Dissecting the Market Pricing of Return Volatility

Transcription

1 Dissecting the Market Pricing of Return Volatility Torben G. Andersen Kellogg School, Northwestern University, NBER and CREATES Oleg Bondarenko University of Illinois at Chicago Measuring Dependence in Finance CEA-ESRC Conference Cass Business School December 8, 27

2 Introduction Growing Interest in Equity-Market Volatility Index, VIX, published by the CBOE Uses Notion of Model-Free Implied Volatility (MFIV) for S&P5 Cash Index MFIV is Expected (Q-) Value (Price) of One-Month-Ahead Integrated Variance, Reflecting: - Future Expected Return Volatility, i.e., a Volatility Forecast (Jiang & Tian) - Market Pricing of Volatility Risk (Investor Fear Gauge ) We Argue/Demonstrate, - VIX Truncated, not pure MFIV, like Corridor Implied Volatility (CIV) - MFIV Not Best IV Forecasts for (P-measure) Return Volatility - CIV Measures Allow Refined Info Extraction from RND - E.g., Up-Variance and Down-Variance Pricing Identified Separately

3 Model-Free Implied Volatility (VIX) Conceptually, MFIV Prices the Expected Return Variation Given an Arbitrage-Free Setting (Special Semi-Martingale) w/ df t = : t dt + F t dw t (+Jumps of Finite Activity) (+ cumulative squared jumps) (+ cum sq d jumps) (+ cum sq d jumps)

4 Barrier & Corridor Variance Contracts - Notation Current Time is t, with # t # T < T. Assume zero risk-free rate. F t : Value at time t of S&P5 Futures Contract Expiring at T. European Options with Strike K and Expiration Date T: ; Other No-Arbitrage Cond ns [Ross (1976, 1978), Breeden & Litzenberger (1978), Banz & Miller (1978)] Practical Implement n, Bondarenko (23), Positive Convolution Approximation

5 Barrier & Corridor Contracts - General Payoffs For any x $ and arbitrary Payoff function g(f) w/ finite 2 nd Derivative, Payoff-Spanning via Position in Options [Carr & Madan (1998)] Let x = F, take Expectation, and note F t is a Q-martingale, so (1) where M t (K) is Value of OTM Put or Call Option w/ Strike K at time t.

6 Barrier & Corridor Contracts - General Payoffs Futures Price: df t / F t = F t dw t (Q-Martingale) Use Ito s Lemma for g(f t ) and take expectation, (2) From (2) and (1),

7 Barrier Variance Contracts - Definition Introduce Down-Barrier Indicator Function, I t = I t (B) = 1[ F t # B] Consider Contract w/ Payoff equal to Barrier Integrated Variance, This Payoff accumulates Variance only when the underlying Futures Price Lies below the pre-specified Barrier. Note,

8 Barrier Volatility Contracts - Pricing From above, Choose It follows, Barrier Implied Volatility and lim B64 BIV (B) = MFIV 64

9 Corridor Variance Contracts - Pricing Introduce Corridor Indicator Function, I t = I t (B 1, B 2 ) = 1[ B 1 # F t # B 2 ] Consider the Contract w/ Payoff equal to the Corridor Integrated Variance, Pricing, Corridor Implied Volatility ; Plot

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11 Volatility Risk Premium Discrete )-period (Intraday) Returns r t,) / p(t) - p(t- ) Realized Volatility/Variation Converges Uniformly in Probability (5 min plus Overnight or Daily) (Model-Free Measure of Realization) Andersen and Bollerslev (1998); ABDL (21, 23); BNS (22)

12 Data Sources 17 year Sample Period: January December 26 CME: Futures on S&P 5 and S&P 5 Futures Option Prices CBOE: New VIX (on S&P 5 Cash Index) Federal Reserve: Treasury Rates (Proxy for Risk-free rate) CME Options and Futures more suitable than CBOE options and S&P 5 1) 15 minute lag b/w Close of CBOE and NYSE, NASDAQ, AMEX 2) Futures Contracts much easier for hedging 3) No need to deal with Dividend Stream 4) Only Disadvantage is American feature, but irrelevant for OTM Every Trading Day, Obtain Nine Volatility Measures (use Bondarenko (,3)): CIV1, CIV2, CIV3, CIV4, BSIV, MFIV, VIX, RVD, RVH (5min) CIV1-CIV4: CIV (B 1, B 2 ) w/ B 1 = H -1 (p) and B 2 = H -1 (1-p), Fractiles of RND, p =.25,.1,.5,.25 so [.25,.75] through [.25,.975]

13 Implied Volatility Risk Neutral Density Moneyness k Moneyness k.3 Normalized Option Price 1 Cumulative Risk Neutral Density Function Moneyness k Moneyness k S&P 5 option data for 4/19/2 when τ = 21 trading days.

