By: Andrey Fradkin Date: 1/22/08 The Informational Content of Implied Volatility in Individual Stocks and the Market By: Andrey Fradkin Date: 1/22/08

Andrey Fradkin: Informational Content of Volatility 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. Andrey Fradkin: Informational Content of Volatility

Andrey Fradkin: Informational Content of Volatility Outline Jump Detection HAR-RV-CJ Models Data Preparation Summary Statistics Summary Plots Significance of Jump Statistics in HAR-RV-CJ Models Andrey Fradkin: Informational Content of Volatility

Background Mathematics Andrey Fradkin 1/31/2007 Background Mathematics Realized Variation: Realized Bi-Power Variation: Andrey Fradkin: Informational Content of Volatility

Background Mathematics Part 2 Andrey Fradkin 1/31/2007 Background Mathematics Part 2 Tri-Power Quarticity Quad-Power Quarticity Andrey Fradkin: Informational Content of Volatility

Background Mathematics Part 3 Andrey Fradkin 1/31/2007 Background Mathematics Part 3 Z-statistics (max version) – Paper uses .999 significance level Andrey Fradkin: Informational Content of Volatility

Andrey Fradkin: Informational Content of Volatility 1/31/2007 Original HAR-RV-J Model (Developed in Andersen, Bollerslev, Diebold 2006, Extended for use with RAV measure in Forsberg and Ghysels(2007)) Andrey Fradkin: Informational Content of Volatility

Andrey Fradkin: Informational Content of Volatility 1/31/2007 The HAR-RV-CJ Model Andrey Fradkin: Informational Content of Volatility

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. Andrey Fradkin: Informational Content of Volatility

Andrey Fradkin: Informational Content of Volatility 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 Andrey Fradkin: Informational Content of Volatility

Andrey Fradkin: Informational Content of Volatility Combined Regressions Andrey Fradkin: Informational Content of Volatility

Andrey Fradkin: Informational Content of Volatility 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 Andrey Fradkin: Informational Content of Volatility

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 Andrey Fradkin: Informational Content of Volatility

Andrey Fradkin: Informational Content of Volatility Summary Statistics Variable Mean Std. Dev. Min Max rv 3.17 4.11 0.27 45.28 imp vol 4.04 3.25 0.45 21.27 BMY rav 1.42 0.73 0.44 6.32 jump 0.04 0.00 6.70 2.98 5.26 0.17 97.38 3.60 3.40 0.10 34.81 Citigroup 1.39 0.84 0.30 9.29 0.02 0.21 5.90 2.65 3.69 0.11 53.55 3.54 2.92 0.40 17.24 GE 1.34 0.75 0.23 6.21 0.01 4.29 2.54 2.88 39.50 4.30 3.35 0.80 20.99 GS 1.36 0.61 4.58 0.03 0.70 24.09 3.26 4.01 50.42 4.43 3.42 1.00 20.56 HD 1.49 0.74 0.26 6.06 0.24 4.83 Andrey Fradkin: Informational Content of Volatility

Andrey Fradkin: Informational Content of Volatility Variable Mean Std. Dev. Min Max rv 1.69 1.90 0.15 25.48 imp vol 2.13 1.56 0.29 11.41 KO rav 1.11 0.49 0.25 4.98 jump 0.02 0.18 0.00 4.18 2.24 2.70 0.22 34.90 2.89 1.86 0.59 12.29 MDT 1.23 0.57 0.30 5.40 0.05 0.56 17.33 6.99 8.89 0.34 175.37 9.90 7.18 1.17 46.31 MOT 2.17 1.04 0.41 10.05 0.09 0.80 19.92 4.17 0.20 43.54 9.21 6.86 1.38 38.54 NOK 0.91 0.32 6.41 0.04 0.65 21.50 7.80 8.35 0.51 62.92 10.14 7.10 1.39 41.08 TXN 2.33 1.14 8.07 0.12 0.90 17.12 1.03 1.30 16.25 1.89 0.42 8.06 SPY 0.85 0.43 0.16 3.54 0.01 0.23 7.94 Andrey Fradkin: Informational Content of Volatility

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*** (0.03) (0.02) (0.04) (0.06) c5 0.449*** 0.125 0.256* 0.257** 0.266** 0.171* 0.338** 0.313*** 0.398*** 0.439*** 0.328*** (0.13) (0.08) (0.11) (0.09) (0.12) (0.05) (0.10) (0.07) c22 0.236 0.348*** 0.277** 0.407*** 0.191** 0.333*** 0.214 0.260** 0.242** 0.112 0.212*** j 0.117 0.378** 0.237 -0.064*** 0.013 0.185** -0.094** 0.034 0.052 0.15 -0.135*** (0.29) (0.01) (0.17) j5 -1.284 -0.052 1.472 -0.148 3.444* 0.876 0.509* -0.507 -0.590* -0.631 -0.424** (1.11) (1.08) (2.23) (1.63) (0.56) (0.20) (0.42) (0.30) (0.85) (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 (6.47) (2.64) (10.78) (0.18) (4.72) (0.59) (0.35) (1.76) (0.60) (3.88) (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** (0.21) (0.41) (0.28) (0.16) (0.71) (0.31) (0.61) * p<0.05, ** p<0.01, *** p<0.001 Andrey Fradkin: Informational Content of Volatility

HAR-RV-CJ – 1 week Newey-West Standard Errors Lag(60) 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** c5 0.499*** 0.161* 0.317*** 0.259*** 0.455*** 0.325** 0.553*** 0.501*** 0.468*** 0.544*** 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** j 0.315** 0.884* -0.09 -0.143*** -0.039 0.329** -0.204*** 0.015 0.166*** 0.336 -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*** j22 2.491 -0.299 3.539 -0.678* 6.846 -0.669 1.413*** 0.034 -0.78 1.83 -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** Andrey Fradkin: Informational Content of Volatility

HAR-RV-CJ – Level Day Ahead – Newey-West Standard Errors Lag(60) 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 0.394*** 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* c22 0.278** 0.221*** 0.244*** 0.339** 0.086 0.105 0.15 0.217* 0.250* 0.181* 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 j5 0.305 1.351* -0.965 -0.482*** 1.308 1.289*** 0.278 -0.051 -0.415 -0.091 -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 Constant 0.213* 0.330* 0.259* 0.078 0.238* 0.167** 0.224*** 0.857** 0.209* 0.463** 0.097** Andrey Fradkin: Informational Content of Volatility

Realized Variance and IV Andrey Fradkin: Informational Content of Volatility

Realized Variance and RAV Andrey Fradkin: Informational Content of Volatility

Andrey Fradkin: Informational Content of Volatility SPY RV and RAV Andrey Fradkin: Informational Content of Volatility