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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