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 SchoolDecember 8, 2007

Introduction

Growing Interest in Equity-Market Volatility Index, VIX, published by the CBOE

Uses Notion of Model-Free Implied Volatility (MFIV) for S&P500 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

Model-Free Implied Volatility (VIX)__________________________________________________________

Conceptually, MFIV Prices the Expected Return Variation

Given an Arbitrage-Free Setting (Special Semi-Martingale) w/

dFt = :t dt + Ft dWt (+Jumps of Finite Activity)

(+ cumulative squared jumps)

(+ cum sq’d jumps)

(+ cum sq’d jumps)

Barrier & Corridor Variance Contracts - Notation_________________________________________________________________

Current Time is t, with 0 #### t #### T < T’. Assume zero risk-free rate.

Ft : Value at time t of S&P500 Futures Contract Expiring at T’.

European Options with Strike K and Expiration Date T:

; Other No-Arbitrage Condns

[Ross (1976, 1978), Breeden & Litzenberger (1978), Banz & Miller (1978)]

Practical Implementn, Bondarenko (2003), Positive Convolution Approximation

Barrier & Corridor Contracts - General Payoffs________________________________________________________________

For any x $$$$ 0 and arbitrary Payoff function g(F) w/ finite 2nd Derivative,

Payoff-Spanning via Position in Options [Carr & Madan (1998)]

Let x = F0 , take Expectation, and note Ft is a Q-martingale, so

(1)

where Mt(K) is Value of OTM Put or Call Option w/ Strike K at time t.

Barrier & Corridor Contracts - General Payoffs________________________________________________________________

Futures Price: dFt / Ft = F t dWt (Q-Martingale)

Use Ito’s Lemma for g(Ft ) and take expectation,

(2)

From (2) and (1),

Barrier Variance Contracts - Definition_________________________________________________________________

Introduce Down-Barrier Indicator Function, It = It (B) = 1[ Ft #### 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,

Barrier Volatility Contracts - Pricing_________________________________________________________________

From above,

Choose

It follows,

Barrier Implied Volatility

and limB64646464 BIV0 (B) = MFIV0

Corridor Variance Contracts - Pricing_________________________________________________________________

Introduce Corridor Indicator Function, It = It (B1 ,B2 ) = 1[ B1 #### Ft #### B2 ]

Consider the Contract w/ Payoff equal to the Corridor Integrated Variance,

Pricing,

Corridor Implied Volatility

; Plot

Volatility Risk Premium__________________________________________________________

Discrete ))))-period (Intraday) Returns

rt,) / 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 (2001, 2003); BNS (2002)

Data Sources

17 year Sample Period: January 1990 - December 2006

CME: Futures on S&P 500 and S&P 500 Futures Option Prices

CBOE: New VIX (on S&P 500 Cash Index)

Federal Reserve: Treasury Rates (Proxy for Risk-free rate)

CME Options and Futures more suitable than CBOE options and S&P 500

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 (00,03)):

CIV1, CIV2, CIV3, CIV4, BSIV, MFIV, VIX, RVD, RVH (5min)

CIV1-CIV4:

CIV0 (B1 , B2 ) w/ B1 = H0-1(p) and B2 = H0

-1(1-p), Fractiles of RND,

p = 0.25, 0.10, 0.05, 0.025 so [0.25, 0.75] through [0.025, 0.975]

0.7 0.8 0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

Moneyness k

Implied Volatility

0.7 0.8 0.9 1 1.1 1.20

0.5

1

1.5

2

Moneyness k

Risk Neutral Density

0.7 0.8 0.9 1 1.1 1.20

0.005

0.01

0.015

0.02

0.025

0.03

Moneyness k

Normalized Option Price

0.7 0.8 0.9 1 1.1 1.20

0.2

0.4

0.6

0.8

1

Moneyness k

Cumulative Risk−Neutral Density Function

S&P 500 option data for 04/19/2000 when τ = 21 trading days.

