Implied Volatility Index Kyu Won Choi March 2

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

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

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

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

+ n Returns and Volatility Negative and asymmetric relationship btw returns and volatility n n Leverage Effect: negative (positive) returns increase financial leverage, stocks riskier, driving up volatility (down) n n 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 n impact of the current volatilities on the future returns n Time-varying risk premiums n 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

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