Option Prices and the Cross Section ofEquity Returns
Peter ChristoffersenRotman School of Management, University of Toronto,
Copenhagen Business School, andCREATES, University of Aarhus
12nd Lectureon Friday
Overview of Topics• Earlier
– Portfolio allocation with RV and RCov– Realized beta form RV and RCov– Firm-specific higher moments from intraday data.
• Today– 1) Extracting option implied higher moments– 2) Portfolio allocation with higher moments– 3) VIX and SKEW as equity market factors– 4) Option-implied beta– 5) Factor structure revealed by equity option prices
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First Topic:Option Spanning and Forecasting
• Bakshi and Madan (JFE, 2000) and Carr andMadan (QF, 2001) show that any twice-differentiable (date T) payoff function can bereplicated by a portfolio of bonds, stocks, andEuropean OTM calls and puts.
• If we choose the payoff function to be returnsto the power of 2, 3 and 4 then we get optionimplied vol, skew and kurtosis, respectively.
• Many other payoffs are of course possible…3
The Replication (or Spanning) ResultCarr and Madan (QF, 2001)
• Any twice differentiable function H(ST) can bereplicated by positions in bond, stock and options:
• The discounted risk neutral expectation is:
• Think of forecasting applications…4
Quadratic, Cubic and Quartic PayoffsNote: Simple Returns Here
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Now get the Moments from the Quad,Cube and Quartic Contracts
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Compute SKEW andKURT daily using 1Mmaturity.
Use VIX for secondmoment as in theliterature.
Our Vol estimate hasa correlation of 0.99with VIX.
Implementation:Estimate cubicsplines on discretestrike prices andintegrate on spline.Use linearinterpolation to getfixed 1M maturity
Second Topic: Firm SpecificOption Implied Moments (OIMs)
• Conrad, Dittmar, Ghysels (JF, 2013). Monthly tercilereturns from sorting stocks on their OIMs
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Portfolio Allocationwith Firm-Specific Higher Moments
• Brandt, Santa-Clara and Valkanov (RFS, 2009)• Ghysels, Valkanov and Plazzi (WP, 2011).• Realized moments could be used as well (ACJV)
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ToughProblem:
Unless you assumeparametric portfolio weights:
Third Topic:VIX and SKEW as Market Factors
• For each stock i, run a time series regression onmonthly data of the form
• Then sort the stocks into quintiles based on thesize of their regression coefficients β∆vol
• Finally compute the average return for eachquintile.
• Use change in VIX for ∆VOL as a measure ofunexpected volatility.
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∆VIX as an Equity Market FactorAng, Hodrick, Xing and Zhang (JF,2006).
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CBOE SKEW IndexOption Implied (Negative of) Skewness
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Market SKEW as a FactorChang, Christoffersen and Jacobs (JFE, 2013)
Regress returns on market skew. Sort on skew beta
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Fourth Topic: Option Implied Betas
• Let us assume a single factor CAPM style modelwith the market return (S&P500) being the factor
• We assume that the idiosyncratic shock has zeromean and is independent of the market factor.
• The conventional estimator of market beta is
Deriving Option-Implied Beta
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Solve for beta:
Then use Carr and Madan (2001) to get moments.
Option-Implied Beta Across Firms
We scatter plotthe mean OIbeta on the X-axis against themean realizedbeta on the Y-axis for S&P100firms. 6-monthsoptions. Dailyobs. 1996-2005.
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Pfizer/Warner-Lambert Merger:Option–Implied vs Historical Beta
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Fifth Topic:Evidence of Equity Market Factor
Structure Using Equity Option Prices• Black Scholes versus CAPM (MBA Teaching)• Is there a factor structure in equity options• CAPM is dead?• Options are informative about equity risk
– Volatility– Skewness– Beta
• Equity option risk management• Equity option returns
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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– 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.– 1996-2010– Various standard data filters
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Table 1:
Companies,Tickers andOptionContracts,1996-2010
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Table 2:
SummaryStatistics onImpliedVolatility(IV).
Puts (left)Calls (right)
1996-2010
Figure 1:
Short-Term, At-the-moneyimpliedvolatility.
Simple averageof availablecontracts eachday.
Sub-sample ofsix large firms
1996-2010
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PCA Analysis• On each day run the following regression for
each firm
• For the set of 29 firms do principal componentanalysis (PCA) on 10-day moving average ofslope coefficients.
• Also do PCA on the short-term at-the-moneyIVs.
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Figure 2:
Does thecommonfactor inthe timeseries ofequity IVlevels lookanythinglikeS&P500index IV?
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Figure 3:
Does thecommonfactor inthe firmmoneynessslopes lookanythinglikeS&P500indexslope?
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Figure 4:
Does thecommonfactor in thefirm IV termstructurelookanything likeS&P500index termstructure?
Part II: Theoretical Model
• Idea: Stochastic volatility (SV) in index andequity volatility gives you identification ofbeta.
• Black-Scholes-Merton: Impossible to identifybeta.
• SV is a strong stylized fact in equity and indexreturns.
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Market Index Specification
• Assume the market factor index level evolvesas
• With affine stochastic volatility
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Individual Equities• The stock price is assumed to follow these price
and volatility dynamics:
• Beta is the firm’s loading on the index.• Note that idiosyncratic vol 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 on theindex but not on the idiosyncratic volatility.
• The firm will inherit equity risk premium via itsbeta with the market.
• The firm will inherit the volatility risk premiumfrom 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 riskpremium < 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 factormodel, we derive the closed-form solution for theconditional characteristic function of the stock price.
• From this we can price equity options using Fourierinversion which requires numerical integration. Callprice:
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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: IV Levels
• When the market risk premium is negative wehave that
• We can show that for two firms with samelevels of total physical variance we have
• Upshot: Beta matters for total RN variance.
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Model Property 2: IV Slope
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Figure 5:
Beta andmodel basedBS IV acrossmoneyness
Unconditionaltotal varianceis held fixed.
Index ρ =-0.8and firm-specific ρ =0.
Model Property 3: IV Term
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Figure 6:
Beta andmodel basedBS IV acrossmaturity
Unconditionaltotal varianceis held fixed.
Index κ = 5and firm-specific κ = 1.
Model Property 4: Risk Management
• Equity option sensitivity “Greeks” with marketlevel and volatility
• Market “Delta”:
• Market “Vega”:
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Model Property 5: Expected Returns
• The model implies the following simplestructure for expected equity option returns
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Part IV: Estimation and Fit
• We need to estimate the structuralparameters
• We also need on each day to estimate/filterthe 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 twooptimizations.
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Estimation Step 2: Each Equity
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• Take index parameters as given. For a fixed setof starting values for the structural equityparameters, on each day solve
• Then keep sequence of vols fixed and solve
• Then iterate between these twooptimizations. Do this for each equity…
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Table 6:
ModelParameters andProperties.
Modelsestimated on2002-2005 dataonly.
Preliminaryresults.
Model Fit
• To measure model fit we compute
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Table 8:
ModelFit
Patternin Bias.
Summary of Findings
• Model-free PCA analysis reveals strong factorstructure in equity index option impliedvolatility and thus price.
• We develop a market-factor model based ontwo SV processes: Market and idiosyncratic.
• Theoretical model properties broadlyconsistent with market data.
• Model fits data reasonably well.
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Discussion• Specifically
– Study cross-sectional properties of beta estimates.– Add a second volatility factor to the market index.– Add jumps to index and/or to idiosyncratic
process.• Generally
– Think of alternate uses of option impliedinformation using Carr and Madan (2001).
– CJC survey chapter in Handbook of EconomicForecasting, Volume 2.
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