The Informational Content of Implied Volatility in Individual Stocks

and the Market

By: Andrey FradkinDate: 1/22/08

Introduction • Examine relative informational content of implied volatility

forecasts against historical volatility forecasts using high frequency data for 10 large cap stocks trading on NYSE and for the SPY.

• Important prior studies on this subject are Andersen et al(2007) and Jiang and Tian(2006). Find contradictory evidence as to which measure is better.

• This study is first to use individual stocks, HAR-RV CJ model, RAV, and robust regression in examining this question.

• HAR-RV-CJ model allows for the separation of the jump component of variance from continuous component of volatility.

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Outline

• Jump Detection• HAR-RV-CJ Models• Data Preparation• Summary Statistics• Summary Plots• Significance of Jump Statistics in HAR-RV-CJ

Models

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Background MathematicsRealized Variation:

Realized Bi-Power Variation:

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Background Mathematics Part 2

• Tri-Power Quarticity

• Quad-Power Quarticity

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Background Mathematics Part 3

• Z-statistics (max version) – Paper uses .999 significance level

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Original HAR-RV-J Model (Developed in Andersen, Bollerslev, Diebold 2006, Extended for use

with RAV measure in Forsberg and Ghysels(2007))

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The HAR-RV-CJ Model

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Mincer-Zarnowitz Regressions

• Standard framework for evaluating implied volatility forecasts.

• Implied volatilities are obtained from the OptionMetrics database. One implied volatility is used per day. This implied volatility is taken from an at-the-money call option expiring about a month ahead.

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Combined Regressions

• Combine implied forecasts with historical forecasts.

• Compare marginal improvements from historical to combined and from implied to combined.

• Two types of combined forecasts, one using RAV and the other using RV-CJ

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Combined Regressions

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Data Management

• Uses TAQ high-frequency data that was cleaned by Tzuo Law. Prices are sampled at five minute intervals.

• 10 equities and the SPY are used in the study. Equities chosen are based on the high open interest in their options.

• In-Sample data ranges from 2001 through 2004• The Out-of-Sample data encompasses all of 2005

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10 Stocks Referred to from now on by Ticker

• BMY - Bristol-Meyers• C - Citigroup • GE - General Electric • GS -Goldman Sachs • HD - Home Depot • KO – Coca Cola• MDT - Medtronic• MOT – Motorola• NOK – Nokia• TXN – Texas Instruments• SPY – SPY RV’s with Vix Implied Volatility• SPX – SPY RV’s with SPX Option Implied Volatility

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Summary Statistics

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14

Variable Mean Std. Dev. Min Max

rv 3.17 4.11 0.27 45.28imp vol 4.04 3.25 0.45 21.27

BMY rav 1.42 0.73 0.44 6.32jump 0.04 0.27 0.00 6.70

rv 2.98 5.26 0.17 97.38imp vol 3.60 3.40 0.10 34.81

Citigroup rav 1.39 0.84 0.30 9.29jump 0.02 0.21 0.00 5.90

rv 2.65 3.69 0.11 53.55imp vol 3.54 2.92 0.40 17.24

GE rav 1.34 0.75 0.23 6.21jump 0.01 0.17 0.00 4.29

rv 2.54 2.88 0.10 39.50imp vol 4.30 3.35 0.80 20.99

GS rav 1.36 0.61 0.23 4.58jump 0.03 0.70 0.00 24.09

rv 3.26 4.01 0.21 50.42imp vol 4.43 3.42 1.00 20.56

HD rav 1.49 0.74 0.26 6.06jump 0.03 0.24 0.00 4.83

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Variable Mean Std. Dev. Min Maxrv 1.69 1.90 0.15 25.48

imp vol 2.13 1.56 0.29 11.41KO rav 1.11 0.49 0.25 4.98

jump 0.02 0.18 0.00 4.18

rv 2.24 2.70 0.22 34.90imp vol 2.89 1.86 0.59 12.29

MDT rav 1.23 0.57 0.30 5.40jump 0.05 0.56 0.00 17.33

rv 6.99 8.89 0.34 175.37imp vol 9.90 7.18 1.17 46.31

MOT rav 2.17 1.04 0.41 10.05jump 0.09 0.80 0.00 19.92

rv 4.17 4.98 0.20 43.54imp vol 9.21 6.86 1.38 38.54

NOK rav 1.69 0.91 0.32 6.41jump 0.04 0.65 0.00 21.50

rv 7.80 8.35 0.51 62.92imp vol 10.14 7.10 1.39 41.08

TXN rav 2.33 1.14 0.51 8.07jump 0.12 0.90 0.00 17.12

rv 1.03 1.30 0.05 16.25imp vol 1.89 1.39 0.42 8.06

SPY rav 0.85 0.43 0.16 3.54jump 0.01 0.23 0.00 7.94

HAR-RV-CJ – Level Month Ahead – Newey-West Standard Errors Lag(60)