1990 1992 1994 1996 1998 2000 2002 2004 2006

400

800

12001600

Level of S&P 500

1990 1992 1994 1996 1998 2000 2002 2004 2006

−0.05

0

0.05

Daily Return on S&P 500

1990 1992 1994 1996 1998 2000 2002 2004 20060

0.2

0.4

VIX and Realized Volatility

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 ??

Descriptive statistics

Panel A: Full sample 01/1990-12/2006

RVD RVH VIX BSIV CIV1 CIV2 CIV3 CIV4 MFIV

Mean 0.149 0.150 0.191 0.165 0.134 0.163 0.172 0.178 0.185StDev 0.071 0.066 0.064 0.058 0.048 0.058 0.062 0.063 0.064

Skewness 1.500 1.428 0.983 1.017 1.009 1.042 1.064 1.069 1.080Kurtosis 5.913 5.254 3.795 3.955 3.958 4.046 4.096 4.116 4.137

ρ1 0.991 0.997 0.983 0.981 0.981 0.980 0.980 0.980 0.980ρ21 0.650 0.758 0.827 0.830 0.837 0.827 0.823 0.821 0.822ρ63 0.497 0.519 0.652 0.678 0.694 0.670 0.660 0.654 0.652

Panel B: Subsample 01/1990-12/1999

RVD RVH VIX BSIV CIV1 CIV2 CIV3 CIV4 MFIV

Mean 0.138 0.139 0.185 0.157 0.126 0.155 0.164 0.170 0.177StDev 0.063 0.057 0.059 0.051 0.041 0.052 0.055 0.057 0.058

Skewness 1.878 1.629 1.134 1.117 1.025 1.171 1.230 1.258 1.284Kurtosis 8.983 7.132 4.526 4.830 4.590 5.048 5.182 5.277 5.327

ρ1 0.988 0.996 0.979 0.974 0.974 0.973 0.974 0.975 0.975ρ21 0.586 0.736 0.806 0.799 0.805 0.796 0.792 0.790 0.794ρ63 0.413 0.467 0.635 0.651 0.667 0.643 0.633 0.626 0.627

Panel C: Subsample 01/2000-12/2006

RVD RVH VIX BSIV CIV1 CIV2 CIV3 CIV4 MFIV

Mean 0.163 0.164 0.199 0.177 0.145 0.175 0.183 0.189 0.196StDev 0.079 0.075 0.070 0.065 0.054 0.065 0.068 0.069 0.070

Skewness 1.091 1.107 0.760 0.775 0.774 0.793 0.804 0.799 0.803Kurtosis 3.824 3.698 3.063 3.033 3.029 3.080 3.114 3.103 3.116

ρ1 0.993 0.997 0.986 0.985 0.986 0.985 0.984 0.984 0.984ρ21 0.696 0.761 0.844 0.847 0.850 0.846 0.844 0.844 0.843ρ63 0.550 0.527 0.658 0.680 0.691 0.675 0.667 0.664 0.658

Time-Series Correlations

RVD RVH VIX BSIV CIV1 CIV2 CIV3 CIV4 MFIV

RVD 1.000 0.955 0.855 0.857 0.854 0.857 0.858 0.859 0.859RVH 0.955 1.000 0.898 0.899 0.896 0.899 0.899 0.899 0.899VIX 0.855 0.898 1.000 0.988 0.981 0.990 0.992 0.993 0.993

BSIV 0.857 0.899 0.988 1.000 0.998 1.000 0.998 0.997 0.995CIV1 0.854 0.896 0.981 0.998 1.000 0.997 0.994 0.991 0.988CIV2 0.857 0.899 0.990 1.000 0.997 1.000 0.999 0.998 0.996CIV3 0.858 0.899 0.992 0.998 0.994 0.999 1.000 1.000 0.998CIV4 0.859 0.899 0.993 0.997 0.991 0.998 1.000 1.000 0.999

MFIV 0.859 0.899 0.993 0.995 0.988 0.996 0.998 0.999 1.000

1990 1992 1994 1996 1998 2000 2002 2004 2006

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Corridor Volatility scaled by MFIV

1990 1992 1994 1996 1998 2000 2002 2004 20060.7

0.8

0.9

1

1.1

1.2

Corridor Volatility scaled by BSIV

Top panel: corridor variances CIV1-CIV4 scaled by MFIV. Bottom panel: corridor variances CIV1-CIV4and MFIV scaled by BSIV.