14 Level of S&P Daily Return on S&P VIX and Realized Volatility

15 Descriptive Statistics 1) VIX, MFIV Higher Mean than RVH 6 Large (Neg) Vol Risk Premium 2) VIX, MFIV, CIV4 very Highly Correlated, but not Identical 3) Moving CIV4 6 CIV1 yields Lower, more Stable Measures 4) Even CIV2 much Larger than RVH, RVD 5) ATM BSIV extremely Highly Correlated w/ CIV2, but not Identical 6) RV Measures most Erratic - Realizations vs. Expectations 7) All Vol Measures, RV and IV, have Long Memory like Persistence Obvious all Implied Vol Measures are Biased as Forecasts for RV But still Type of Scaled Volatility (log Vol, Variance) Forecast??

16 Descriptive statistics Panel A: Full sample 1/199-12/26 RVD RVH VIX BSIV CIV1 CIV2 CIV3 CIV4 MFIV Mean StDev Skewness Kurtosis ρ ρ ρ

17 Panel B: Subsample 1/199-12/1999 RVD RVH VIX BSIV CIV1 CIV2 CIV3 CIV4 MFIV Mean StDev Skewness Kurtosis ρ ρ ρ Panel C: Subsample 1/2-12/26 RVD RVH VIX BSIV CIV1 CIV2 CIV3 CIV4 MFIV Mean StDev Skewness Kurtosis ρ ρ ρ

18 RVD RVH Time-Series Correlations VIX BSIV CIV1 CIV2 CIV3 CIV4 MFIV RVD RVH VIX BSIV CIV CIV CIV CIV MFIV

19 Corridor Volatility scaled by MFIV Corridor Volatility scaled by BSIV Top panel: corridor variances CIV1-CIV4 scaled by MFIV. Bottom panel: corridor variances CIV1-CIV4 and MFIV scaled by BSIV.

20 Forecast Performance Evaluation for IV Measures In-Sample Predictive Regressions, for One-Month RV Predictors, x j, j = 1,...,J RV t+1 = " j + $ j x j,t + u j,t, j=1,...,j=9 and Encompassing Regressions, RV t+1 = " jk + $ j x j,t + $ k x k,t + u jk,t, j k Out-of-Sample RMSE w/ Regression Coeff ts Fixed at In-Sample Point Estimates. Out-of-Sample RMSE over Subsamples sorted in Ascending Vol Levels Switch In-Sample and Out-of-Sample Classification, again w/ Vol Subsamples Analysis for Volatility, Log-Volatility and Variance

21 Volatility Regressions: Estimation = 1/9 12/99, Forecast = 1/ 12/6 In-Sample Estimation Out-of-Sample RMSE α β 1 β 2 R 2 All days Low Medium High RVD ( 5.49) ( 8.15) RVH ( 3.65) ( 8.92) VIX (.8) ( 9.9) BSIV (.36) ( 1.79) CIV (.22) ( 11.1) CIV (.55) ( 1.79) CIV (.71) ( 1.46) CIV (.7) ( 1.34) MFIV (.45) ( 1.31) RVD + RVH ( 3.4) ( -.39) ( 3.25) RVH + VIX (.43) ( 2.19) ( 6.46) RVH + CIV (.6) ( 2.4) ( 8.26) VIX + CIV (.5) ( 1.1) ( 1.93)

