Expected Stock Returns and Variance Risk Premia (joint paper with Hao Zhou)

Transcription

1 Expected Stock Returns and Variance Risk Premia (joint paper with Hao Zhou) Tim Bollerslev Duke University NBER and CREATES Cass Business School December 8, 2007

2 Much recent work on so-called model-free volatilities... C Realized volatilities based on high-frequency data C Implied volatilities based on options prices This Paper C Relationship between the two volatility concepts and implications for return predictability C Largely empirical Cass 08/12/07-1

3 Realized Volatilities C Stochastic volatility model Variance of given Integrated volatility/quadratic variation Integrated volatility/quadratic variation is the right way to think about ex-post variation Andersen and Bollerslev (1998, IER) But, IV t+1 is latent Cass 08/12/07-2

4 C Realized volatility from high(er)-frequency data Earlier precedent in finance French, Schwert and Stanbaugh (1987, JFE) Hsieh (1998, JF), Schwert (1989, JF; 1990, RFS) C Theory of quadratic variation High-frequency data crucial C Lot s of exciting current work... Market microstructure noise Jumps and cojumps Distributional implications Multivariate measures Interface with option implied volatilities Andersen and Bollerslev (1998, IER) Andersen, Bollerslev, Diebold and Labys (2001, JASA) Barndorff-Nielsen and Shephard (2001, JRSS) Cass 08/12/07-3

5 Option Implied Volatilities C Options trading is essentially equivalent to trading volatility C Black-Scholes-Merton pricing formula dp(t) = µ + σ dw(t) Implied volatility C BS t ( p(t), K i, T i, r, σ ) = C Market t ( p(t), K i, T i, r ) Market-based forecast of (average) volatility over the life of the option Implied volatilities varies across strikes, K i, and maturities, T i Cass 08/12/07-4

6 C Extensive literature devoted to reconciling smiles/smirks and term structure in BS implied volatilities by allowing for stochastic volatility and/or jumps Continuous-time jump-diffusions dp(t) = µ(t)dt + σ(t)dw(t) + κ(t)dq(t) More general Lévy-driven processes Andersen, Benzoni and Lund (2002, JF) Bakshi, Cao and Chen (1997, JF), Bates (1996, RFS; 2003, JoE) Chernov, Gallant, Ghysels and Tauchen (2002, JoE) Duffie, Pan and Singleton (2000, Ect) Eraker (2004, JF), Eraker, Johannes and Polson (2003, JF) Pan (2002, JFE) Barndorff-Nielsen and Shephard (2001, JRSS; 2002, JRSS) Brockwell (2001, Ann. ISM), Carr and Wu (2004, JFE) Carr, Geman, Madan and Yor (2003, Math. Fin.) Tauchen and Todorov (2006, JBES), Todorov (2007, wp) Cass 08/12/07-5

7 C Extensive literature devoted to testing whether option implied volatilities are unbiased forecasts of future (realized) volatilities Canina and Figlewski (1993, RFS) Day and Lewis (1998, JFE) Lamoureux and Lastrapes (1993, RFS) Typically b 0 >0 and b 1 <1 Bollerslev and Zhou (2006, JoE) Regression plagued by errors-in-variables, strong persistence, overlapping date,... Andersen, Frederiksen and Staal (2007, wp) Bandi and Perron (2006, J.Fin.Ect.) Chernov (2007, JBES) Christensen and Prabhala (1998, JFE) But what to expect in population? Cass 08/12/07-6

8 C One-factor affine SV model Heston (1993, RFS) Corresponding risk-neutral dynamics Volatility risk premium λ (<0) Corresponding integrated-implied volatility regression Bollerslev and Zhou (2006, JoE) Cass 08/12/07-7

9 Integrated-implied volatility relationship allows for the estimation of the volatility risk premium λ (and other model parameters) Bollerslev and Zhou (2006, JoE) Bollerslev, Gibson and Zhou (2006, wp) Carr and Wu (2008, RFS) Todorov(2007, wp) IV t+1 is accurately measured by RV t+1 ( ) for 64 But how do you measure E t * (IV t+1 )? Cass 08/12/07-8

10 C New model-free implied volatility Carr and Madan (1998, Book Ch.) Demeterfi, Derman, Kamal and Zou (1999, J.Deriv.) Britten-Jones and Neuberger (2000, JF) Risk-neutral density Also works with jumps Breeden and Litzenberger (1978, J.Buss.) Carr and Wu (2008, RFS) Jiang and Tian (2005, RFS) S&P500 VIX index and many other recent (now actively traded) indexes are based on this idea approximating the integral with a sum over finitely many strikes Andersen and Bondarenko (2007, wp) Carr and Wu (2008, RFS) Cass 08/12/07-9

11 C Monthly S&P500 VIX and realized volatility Bollerslev, Gibson and Zhou (2006, wp) Cass 08/12/07-10

12 C Estimated volatility risk premium -λ t based on macro/finance explanatory variables and one-factor affine SV model Bollerslev, Gibson and Zhou (2006, wp) Higher premium associated with financial market crises or bad times Premium may be linked to notions of aggregate risk aversion Bakshi and Kapadia (2003, RFS) Bakshi and Madan (2006, wp), Bates (1988, wp) Gordon and St-Amour (2004, JBES) Rosenberg and Engle (2002, JFE) What about return predictability? Cass 08/12/07-11