Forecast Performance Evaluation for IV Measures

In-Sample Predictive Regressions, for One-Month RV Predictors, xj , j = 1, ...,J

RVt+1 = """"j + $$$$j xj,t + uj,t , j=1, ... ,J=9

and Encompassing Regressions,

RVt+1 = """"jk + $$$$j xj,t + $$$$k xk,t + ujk,t , j ………… k

Out-of-Sample RMSE w/ Regression Coeffts 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

Volatility Regressions: Estimation = 01/90–12/99, Forecast = 01/00–12/06

In-Sample Estimation Out-of-Sample RMSE

α β1 β2 R2 All days Low Medium High

RVD 0.05 0.62 46.75 31.91 16.47 26.81 44.15

( 5.49) ( 8.15)

RVH 0.04 0.74 54.05 30.23 13.91 25.62 42.12( 3.65) ( 8.92)

VIX 0.00 0.75 60.25 27.98 10.52 26.45 38.08

( 0.08) ( 9.90)

BSIV 0.00 0.87 60.83 26.53 10.74 25.70 35.62

( 0.36) ( 10.79)

CIV1 0.00 1.09 60.79 26.11 10.78 25.39 34.97( 0.22) ( 11.01)

CIV2 0.01 0.86 60.88 26.76 10.73 25.90 35.95

( 0.55) ( 10.79)

CIV3 0.01 0.80 60.59 27.15 10.72 26.21 36.56

( 0.71) ( 10.46)

CIV4 0.01 0.78 60.28 27.39 10.72 26.41 36.91

( 0.70) ( 10.34)

MFIV 0.01 0.76 60.34 27.58 10.83 26.58 37.18

( 0.45) ( 10.31)

RVD + RVH 0.04 -0.08 0.82 54.13 30.28 13.88 25.72 42.17

( 3.40) ( -0.39) ( 3.25)

RVH + VIX 0.01 0.19 0.59 60.91 27.42 10.53 25.70 37.40

( 0.43) ( 2.19) ( 6.46)

RVH + CIV1 0.01 0.20 0.84 61.62 26.08 10.81 24.87 35.19( 0.60) ( 2.04) ( 8.26)

VIX + CIV1 0.00 0.30 0.67 61.15 26.47 10.51 25.54 35.64

( 0.05) ( 1.01) ( 1.93)

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

Log-Volatility Regressions: Estimation = 01/90–12/99, Forecast = 01/00–12/06

In-Sample Estimation Out-of-Sample RMSE

α β1 β2 R2 All days Low Medium High

RVD -0.69 0.65 53.34 12.70 9.39 11.59 16.08

( -6.77) ( 13.97)

RVH -0.44 0.78 61.48 11.60 8.19 10.71 14.84( -4.09) ( 15.62)

VIX -0.31 1.00 65.25 10.93 6.95 10.86 13.80

( -2.79) ( 16.39)

BSIV -0.18 0.98 66.79 10.34 6.82 10.56 12.66

( -1.71) ( 17.71)

CIV1 0.04 0.98 67.21 10.17 6.79 10.44 12.37( 0.30) ( 17.93)

CIV2 -0.19 0.97 66.80 10.42 6.84 10.65 12.79

( -1.79) ( 17.73)

CIV3 -0.26 0.96 66.29 10.58 6.87 10.78 13.05

( -2.48) ( 17.39)

CIV4 -0.29 0.96 65.81 10.68 6.91 10.88 13.20

( -2.75) ( 17.14)

MFIV -0.29 0.98 65.64 10.76 6.90 11.00 13.30

( -2.76) ( 17.10)

