Portfolio Management Using Option Data Peter Christoffersen Rotman School of Management, University of Toronto, Copenhagen Business School, and CREATES, University of Aarhus 2 nd Lecture on Friday 1
Overview of Topics Yesterday 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 2
First Topic: Option Spanning and Forecasting Bakshi and Madan (JFE, 2000) and Carr and Madan (QF, 2001) show that any twicedifferentiable (date T) payoff function can be replicated by a portfolio of bonds, stocks, and European OTM calls and puts. If we choose the payoff function to be returns to the power of 2, 3 and 4 then we get option implied vol, skew and kurtosis, respectively. Many other payoffs are of course possible 3
The Replication (or Spanning) Result Carr and Madan (QF, 2001) Any twice differentiable function H(S T ) can be replicated by positions in bond, stock and options: The discounted risk neutral expectation is: Think of forecasting applications 4
Quadratic, Cubic and Quartic Payoffs Note: Simple Returns Here 5
Now get the Moments from the Quad, Cube and Quartic Contracts 6
Compute SKEW and KURT daily using 1M maturity. Use VIX for second moment as in the literature. Our Vol estimate has a correlation of 0.99 with VIX. Implementation: Estimate cubic splines on discrete strike prices and integrate on spline. Use linear interpolation to get fixed 1M maturity 7
Second Topic: Firm Specific Option Implied Moments (OIMs) Conrad, Dittmar, Ghysels (JF, 2013). Monthly tercile returns from sorting stocks on their OIMs 8
Portfolio Allocation with 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) Tough Problem: Unless you assume parametric portfolio weights: 9
Third Topic: VIX and SKEW as Market Factors For each stock i, run a time series regression on monthly data of the form Then sort the stocks into quintiles based on the size of their regression coefficients vol Finally compute the average return for each quintile. Use change in VIX for VOL as a measure of unexpected volatility. 10
VIX as an Equity Market Factor Ang, Hodrick, Xing and Zhang (JF,2006). 11
CBOE SKEW Index Option Implied (Negative of) Skewness 12
Market SKEW as a Factor Chang, Christoffersen and Jacobs (JFE, 2013) Regress returns on market skew. Sort on skew beta 13
Fourth Topic: Option Implied Betas Let us assume a single factor CAPM style model with the market return (S&P500) being the factor We assume that the idiosyncratic shock has zero mean and is independent of the market factor. The conventional estimator of market beta is 14
Deriving Option Implied Beta Solve for beta: Then use Carr and Madan (2001) to get moments. 15
Option Implied Beta Across Firms We scatter plot the mean OI beta on the X axis against the mean realized beta on the Y axis for S&P100 firms. 6 months options. Daily obs. 1996 2005. 16
Pfizer/Warner Lambert Merger: Option Implied vs Historical Beta 17
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 18
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 19
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 20
Table 1: Companies, Tickers and Option Contracts, 1996 2010 21
Table 2: Summary Statistics on Implied Volatility (IV). Puts (left) Calls (right) 1996 2010 22
Figure 1: Short Term, Atthe money implied volatility. Simple average of available contracts each day. Sub sample of six large firms 1996 2010 23
PCA Analysis On each day run the following regression for each firm For the set of 29 firms do principal component analysis (PCA) on 10 day moving average of slope coefficients. Also do PCA on the short term at the money IVs. 24
Figure 2: Does the common factor in the time series of equity IV levels look anything like S&P500 index IV? 25
Figure 3: Does the common factor in the firm moneyness slopes look anything like S&P500 index slope? 26
Figure 4: Does the common factor in the firm IV term structure look anything like S&P500 index term structure? 27
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. 28
Market Index Specification Assume the market factor index level evolves as With affine stochastic volatility 29
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 30
Risk Premiums We allow for a standard equity risk premium (μ I ) as well as a variance risk premium 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 riskneutral dynamics 31
Risk Neutral Processes (tildes) Variance risk premium < 0 32
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: 33
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 34
Equity Volatility The total spot variance for the firm is The total integrated RN variance is Where 35
Model Property 1: 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. 36
Model Property 2: IV Slope Figure 5: Beta and model based BS IV across moneyness Unconditional total variance is held fixed. Index ρ = 0.8 and firmspecific ρ =0. 37
Model Property 3: IV Term Figure 6: Beta and model based BS IV across maturity Unconditional total variance is held fixed. Index κ = 5 and firmspecific κ = 1. 38
Model Property 4: Risk Management Equity option sensitivity Greeks with market level and volatility Market Delta : Market Vega : 39
Model Property 5: Expected Returns The model implies the following simple structure for expected equity option returns 40
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 41
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. 42
Estimation Step 2: Each Equity 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 43
Table 6: Model Parameters and Properties. Models estimated on 2002 2005 data only. Preliminary results. 44
Model Fit To measure model fit we compute 45
Table 8: Model Fit Pattern in Bias. 46
Summary of Findings 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. 47
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 implied information using Carr and Madan (2001). CJC survey chapter in Handbook of Economic Forecasting, Volume 2. 48