22 Summary of Forecast Analysis Lagged One-Month RV Measures (Historical Vol) have Predictive Value, but they are - as expected - Worst Forecasts RVH Vastly Dominates RVD. No contest in terms of Use as RV Realization VIX is Worst IV Forecast, Followed closely by MFIV and CIV4, CIV3. CIV1, BSIV and CIV2 are Best IV Forecasts Results are Extremely Robust over Subsamples, Vol Sorting, Vol Definition Encompassing Regressions find Added Value for RVH r.t. CIV1, not for VIX VIX (MFIV) Forecast Info Subsumed by CIV1, while RVH not Better Use of Past Daily RV Measures may be Competitive w/ IV Measures No Direct Implication for Value of MFIV - many Uses as Risk Price Gauge

23 Log-Volatility Regressions: Estimation = 1/9 12/99, Forecast = 1/ 12/6 In-Sample Estimation Out-of-Sample RMSE α β 1 β 2 R 2 All days Low Medium High RVD ( -6.77) ( 13.97) RVH ( -4.9) ( 15.62) VIX ( -2.79) ( 16.39) BSIV ( -1.71) ( 17.71) CIV (.3) ( 17.93) CIV ( -1.79) ( 17.73) CIV ( -2.48) ( 17.39) CIV ( -2.75) ( 17.14) MFIV ( -2.76) ( 17.1) RVD + RVH ( -4.2) ( -.21) ( 6.14) RVH + VIX ( -2.59) ( 4.18) ( 7.18) RVH + CIV ( -.21) ( 3.42) ( 9.67) VIX + CIV (.3) (.49) ( 4.12)

24 Variance Regressions: Estimation = 1/9 12/99, Forecast = 1/ 12/6 In-Sample Estimation Out-of-Sample RMSE α β 1 β 2 R 2 All days Low Medium High RVD ( 5.12) ( 5.14) RVH ( 3.7) ( 5.31) VIX (.8) ( 5.66) BSIV (.38) ( 6.25) CIV (.27) ( 6.35) CIV (.57) ( 6.28) CIV (.69) ( 6.8) CIV (.69) ( 6.6) MFIV (.5) ( 6.1) RVD + RVH ( 3.26) ( -.4) ( 1.64) RVH + VIX (.32) ( 1.21) ( 4.44) RVH + CIV (.69) ( 1.8) ( 5.41) VIX + CIV (.6) ( 1.5) (.39)

25 Volatility Regressions: Estimation = 1/ 12/6, Forecast = 1/9 12/99 In-Sample Estimation Out-of-Sample RMSE α β 1 β 2 R 2 All days Low Medium High RVD ( 5.46) ( 13.85) RVH ( 3.88) ( 11.99) VIX ( -1.62) ( 15.7) BSIV ( -.91) ( 15.19) CIV ( -.66) ( 15.7) CIV ( -.74) ( 15.16) CIV ( -.71) ( 15.18) CIV ( -.79) ( 15.14) MFIV ( -1.28) ( 14.97) RVD + RVH ( 3.92) (.36) ( 2.57) RVH + VIX ( -1.27) (.88) ( 5.9) RVH + CIV ( -.45) (.81) ( 5.34) VIX + CIV ( -.88) (.47) ( 1.37)

26 Relevance of (Up- and Down-) Variance Contracts Huge Literature seeking to understand OTM Equity-Index Put Option Prices Pronounced Skew in BSIV for Equity Options (Q) Return-Volatility Asymmetry (P) f/ Leverage Effect or Volatility Feedback Generally, Risk naturally associated w/ Down-States (Semi-Variance) Large Negative Volatility Risk Premium in MFIV - Differ for Down vs. Up? No Direct Quantitative Measure - BSIV Self-Contradictory - MFIV provides only One overall Measure - Model based Estimates are, well, Model-Dependent

27 Down-Variance and Up-Variance Contracts Corridors determined by Beginning-of-Month Futures Price (F ) as Down: [, F ] and Up: [ F, 4 [ Induces simple Decomposition of Monthly Realized Return Variation and MFIV, RV t = RVD t + RVU t MFV t-1 = MFVD t-1 + MFVU t-1 Monthly Returns on Variance, Down-Variance and Up-Variance Contracts

28 Realized and Model-Free Variances Panel A: Summary Statistics Prop. of Realized Variance Model-Free Variance returns Mean StDev Skew. Kurt. Mean StDev Skew. Kurt. Down Up Total Panel B: Correlations RVD RVU RV MFVD MFVU MFV RVD RVU RV MFVD MFVU MFV