RVD + RVH -0.44 -0.02 0.81 61.48 11.61 8.19 10.72 14.85

( -4.02) ( -0.21) ( 6.14)

RVH + VIX -0.28 0.29 0.67 66.78 10.52 6.86 10.37 13.25

( -2.59) ( 4.18) ( 7.18)

RVH + CIV1 -0.02 0.23 0.73 68.13 10.07 6.79 10.17 12.35( -0.21) ( 3.42) ( 9.67)

VIX + CIV1 0.00 0.12 0.86 67.23 10.19 6.75 10.43 12.44

( 0.03) ( 0.49) ( 4.12)

Variance Regressions: Estimation = 01/90–12/99, Forecast = 01/00–12/06

In-Sample Estimation Out-of-Sample RMSE

α β1 β2 R2 All days Low Medium High

RVD 0.01 0.51 37.51 68.27 24.04 52.52 97.04

( 5.12) ( 5.14)

RVH 0.01 0.66 43.07 65.98 19.17 51.81 93.88( 3.70) ( 5.31)

VIX 0.00 0.60 50.82 59.21 11.27 53.28 82.44

( 0.08) ( 5.66)

BSIV 0.00 0.80 50.18 56.95 12.00 52.14 78.89

( 0.38) ( 6.25)

CIV1 0.00 1.26 49.18 56.36 12.10 51.61 78.05( 0.27) ( 6.35)

CIV2 0.00 0.80 50.45 57.39 12.07 52.50 79.52

( 0.57) ( 6.28)

CIV3 0.00 0.70 50.74 58.06 12.07 52.99 80.50

( 0.69) ( 6.08)

CIV4 0.00 0.65 50.81 58.42 12.03 53.25 81.04

( 0.69) ( 6.06)

MFIV 0.00 0.62 51.43 58.66 12.20 53.39 81.39

( 0.50) ( 6.10)

RVD + RVH 0.01 -0.01 0.67 43.05 65.99 19.13 51.84 93.89

( 3.26) ( -0.04) ( 1.64)

RVH + VIX 0.00 0.13 0.50 51.20 58.92 11.42 52.51 82.21

( 0.32) ( 1.21) ( 4.44)

RVH + CIV1 0.00 0.23 0.92 50.63 57.08 12.35 50.80 79.57( 0.69) ( 1.80) ( 5.41)

VIX + CIV1 0.00 0.50 0.21 50.87 58.33 11.29 52.82 81.08

( 0.06) ( 1.50) ( 0.39)

Volatility Regressions: Estimation = 01/00–12/06, Forecast = 01/90–12/99

In-Sample Estimation Out-of-Sample RMSE

α β1 β2 R2 All days Low Medium High

RVD 0.05 0.71 54.70 26.00 30.04 21.83 19.83

( 5.46) ( 13.85)

RVH 0.04 0.77 57.94 23.86 27.99 19.23 18.24( 3.88) ( 11.99)

VIX -0.01 0.89 67.96 23.46 27.05 19.73 17.98

( -1.62) ( 15.07)

BSIV -0.01 0.96 68.39 22.16 26.26 18.07 16.18

( -0.91) ( 15.19)

CIV1 -0.01 1.16 68.52 21.93 26.21 17.61 15.87( -0.66) ( 15.07)

CIV2 -0.01 0.97 68.15 22.24 26.33 18.20 16.24

( -0.74) ( 15.16)

CIV3 -0.01 0.92 67.93 22.57 26.66 18.56 16.47

( -0.71) ( 15.18)

CIV4 -0.01 0.90 67.74 22.79 26.82 18.86 16.72

( -0.79) ( 15.14)

MFIV -0.01 0.89 67.42 22.82 26.79 18.92 16.85

( -1.28) ( 14.97)

RVD + RVH 0.04 0.09 0.68 57.98 23.94 28.09 19.37 18.23

( 3.92) ( 0.36) ( 2.57)

RVH + VIX -0.01 0.11 0.79 68.18 23.05 26.78 19.30 17.35

( -1.27) ( 0.88) ( 5.90)

