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Factor Structure in Equity Options Peter Christoffersen
(Rotman School, CBS and CREATES) Mathieu Fournier
(University of Toronto, PhD student) Kris Jacobs
(University of Houston)
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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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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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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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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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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Table 1: Companies, Tickers and Option Contracts, 1996-2010
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Table 2: Summary Statistics on Implied Volatility (IV). Puts (left) Calls (right) 1996-2010
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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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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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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Table 3: Firms’ loadings on the first 3 PCs of the matrix of constant terms from the IV regressions
Moments of PC Loadings IV Levels (Table 3)
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Figure 3: Moneyness slopes: S&P500 index versus 1st Principal Component. - Need firm-specific variation.
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Table 4: Firms’ loadings on the first 3 PCs of the matrix of moneyness slopes from the IV regressions
Moments of PC Loadings IV Moneyness Slopes (Table 4)
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Figure 4: IV term structure: common factor versus S&P500 index term structure?
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Table 5: Firms’ loadings on the first 3 PCs of the matrix of maturity slopes from the IV regressions
Moments of PC Loadings IV Maturity Slopes (Table 5)
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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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Market Index Specification
• Assume the market factor index level evolves as
• With affine stochastic volatility
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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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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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Risk Neutral Processes (tildes)
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Variance risk premium < 0
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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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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Equity Volatility
• The total spot variance for the firm is
• The total integrated RN variance is
• Where
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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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Model Property 2: Beta Matters for the IV Slope across Moneyness
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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.
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.
Model Property 4: Risk Management
• Equity option sensitivity “Greeks” with market level and volatility
• Market “Delta”:
• Market “Vega”:
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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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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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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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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…
Parameter Estimates
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Definitions
• Average total spot volatility (ATSV)
• Systematic risk ratio (SSR)
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Model Fit
• To measure model fit we compute
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IV Smiles. Market (solid) and Model (dashed). High Vol (black) and Low Vol (grey) Days.
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• Conclusion: The “smiles” vary considerably across firms and we fit them quite well.
• We also fit index quite well. 46
IV Term Slopes: Up and Down. Market (solid) and Model (dashed)
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• Conclusion: IV term structures vary considerably across firms. Model seems to adequately capture persistence.
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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:
Betas versus IV Levels
• Regress time-averaged constant terms from daily IV regressions on betas.
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Betas versus Moneyness Slopes
• Regress time-averaged moneyness slopes from daily IV regressions on betas.
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Beta and Maturity Slopes
• Regress time-averaged maturity slopes from daily IV regressions on betas.
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OLS Beta versus Option Beta
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OLS betas are estimated on daily Returns. 1996-2010. Regression line 45 degree line
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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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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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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