+ Implied Volatility Index Kyu Won Choi March 2, 2011 Econ 201FS.

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+ Implied Volatility Index Kyu Won Choi March 2, 2011 Econ 201FS

+ Implied Volatility Index With observed option prices, market’s estimate of the volatility is found Black-Scholes-Merton pricing formula C t observed = C t BSM (p(t), K, T-t, r,  t ) Depending on the validity of model Chicago Board Options Exchange (CBOE)’s Market Volatility Index VIX: Model-free implied volatility for S&P 500 index Developed by Whaley (1993) VXN: Model-free implied volatility for Nasdaq 100 index Since September 2003 Expected future market volatility over the next 30-day of risk-neutral world

+ Contents Leverage Effect & Volatility Feedback Effect S&P 500 and VIX Nasdaq 100 and VXN Jump Detection using RV and BV Difference between Annualized RV and Annualized VIX Volatility Risk Premium Relationship between VIX and VXN

+ Data Set Daily closing values of the VIX from 1/3/2000 to 12/31/2010 Total of 2767 days S&P 500 Prices from 1/3/2000 to 12/31/2010 Nasdaq 100 Daily Closing Prices from 9/22/2003 to 12/31/2010 Total of 1834 days Daily closing values of VXN from 9/22/2003 to 12/31/2010

+ S&P 500 Index and VIX

+ S&P 500 Index Returns

+ Returns and Volatility Negative and asymmetric relationship btw returns and volatility Asymmetric effect when returns decline/volatility increases Leverage Effect: negative (positive) returns increase financial leverage, stocks riskier, driving up volatility (down) impact of the lagged returns on the current volatilities (current returns on future volatilities) Volatility Feedback hypothesis: an increase in volatility leads to a decrease in return impact of the current volatilities on the future returns Time-varying risk premiums Can use GARCH model

+ Correlation between S&P 500 Index Returns and VIX (negative)

+ Between return and change in VIX (asymmetry)

+ Realized Volatility of S&P 500

+ Bipower Volatility of S&P 500 Index

+ Relative Jump Contribution

+ Annualized VIX

+ The Difference btw Annualized RV and Annualized VIX

+ Nasdaq 100 Index and VXN

+ Nasdaq 100 Returns

+ Correlation between NDX Returns and VXN (negative)

+ Movement of VIX and VXN

+ Scatter Plot of VIX and VXN VIXVXN Mean Standard Deviation Skewness Kurtosis

+ Further study VXD (based on DJIA), VSTOXX in France, VDAX-NEW in Germany Frequency data of them Look for the relationship Jump option pricing models Co-jumping process ? An implied volatility index follows a stochastic process Option valuation for stochastic volatility Time-varying risk premium?