.82) (10.01)
1.64
1.61
1.54
1.59
3.12
2.52
2.10
1.70
1.53
(11.28) (11.87) (10.38) (10.85) (11.40) (7.00) (6.04) (5.83) (6.32)
1.65
1.73
1.81
1.87
3.40
2.92
2.51
1.64
1.53
(10.25) (9.06) (11.15) (10.03) (9.17) (5.44) (4.97) (3.77) (3.44)
14/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Portfolio Returns Sorted on VOL: Subsamples
Subsample Evidence: Equal-weighted Portfolio Returns (%) Sorted on VOL
1-Low
2
3
4
5-High 5-1
t-stat
Size Quintile 1
2.78
3.52
3.67
4.15
5.00
2.22
(8.15)
Size Quintile 2
2.15
2.59
2.91
3.04
3.94
1.79
(6.90)
Size Quintile 3
1.78
2.18
2.28
2.64
3.39
1.61
(6.34)
Size Quintile 4
1.43
1.77
2.04
2.39
2.92
1.49
(5.59)
Size Quintile 5
1.36
1.56
1.59
1.83
2.58
1.23
(5.74)
January
Feb-Dec
1996 - 1999
2000 - 2003
2004 - 2006
1.85
1.60
1.62
1.91
1.25
2.24
2.03
1.98
2.43
1.63
2.54
2.56
2.45
2.98
2.15
3.31
2.95
2.95
3.38
2.49
4.62
3.89
4.02
4.12
3.64
2.76
2.29
2.40
2.22
2.38
(3.87)
(9.03)
(6.69)
(4.82)
(14.76)
15/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Time-Series of the (5-1) Spreads Sorted on VOL
1996-1999
2000-2003
2004-2006
10
8
6
4
2
0
-2
-4
-6
16/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Impact of Transaction Costs and Liquidity


Option bid-ask spread is relatively high
The effective bid-ask spread (ESPR) is a subset of quoted bid-ask
spread (QSPR)
Equal-Weighted Portfolio Returns (%) Sorted on Total Volatility (VOL)
5-1
10-1
ESPR/QSPR
ESPR/QSPR
Sorted on
MidP
50%
75%
100%
MidP
50%
75%
100%
Average Return
2.33
1.25
0.73
0.22
2.87
1.61
1.00
0.41
(10.37)
(5.58)
(3.22)
(0.95)
(10.42)
(5.91)
(3.65)
(1.46)
17/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Economic Interpretations of Negative Risk Premium

Compensation for option sellers



Volatility premium could be compensation for option sellers
who are unable to eliminate volatility risk through hedging and
diversification
Even if they could perfectly hedge the option’s exposure to
underlying stock, they are exposed to volatility risk, which are
higher for more volatile stocks
Options on high volatility stocks are overvalued


These options attract investors who like to gamble or who
prefer positive skewness in payoffs
Overconfident investors overreact to recent increase in
volatility of these stocks
18/19
Motivation
Framework
Data
Regressions
Portfolio Sorts
Conclusion
Conclusion

This paper provides a comprehensive study of individual option
returns, after delta-hedging their exposure to the underlying
stock returns

The average delta-hedged stock option returns are negative

These returns decrease with the volatility of the underlying stock.
The result is driven by idiosyncratic volatility

Individual stock options embed a negative premium for the
underlying stock’s stochastic volatility

This premium could be compensation for option sellers, or
reflect the overvaluation created by option investors
19/19