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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