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Implied Volatility Index Kyu Won Choi

March 2, 2011Econ 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 = Ct

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

VIX VXNMean 20.66

2524.1188

Standard Deviation

10.7854

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?