RVH + CIV1 -0.00 0.11 1.03 68.74 21.74 26.01 17.52 15.62( -0.45) ( 0.81) ( 5.34)

VIX + CIV1 -0.01 0.23 0.87 68.58 22.00 26.10 17.86 16.08

( -0.88) ( 0.47) ( 1.37)

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

Down-Variance and Up-Variance Contracts

Corridors determined by Beginning-of-Month Futures Price (F0) as

Down: [ 0, F0 ] and Up: [ F0 , 4444 [

Induces simple Decomposition of Monthly Realized Return Variation and MFIV,

RVt = RVDt + RVUt

MFVt-1 = MFVDt-1 + MFVUt-1

Monthly Returns on Variance, Down-Variance and Up-Variance Contracts

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 0.377 0.014 0.024 3.711 21.326 0.026 0.021 2.975 18.052Up 0.623 0.013 0.016 4.505 36.158 0.013 0.009 2.151 10.506

Total 1.000 0.027 0.028 2.857 13.377 0.039 0.030 2.685 15.408

Panel B: Correlations

RVD RVU RV MFVD MFVU MFV

RVD 1.000 -0.095 0.821 0.343 0.374 0.356RVU -0.095 1.000 0.491 0.737 0.743 0.746

RV 0.821 0.491 1.000 0.724 0.754 0.740MFVD 0.343 0.737 0.724 1.000 0.959 0.996MFVU 0.374 0.743 0.754 0.959 1.000 0.981

MFV 0.356 0.746 0.740 0.996 0.981 1.000

Monthly Returns for S&P 500, Down Variance, Up Variance, Total

Variance, and Mean-Variance Portfolio

Panel A: Return Distribution

Min. 1% 5% 10% Med. 90% 95% 99% Max.

rm -15.05 -10.52 -6.67 -4.53 0.90 5.63 6.57 8.96 10.75

rvd -99.99 -99.86 -99.36 -98.64 -67.25 35.89 99.49 260.30 349.43

rvu -100.00 -99.94 -97.90 -93.06 -2.65 67.90 110.51 174.06 306.26

rv -82.61 -71.90 -65.75 -63.00 -42.10 1.57 43.66 136.13 200.23

rmv -38.33 -24.63 -7.17 -0.00 8.59 13.23 13.76 16.14 17.06

Panel B: Risk Characteristics

Mean SD Skew. Kurt. α β SR TM M2

rm 0.55 4.02 -0.51 3.92 0.00 1.00 0.14 0.00 0.55

rvd -45.55 71.23 2.52 11.14 -38.88 -12.21 -0.64 3.18 -2.57

rvu -4.95 65.68 0.77 4.75 -10.31 9.81 -0.08 -1.05 -0.30

rv -31.97 38.42 2.82 13.58 -29.40 -4.72 -0.83 6.22 -3.34

rmv 6.71 7.58 -2.64 12.62 6.55 0.29 0.89 22.63 3.56

OLS Regressions for variance returns rvd, rvu, and rv

Const SP500 SMB HML UMD ATMP OTMP ATMC OTMC R2

-38.88 -12.21 0.47(-10.63) (-13.51)

-35.65 -13.40 -5.77 -3.77 0.54(-10.08) (-14.10) ( -5.43) ( -2.96)

rvd -34.24 -13.85 -5.66 -4.04 -1.30 0.54( -9.48) (-14.12) ( -5.34) ( -3.16) ( -1.74)

-27.38 0.31 0.31 0.11 -0.15 0.02 0.58( -6.97) ( 0.11) ( 4.28) ( 2.33) ( -2.27) ( 0.94)

-24.81 -2.72 -4.30 -3.07 -0.95 0.24 0.12 -0.13 0.02 0.62( -6.25) ( -0.91) ( -4.30) ( -2.58) ( -1.32) ( 3.32) ( 2.60) ( -2.05) ( 0.75)

-10.31 9.81 0.36( -2.77) ( 10.67)