29 Monthly Returns for S&P 5, Down Variance, Up Variance, Total Variance, and Mean-Variance Portfolio Panel A: Return Distribution Min. 1% 5% 1% Med. 9% 95% 99% Max. r m r vd r vu r v r mv Panel B: Risk Characteristics Mean SD Skew. Kurt. α β SR T M M 2 r m r vd r vu r v r mv

30 OLS Regressions for variance returns r vd, r vu, and r v Const SP5 SMB HML UMD ATMP OTMP ATMC OTMC R (-1.63) (-13.51) (-1.8) (-14.1) ( -5.43) ( -2.96) r vd ( -9.48) (-14.12) ( -5.34) ( -3.16) ( -1.74) ( -6.97) (.11) ( 4.28) ( 2.33) ( -2.27) (.94) ( -6.25) ( -.91) ( -4.3) ( -2.58) ( -1.32) ( 3.32) ( 2.6) ( -2.5) (.75) ( -2.77) ( 1.67) ( -2.79) ( 9.57) ( 1.26) (.17) r vu ( -2.72) ( 9.21) ( 1.25) (.18) (.4) ( -2.25) (.73) ( -3.3) ( 1.72) ( 2.19) ( -1.6) ( -1.77) (.29) (.69) ( -.63) ( -.88) ( -3.37) ( 1.82) ( 2.33) ( -.79) (-12.42) ( -8.8) (-11.8) ( -8.72) ( -4.6) ( -2.75) r v (-11.15) ( -8.93) ( -4.5) ( -2.97) ( -1.81) ( -8.27) (.41) ( 2.3) ( 3.22) ( -.93) (.37) ( -7.29) ( -.84) ( -3.69) ( -2.81) ( -1.93) ( 1.21) ( 3.58) ( -.55) (.33)

31 .5 Model Free Volatility: Down, Up, and Total Normalized Model Free Variance: Down and Up Top panel: the 1-month option-implied volatility (down, up, and total). Bottom panel: the model-free down and up variances scaled by the model-free total variance.

32 Realized and Model Free Down Volatility Realized and Model Free Up Volatility Realized and Model Free Total Volatility

33 3 2 1 Aug 199 Down Variance Return Jul 1996 Apr 2 Sep 21 Jul 22 Jun Up Variance Return Feb 1991 Feb 1996 Mar Total Variance Return Aug 199 Sep 21 Jul

34 r vd versus r m r vd versus r v r vu versus r m r vu versus r v

35 Realized and Model-Free Variances V1-V4 Panel A: Summary Statistics Prop. of Realized Variance Model-Free Variance returns Mean StDev Skew. Kurt. Mean StDev Skew. Kurt. V V V V Panel B: Correlations RV1 RV2 RV3 RV4 MFV1 MFV2 MFV3 MFV4 RV RV RV RV MFV MFV MFV MFV

36 Future Inquiry and Research Only Scratching Surface in terms of Use of CIV Measures Studied Centered CIV and Specific Location of Risk Premiums (Up vs. Down) Can Check for Constancy of Relative Premiums Investigate how Premiums Respond across Range of RND conditional on Events See What Pieces are Correlated w/ other Macro Risk Measures (Spreads) How are Jumps Related to the Risk Premiums More on How Return-Volatility Asymmetries Appear Explore other Markets, Maturities

37 1-Month RV Forecast for S&P 5: Forecast period = 1/ /26 Log-volatility Volatility Variance R 2 RMSE MAE R 2 RMSE MAE R 2 RMSE MAE RV RVH CCAR FI VIX BSIV CMF CMF CMF MFIV CCAR+BSIV CCAR+CMF CCAR+MFIV

38 1-Month RV Forecast for T-bond: Forecast period = 1/21 12/26 Log-volatility Volatility Variance R 2 RMSE MAE R 2 RMSE MAE R 2 RMSE MAE RV RVH CCAR FI BSIV CMF CMF CMF MFIV CCAR+BSIV CCAR+CMF CCAR+MFIV

39 S&P 5: 1 month RV and MFIV Tbond: 1 month RV and MFIV Sample averages for the square root of 1-month RV and MFIV: RV MFIV S&P Tbond.91.98