1. Factor Structure in Equity Options
Peter Christoffersen
(Rotman School, CBS and CREATES)
Mathieu Fournier
(University of Toronto, PhD student)
Kris Jacobs
(University of Houston)

2. Motivation
• Black and Scholes (1973) derive their famous formula
in several ways including one in which the underlying
assets (the stock) obey a CAPM-type factor structure.
• They show that in their setting the beta of the stock
does not matter for the price of the option.
• They of course assume constant volatility.
• We show that under SV the beta of the stock matters.
– Equity option valuation
– Equity and index option risk management
– Equity option expected returns
• We find strong empirical evidence for factor structure
in equity option IV.
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3. Scale of Empirical Study
• Principal component analysis
– 775,000 Index Options
– 11 million Equity Options
• Estimation of structural model parameters
– 6,000 Index Options
– 150,000 Equity Options
• Estimation of spot variance processes
– 130,000 Index Options
– 3.1 million Equity Options
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4. Related Literature (Selective)
• Bakshi, Kapadia and Madan (RFS, 2003)
• Serban, Lehoczky and Seppi (WP, 2008)
• Driessen, Maenhout and Vilkov (JF, 2009)
• Duan and Wei (RFS, 2009)
• Elkamhi and Ornthanalai (WP, 2010)
• Buss and Vilkov (RFS, 2012)
• Engle and Figlewski (WP, 2012)
• Chang, Christoffersen, Jacobs and Vainberg
(RevFin, 2012)
• Kelly, Lustig and Van Nieuwerburgh (WP, 2013)
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5. Paper Overview
• Part I: A model-free look at option data
• Part II: Specifying a theoretical model
• Part III: Properties of the model
• Part IV: Model estimation and fit
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6. Part I: Data Exploration
• Option Data from OptionMetrics
– Use S&P500 options for market index
– Equity options on 29 stocks from Dow Jones 30
Index.
– Kraft Foods only has data from 2001 so drop it.
– Volatility surfaces.
– 1996-2010
– Various standard data filters (IV <5%, IV>150%,
DTM<30, DTM>365, S/K<0.7, S/K>1.3, PV
dividends > .04*S)
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7. 7
Table 1:
Companies,
Tickers and
Option
Contracts,
1996-2010

8. 8
Table 2:
Summary
Statistics on
Implied
Volatility
(IV).
Puts (left)
Calls (right)
1996-2010

9. Figure 1:
Short-Term, At-
the-money
implied
volatility.
Simple average
of available
contracts each
day.
Sub-sample of
six large firms
1996-2010
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10. PCA Analysis
• On each day, t, using standardized regressors,
run the following regression for each firm, j,
• For the set of 29 firms do principal component
analysis (PCA) on 10-day moving average of
slope coefficients.
• Also do PCA index option IVs.
• We use calls and puts here.
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11. Figure 2:
Does the
common
factor in
the time
series of
equity IV
levels look
anything
like
S&P500
index IV?
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12. 12
Table 3:
Firms’
loadings on
the first 3
PCs of the
matrix of
constant
terms from
the IV
regressions

13. Moments of PC Loadings
IV Levels (Table 3)
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14. 14
Figure 3:
Moneyness
slopes:
S&P500
index versus
1st Principal
Component.
- Need firm-
specific
variation.

15. 15
Table 4:
Firms’
loadings on
the first 3
PCs of the
matrix of
moneyness
slopes
from the IV
regressions

16. Moments of PC Loadings
IV Moneyness Slopes (Table 4)
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17. 17
Figure 4:
IV term
structure:
common
factor versus
S&P500
index term
structure?

18. 18
Table 5:
Firms’
loadings on
the first 3
PCs of the
matrix of
maturity
slopes
from the IV
regressions

19. Moments of PC Loadings
IV Maturity Slopes (Table 5)
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20. Part II: Theoretical Model
• Idea: Stochastic volatility (SV) in index and
equity volatility gives you identification of
beta.
• Black-Scholes-Merton: Impossible to identify
beta.
• SV is a strong stylized fact in equity and index
returns.
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21. Market Index Specification
• Assume the market factor index level evolves
as
• With affine stochastic volatility
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22. Individual Equities
• The stock price is assumed to follow these price
and idiosyncratic variance dynamics:
• Beta is the firm’s loading on the index.
• Note that idiosyncratic variance is stochastic also.
• Note that total firm variance has two components:
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23. Risk Premiums
• We allow for a standard equity risk premium (μI)
as well as a variance risk premium (λI) on the
index but not on the idiosyncratic volatility.
• The firm will inherit equity risk premium via its
beta with the market.
• The firm will inherit the volatility risk premium
from the index via beta.
• These assumptions imply the following risk-
neutral dynamics
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24. Risk Neutral Processes (tildes)
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Variance risk
premium < 0

