- A Appendix A.1 Synthetic Variance Swap Rates I compute synthetic variance swap rates from the price of a replicating portfolio that takes a static position in a continuum of out-of-the money European options (Carr and Wu 2009).20 I perform this computation every day for standard expirations between ten calendar days and three years to maturity with at least five out-of-the money call and put options whose Black and Scholes (1973) deltas are greater than or equal to 1%.21 For each date-maturity pair satisfying this filter, I fit a flexible implied volatility function by local linear regression to out-of-the money option prices with positive implied volatility as reported by OptionMetrics.
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- Figure A.6: Observable versus Latent State Variables 1996 1999 2002 2005 2008 2011 2014 2017 Date -6.00 -4.00 -2.00 0.00 2.00 4.00 6.00 8.00 Level Factor: RMSE = 0.24 95% HPD Region Median Posterior Baseline 1996 1999 2002 2005 2008 2011 2014 2017 Date -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 Slope Factor: RMSE = 0.09 This figure plots the standardized logarithm of the first two principal components of variance swap rates denoted as level and slope that are used in the baseline estimation The plot also reports the smoothed estimates of the corresponding latent variables from the Bayesian MCMC estimation including the posterior median and the 95% highest posterior density region. The smoothed median is very close to the observed principal components. In addition, the highest posterior density regions fall very tightly around the observed principal components, particularly for the level factor.
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- I determine which options are out-of-the money using the forward rate implied by put-call parity.22 To extrapolate beyond the observed strikes and for deep out-of-the money options with a delta less than 1%, I append log-normal tails using flat implied volatility functions. I then compute synthetic variance swap rates as a weighted average of out-of-the money option prices, using the fitted implied volatility functions to compute option prices. To obtain the term-structure on a constant grid, I interpolate the synthetic variance swap rates at the observed maturities onto a monthly grid from one-month to two-years. The interpolation is linear in total variance following Carr and Wu (2009) and the CBOE volatility indexes.
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- Panel A: Physical Parameters Φ1,1 Φ1,2 Φ1,3 ln(RV ) 0.00 0.23∗∗∗ 0.21∗∗∗ 0.40∗∗∗ [0.01] [2.95] [8.43] [5.76] PClevel 0.01 0.07 0.88∗∗∗ -0.07 [0.15] [0.55] [23.73] [-0.69] PCslope-0.01 0.02-0.04∗∗∗ 0.78∗∗∗ [-0.33] [0.58] [-3.41] [23.89] Panel B: Prices of Risk Λ0 Λ1,1 Λ1,2 Λ1,3 ln(RV ) -0.68∗∗∗ 0.21∗∗∗ -0.07∗∗∗ -0.12∗∗ [-13.62] [3.58] [-3.23] [-1.97] PClevel-0.06 0.11-0.08∗ 0.37∗∗∗ [-0.79] [1.01] [-1.89] [3.38] PCslope-0.04 0.05-0.03 0.02 [-0.80] [0.48] [-0.76] [0.23] Table 4: Variance Swap Pricing Errors This table summarizes the model fitting errors for variance swap rates and monthly returns from 1996 to 2016 using daily data. The errors are small and unbiased overall. For example, the standard deviation of the pricing errors in Panel A are below the average bid-ask spreads reported by Markit.
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- Straddle Return Predictability from 1996 to 2016: Rstraddle t+h,b = β0 + β1Êt[Rt+h,n] + t+h,b Straddle Maturity Bucket b (1,3] (3,6] (6,9] (9,15] (15,24] Avg. Expected Return Maturity n 3 6 9 12 18 Avg.
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- The gray box for the November 2008 plot highlights the scale of the other subplots, illustrating the increase in magnitude of the partial derivatives during the financial crisis.
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- The out-of-sample analysis is performed by estimating the models with an expanding window using 1996 to 1998 as the initial estimation period. Panel I reports variance swap pricing errors measured by the root mean squared error RMSE, mean absolute error MAE, and median absolute deviation MAD. Panels II-V report variance swap return forecast errors measured by explanatory power R2 adj, mean squared error MSE, and mean absolute error MAE.
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Van Tassel, P. and E. Vogt (2016). Global variance term premia and intermediary risk appetite. Working Paper.
- VIX Futures Pricing Errors 2007-2016: et,n = Futo t,n − Futt,n Contract (n) 1 2 3 4 5 6 Avg.
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Zhang, L., P. A. Mykland, and Y. Aït-Sahalia (2005). A tale of two time scales: Determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association 100(472), 1394–1411.