Report

+ Implied Volatility Index Kyu Won Choi March 2, 2011 Econ 201FS + Implied Volatility Index Implied Volatility Index With observed option prices, market’s estimate of the volatility is found Black-Scholes-Merton pricing formula Ctobserved = CtBSM (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 VXN: Model-free implied volatility for Nasdaq 100 index Developed by Whaley (1993) 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 Leverage Effect: negative (positive) returns increase financial leverage, stocks riskier, driving up volatility (down) Asymmetric effect when returns decline/volatility increases 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 VIX VXN Mean 20.662 5 24.1188 Standard Deviation 10.785 4 9.6296 Skewness 2.2256 2.3771 Kurtosis 9.0324 10.3726 + 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?