13 Expected Stock Returns and Variance Risk Premia Tim Bollerslev Duke University, NBER and CREATES Hao Zhou Federal Reserve Board Cass, December

14 Stock Return Predictability P/E and DY: Shiller (1984), Keim and Stambaugh (1986), Fama and French (1988), Campbell and Shiller (1988) Risk-Free Rate: Campbell (1991), Hodrick (1992) Term Structure and Default Premia: Fama and Schwert (1977), Fama and French (1989), Campbell (1987) Dividend Payout Ratio: Lamont (1998) Consumption-Wealth Ratio (CAY): Lettau and Ludvigson (2001) Long-Run Variance: Bandi and Perron (2008) Making the case: Lewellen (2004), Cochrane (2007) 2

15 Critical Issues Data mining Highly persistent regressors Long horizons and overlapping data Stock market bubbles and the 90s In-sample versus out-of-sample performance 3

16 Findings in This Paper Difference between model-free IV and RV yield strong predictability at quarterly horizon Model-free vs Black-Scholes IV High- vs low-frequency data RV Outperform other (traditional) predictor variables CAY, P/E, DFSP, TMSP IV-RV and P/E ratio results in even stronger predictability Why? 4

17 Possible Interpretations IV-RV may be a proxy for time-varying risk aversion or habit Campbell and Cochrane (1999) IV-RV may effectively approximate time-varying long-run consumption and volatility risk Bansal and Yaron (2004), Tauchen (2005) But, these are arguably longer-run phenomena May need a new asset pricing model/theory to satisfactorily explain the results 5

18 Model-Free Implied Variance - Carr and Madan (1998), DDKZ (1999), Britten-Johnes and Neuberger (2000), Bondarenko (2003), Jiang and Tian (2005), Carr and Wu (2005), Jiang and Tian (2005), Bollerslev, Gibson, and Zhou (2006), Andersen and Bondarenko (2007), 6

19 Intuition Risk-neutral distribution - Breeden and Litzenberger (1978) Spanning and replicating risk-neutral moments - Bakshi and Madan (2000) Variance Swap Rate 7

20 Realized Variance Earlier work in finance - Merton (1980), Schwert (1990), Hsieh (1991) High-Frequency based RV - AB (1998), ABDL (2001), BN-S (2002), Meddahi (2002) Market microstructure noise - 5-minute for S&P500 8

21 Variance Risk Premium/Difference Directly observable at time t Expected premium (and other IV and RV combinations) - Rosenberg and Engle (2002), Bakshi and Madan (2006), Carr and Wu (2007), Bollerslev, Gibson, and Zhou (2006), Todorov (2007) 9

22 Data: VIX and S&P500 IV: VIX based on S&P500 options and model-free measure RV: Based on five-minute S&P500 futures 10

23 Table 1: Summary Statistics 11

24 Table 2 Forecast Regressions for Quarterly Stock Market Returns 12

25 Figure 2 13

26 14

27 Table 2 (continued) 15

28 Table 3 Robustness with Alternative Variance Measures 16

29 17

30 Table 5 Robustness with Volatility Risk Premium 18

31 Alternative Predictor Variables Long-run mean reversion lagged return Fama and French (1986), Poterba and Summers (1986) Dividend payout ratio Lamont (1998) Volatility risk premium under Heston SV model Bollerslev, Gibson and Zhou (2006) Realized Quarticity: 19

32 Table 6 Robustness with Alternative Predictor Variables 20

33 Alternative Volatility Premia Variance Premium, VRP t = IV t RV t - Reflects expected changes in risk premia and future variation IV t E t (RV t+1 ) - RV random walk: R 2 15% - HAR-RV forecasts: R 2 10% Model-free Black-Scholes IV - Reflects slope of smirk Andersen and Bondarenko (2007) - IV t BSIV t : R 2 0% 21

34 International Evidence DAX returns and model-free implied VDAX - Sample 1993Q3-2007Q1 - Daily squared-return RV - VRP: R 2 = 4.16, t-stat = IV: R 2 = -1.42, t-stat 0 - RV: R 2 = -1.83, t-stat 0 Similar results for recent (Sept. 3, 2007) VAEX, VBEL and VCAC 22

35 Table 4 Robustness with Monthly Stock Market Return 23

36 Economic Interpretation Wedge between IV and RV Provides a more accurate market-based measure of aggregate risk-aversion Provides a simple proxy for expected business conditions (and subsequent GDP growth) Expected business conditions tend to be negatively correlated with future returns Campbell and Diebold (2007) What about systematic risk? Is it risk or risk-aversion? 24

37 Cross-Sectional Asset Pricing Volatility as a systematic risk factor - Adrian and Rosenberg (2006), Ang, Hodrick, Xing, and Zhang (2006), Bandi, Moise and Russell (2007) VRP as a systematic risk factor 25

38 Table 7 Risk-Factor Test with Fama-French Portfolios 26

39 Conclusion Close link between stock market returns and variance risk premia, IV-RV Economic and statistical significance Use of model-free measures important difficult for traditional predictor variables Risk-aversion interpretation Countercyclical with GDP growth Joint predictability with P/E Long- and short-run changes in risk and risk-aversion are both important 27

40 Key predictor variables Conclusion cont. Quarterly: IV-RV This paper B Cycle: Cay Lettau-Ludvigson (2001) 5+ Years: P/E, DY Shiller (1984), Fama and French (1988) Campbell and Shiller (1988) RV Bandi-Perron (2007) 28