-10.71 9.87 1.46 0.24 0.36( -2.79) ( 9.57) ( 1.26) ( 0.17)

rvu -10.74 9.88 1.45 0.25 0.03 0.35( -2.72) ( 9.21) ( 1.25) ( 0.18) ( 0.04)

-9.76 2.34 -0.26 0.09 0.16 -0.03 0.40( -2.25) ( 0.73) ( -3.30) ( 1.72) ( 2.19) ( -1.06)

-8.13 1.00 0.79 -0.86 -0.73 -0.28 0.10 0.17 -0.02 0.40( -1.77) ( 0.29) ( 0.69) ( -0.63) ( -0.88) ( -3.37) ( 1.82) ( 2.33) ( -0.79)

-29.40 -4.72 0.24(-12.42) ( -8.08)

-27.47 -5.46 -3.22 -2.31 0.31(-11.80) ( -8.72) ( -4.60) ( -2.75)

rv -26.50 -5.77 -3.14 -2.50 -0.89 0.32(-11.15) ( -8.93) ( -4.50) ( -2.97) ( -1.81)

-21.69 0.78 0.11 0.10 -0.04 0.01 0.36( -8.27) ( 0.41) ( 2.30) ( 3.22) ( -0.93) ( 0.37)

-19.40 -1.69 -2.47 -2.25 -0.93 0.06 0.11 -0.02 0.01 0.41( -7.29) ( -0.84) ( -3.69) ( -2.81) ( -1.93) ( 1.21) ( 3.58) ( -0.55) ( 0.33)

1990 1992 1994 1996 1998 2000 2002 2004 20060

0.1

0.2

0.3

0.4

0.5Model−Free Volatility: Down, Up, and Total

1990 1992 1994 1996 1998 2000 2002 2004 20060

0.2

0.4

0.6

0.8

1Normalized Model−Free Variance: Down and Up

Top panel: the 1-month option-implied volatility (down, up, and total). Bottom panel: the model-freedown and up variances scaled by the model-free total variance.

1990 1992 1994 1996 1998 2000 2002 2004 20060

0.2

0.4

Realized and Model−Free Down−Volatility

1990 1992 1994 1996 1998 2000 2002 2004 20060

0.2

0.4

Realized and Model−Free Up−Volatility

1990 1992 1994 1996 1998 2000 2002 2004 20060

0.2

0.4

Realized and Model−Free Total−Volatility

1990 1992 1994 1996 1998 2000 2002 2004 2006−100

0

100

200

300

Down Variance Return

← Aug−1990← Jul−1996

← Apr−2000

← Sep−2001

← Jun−2002

← Jul−2002

1990 1992 1994 1996 1998 2000 2002 2004 2006−100

0

100

200

300

Up Variance Return

← Feb−1991

← Feb−1996

← Mar−2000

1990 1992 1994 1996 1998 2000 2002 2004 2006−100

0

100

200

300

Total Variance Return

← Aug−1990

← Sep−2001

← Jul−2002

−15 −10 −5 0 5 10

−100

0

100

200

300

rvd

versus rm

−50 0 50 100 150 200

−100

0

100

200

300

rvd

versus rv

−15 −10 −5 0 5 10

−100

0

100

200

300

rvu

versus rm

−50 0 50 100 150 200

−100

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100

200

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rvu

versus rv

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.