BMY C GE GS HD KO MDT MOT NOK TXN SPY SPX

cv 0.075** 0.148*** 0.121*** 0.153*** 0.172*** 0.184** 0.108** 0.081* 0.156*** 0.152*** 0.148*** 0.148***

(0.03) (0.02) (0.02) (0.03) (0.04) (0.06) (0.04) (0.03) (0.03) (0.03) (0.04) (0.04)

c5 0.449*** 0.125 0.256* 0.257** 0.266** 0.171* 0.338** 0.313*** 0.398*** 0.439*** 0.328*** 0.328***

(0.13) (0.08) (0.11) (0.08) (0.09) (0.09) (0.12) (0.05) (0.10) (0.07) (0.10) (0.10)

c22 0.236 0.348*** 0.277** 0.407*** 0.191** 0.333*** 0.214 0.260** 0.242** 0.112 0.212*** 0.212***

(0.12) (0.07) (0.09) (0.06) (0.06) (0.09) (0.13) (0.09) (0.09) (0.09) (0.06) (0.06)

j 0.117 0.378** 0.237 -0.064*** 0.013 0.185** -0.094** 0.034 0.052 0.15 -0.135*** -0.135***

(0.07) (0.13) (0.29) (0.01) (0.17) (0.07) (0.03) (0.07) (0.04) (0.09) (0.04) (0.04)

j5 -1.284 -0.052 1.472 -0.148 3.444* 0.876 0.509* -0.507 -0.590* -0.631 -0.424** -0.424**

(1.11) (1.08) (2.23) (0.10) (1.63) (0.56) (0.20) (0.42) (0.30) (0.85) (0.15) (0.15)

j22 8.317 1.388 5.596 -1.300*** 3.114 -1.945** 2.380*** 2.72 -0.684 4.908 0.046 0.046

(6.47) (2.64) (10.78) (0.18) (4.72) (0.59) (0.35) (1.76) (0.60) (3.88) (1.00) (1.00)

Constant 0.500* 1.065** 0.797** 0.454** 0.996** 0.532** 0.633*** 2.187** 0.821** 1.732** 0.326** 0.326**

(0.21) (0.41) (0.28) (0.16) (0.30) (0.20) (0.17) (0.71) (0.31) (0.61) (0.10) (0.10)

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* p<0.05, ** p<0.01, *** p<0.001

HAR-RV-CJ – 1 week Newey-West Standard Errors Lag(60)

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BMY C GE GS HD KO MDT MOT NOK TXN SPY SPX

c 0.097*** 0.296*** 0.190*** 0.257*** 0.245*** 0.318** 0.132** 0.104* 0.192** 0.186*** 0.208** 0.208**

c5 0.499*** 0.161* 0.317*** 0.259*** 0.455*** 0.325** 0.553*** 0.501*** 0.468*** 0.544*** 0.447*** 0.447***

c22 0.286** 0.326*** 0.299*** 0.423*** 0.107 0.192* 0.154 0.217* 0.263** 0.15 0.211** 0.211**

j 0.315** 0.884* -0.09 -0.143*** -0.039 0.329** -0.204*** 0.015 0.166*** 0.336 -0.177** -0.177**

j5 -0.271 -0.008 2.994* -0.244** 0.671 0.719 0.423 0.493 -0.684* -0.606 -0.617*** -0.617***

j22 2.491 -0.299 3.539 -0.678* 6.846 -0.669 1.413*** 0.034 -0.78 1.83 -0.167 -0.167

Constant 0.289* 0.634* 0.416* 0.189 0.422** 0.279** 0.314*** 1.235** 0.365* 0.802** 0.159** 0.159**

HAR-RV-CJ – Level Day Ahead – Newey-West Standard Errors Lag(60)

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BMY C GE GS HD KO MDT MOT NOK TXN SPY SPX

c 0.155*** 0.576*** 0.326*** 0.550*** 0.352*** 0.349* 0.116 0.082 0.352*** 0.394*** 0.329 0.329

c5 0.485*** 0.084 0.307*** 0.096 0.447*** 0.448*** 0.627*** 0.579*** 0.359* 0.355*** 0.400* 0.400*

c22 0.278** 0.221*** 0.244*** 0.339** 0.086 0.105 0.15 0.217* 0.250* 0.181* 0.202* 0.202*

j 0.575* 0.600** 1.583 -0.024 -0.821*** -0.062 -0.372*** 0.443** 0.387*** 0.616 -0.249 -0.249

j5 0.305 1.351* -0.965 -0.482*** 1.308 1.289*** 0.278 -0.051 -0.415 -0.091 -0.541** -0.541**

j22 0.84 -0.392 4.057 -0.374 4.76 -0.584* 1.210*** 0.119 -0.793 0.852 -0.287 -0.287

Constant 0.213* 0.330* 0.259* 0.078 0.238* 0.167** 0.224*** 0.857** 0.209* 0.463** 0.097** 0.097**

Realized Variance and IV

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Realized Variance and RAV

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SPY RV and RAV

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