25. Option Valuation
• Index option valuation follows Heston (1993)
• Using the affine structure of the index variance, the
affine idiosyncratic equity variance, and the linear factor
model, we derive the closed-form solution for the
conditional characteristic function of the stock price.
• From this we can price equity options using Fourier
inversion which requires numerical integration. Call
price:
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26. Part III: Model Properties
• Equity Volatility Level
• Equity Option Skew and Skew Premium
• Equity Volatility Term Structure
• Equity Option Risk Management
• Equity Option Expected Returns
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27. Equity Volatility
• The total spot variance for the firm is
• The total integrated RN variance is
• Where
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28. Model Property 1:
Beta Matters for the IV Levels
• When the market risk premium is negative we
have that
• We can show that for two firms with same
levels of total physical variance we have
• Upshot: Beta matters for total RN variance.
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29. Model Property 2: Beta Matters for the IV
Slope across Moneyness
29
Figure 5:
Beta and model
based BS IV
across
moneyness
Unconditional
total P variance
is held fixed.
Index ρ =-0.8
and firm-
specific ρ =0.

30. Model Property 3: Beta Matters for the IV
Slope across Maturity
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Figure 6:
Beta and model
based BS IV
across maturity
Unconditional
total P variance
is held fixed.
Index κ = 5 and
firm-specific κ =
1.

31. Model Property 4: Risk Management
• Equity option sensitivity “Greeks” with market
level and volatility
• Market “Delta”:
• Market “Vega”:
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32. 32

33. Model Property 5: Expected Returns
• The model implies the following simple
structure for expected equity option returns
• Where we have assumed that αj = 0.
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34. 34

35. Part IV: Estimation and Fit
• We need to estimate the structural
parameters
• We also need on each day to estimate/filter
the latent volatility processes
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36. Estimation Step 1: Index
• For a fixed set of starting values for the
structural index parameters, on each day solve
• Then keep sequence of vols fixed and solve
• Then iterate between these two
optimizations.
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37. Estimation Step 2: Each Equity
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• Take index parameters as given. For a fixed set
of starting values for the structural equity
parameters, on each day solve
• Then keep sequence of vols fixed and solve
• Then iterate between these two
optimizations. Do this for each equity…

38. Parameter Estimates
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39. 39

40. Definitions
• Average total spot volatility (ATSV)
• Systematic risk ratio (SSR)
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41. Model Fit
• To measure model fit we compute
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42. 42

43. IV Smiles. Market (solid) and Model (dashed).
High Vol (black) and Low Vol (grey) Days.
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44. 44

45. 45

46. • Conclusion: The “smiles” vary considerably across
firms and we fit them quite well.
• We also fit index quite well.
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47. IV Term Slopes: Up and Down.
Market (solid) and Model (dashed)
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48. 48

49. 49

50. • Conclusion: IV term structures vary
considerably across firms. Model seems to
adequately capture persistence.
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51. Beta: Cross-Sectional Implications
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• The model fits equity options well.
• What are the cross sectional implications of
the factor structure?
• Recall our IV regression from the model-free
analysis in the beginning:

52. Betas versus IV Levels
• Regress time-averaged constant terms from
daily IV regressions on betas.
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53. Betas versus Moneyness Slopes
• Regress time-averaged moneyness slopes
from daily IV regressions on betas.
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54. Beta and Maturity Slopes
• Regress time-averaged maturity slopes from
daily IV regressions on betas.
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55. OLS Beta versus Option Beta
55
OLS betas are
estimated on daily
Returns. 1996-2010.
Regression line
45 degree line

56. Additional Factor Structure?
• We have modeled a factor structure in returns
which implies a factor structure in equity total
volatility.
• Engle and Figlewski (WP, 2012)
• Kelly, Lustig and Van Nieuwerburgh (WP, 2013)
• Is there a factor structure in the idiosyncratic
volatility paths estimated in our model?
• Yes: The average correlation of idiosyncratic
volatility is 45%.
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57. Conclusions
• Model-free PCA analysis reveals strong factor
structure in equity index option implied volatility
and thus price.
• We develop a market-factor model based on two
SV processes: Market and idiosyncratic.
• Theoretical model properties broadly consistent
with market data.
• Model fits data reasonably well.
• Firm betas are related to IV levels, moneyness
slopes and maturity slopes.
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58. Current / Future Work
• Add firms.
• Study cross-sectional properties of beta
estimates.
• Add a second volatility factor to the market
index.
• Time-varying betas.
• Add jumps to index and/or to idiosyncratic
process.
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