V1 0.062 0.004 0.014 5.903 44.072 0.011 0.010 4.316 34.278

V2 0.315 0.010 0.014 2.380 9.440 0.014 0.012 2.214 9.819

V3 0.340 0.007 0.009 2.579 11.841 0.008 0.006 2.009 8.309

V4 0.283 0.005 0.010 5.191 42.950 0.005 0.004 2.794 16.138

Panel B: Correlations

RV1 RV2 RV3 RV4 MFV1 MFV2 MFV3 MFV4

RV1 1.000 0.456 -0.144 -0.144 0.088 0.094 0.102 0.124

RV2 0.456 1.000 0.171 -0.149 0.396 0.534 0.536 0.437

RV3 -0.144 0.171 1.000 0.468 0.616 0.749 0.754 0.587

RV4 -0.144 -0.149 0.468 1.000 0.586 0.489 0.488 0.608

MFV1 0.088 0.396 0.616 0.586 1.000 0.846 0.817 0.848

MFV2 0.094 0.534 0.749 0.489 0.846 1.000 0.996 0.806

MFV3 0.102 0.536 0.754 0.488 0.817 0.996 1.000 0.816

MFV4 0.124 0.437 0.587 0.608 0.848 0.806 0.816 1.000

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

1-Month RV Forecast for S&P 500: Forecast period = 01/1995–12/2006

Log-volatility Volatility Variance

R2 RMSE MAE R2 RMSE MAE R2 RMSE MAE

RV1 49.19 100.00 100.00 52.48 100.00 100.00 48.91 100.00 100.00

RVH 65.44 81.82 79.71 54.52 95.84 94.72 36.96 107.54 112.89

CCAR 70.79 76.04 73.98 63.66 86.14 83.85 51.47 94.48 92.82

FI 70.50 76.44 74.79 62.41 87.67 85.19 49.40 96.58 94.74

VIX 70.61 76.31 73.84 62.90 87.04 82.42 50.68 95.79 90.10

BSIV 71.70 74.70 72.27 63.85 85.69 81.12 51.00 95.06 89.88

CMF1 72.28 73.92 70.94 64.19 85.25 80.47 50.89 95.06 89.82

CMF2 71.59 74.80 72.40 63.76 85.76 81.27 51.03 95.02 90.13

CMF3 70.95 75.66 73.16 63.25 86.38 82.00 51.01 95.12 90.30

MFIV 69.87 76.97 74.36 62.42 87.40 83.08 50.80 95.53 90.49

CCAR+BSIV 72.24 71.97 69.94 64.90 83.83 81.36 52.54 94.45 92.95

CCAR+CMF1 72.57 72.21 70.02 65.02 83.77 81.07 52.44 94.88 93.73

CCAR+MFIV 71.61 72.90 70.96 64.56 84.48 82.09 52.71 94.44 91.63

1-Month RV Forecast for T-bond: Forecast period = 01/2001–12/2006

Log-volatility Volatility Variance

R2 RMSE MAE R2 RMSE MAE R2 RMSE MAE

RV1 37.92 100.00 100.00 20.05 100.00 100.00 3.85 100.00 100.00

RVH 71.58 69.03 67.34 66.22 67.70 60.84 57.62 66.42 57.67

CCAR 77.71 59.05 55.91 72.18 58.98 51.13 64.46 58.80 47.56

FI 76.10 61.14 57.96 69.08 62.13 53.75 59.46 62.79 50.52

BSIV 71.57 66.79 61.19 63.22 68.14 59.07 52.59 68.41 58.55

CMF1 71.56 66.79 60.78 64.37 66.81 56.95 55.17 66.18 53.72

CMF2 71.86 66.50 60.97 64.66 66.55 57.23 55.48 65.94 54.02

CMF3 72.05 66.29 61.01 64.88 66.34 57.30 55.72 65.76 54.06

MFIV 72.16 66.16 61.26 65.07 66.14 57.38 55.97 65.54 54.10

CCAR+BSIV 78.07 56.53 52.42 72.06 57.38 49.46 64.01 58.00 47.00

CCAR+CMF1 78.07 56.89 52.51 72.08 59.35 51.78 64.03 59.17 47.69

CCAR+MFIV 78.24 56.27 52.38 72.20 59.53 51.96 64.11 59.34 47.82

1990 1992 1994 1996 1998 2000 2002 2004 20060

0.1

0.2

0.3

0.4

0.5

S&P 500: 1−month RV and MFIV

1990 1992 1994 1996 1998 2000 2002 2004 20060

0.05

0.1

0.15

Tbond: 1−month RV and MFIV

Sample averages for the square root of 1-month RV and MFIV:RV MFIV

S&P 500 0.156 0.191Tbond 0.091 0.098