La utilización de HIIT en baloncesto.

We obtain daily S&P 500 index call and put options data from IVolatility.com for the period 1996:01 to 2012:12. Following the standard practice, option prices are calculated as the midpoint between the best bid and best ask price. Expiration time is calculated assuming 360 calendar days per year. Each trading day is matched with the respective dividend yield which is obtained from Bloomberg. Moreover, each option contract is matched with the appropriate continuous risk-free rate that is found after interpolating the 1-, 3-, 6- and 12-month Treasury Constant Maturity rates downloaded from the FRED database of the Federal Reserve Bank of St. Louis. A series of filtering rules are applied to the dataset to eliminate measurement errors and outliers mainly caused by thinly traded options (see for example A¨ıt- Sahalia and Lo, 1998, Han, 2008 and Chang et al., 2013). First, we discard options that do not satisfy standard no-arbitrage conditions. Second, we exclude observa- tions with zero bid prices and midpoint prices that are less than $3/8. Third we filter out options with zero or higher than 1 implied volatility. Finally, we take into consideration only options with non-zero trading volume and maturity between 7 and 270 calendar days.

We use equations (2.37) and (2.38) to estimate implied variance and skewness for constant maturities of 30-, 60-, 90- and 120-days ahead at the end of each month.8

Since interpolation across the time dimension is needed for this exercise, we make sure that we consider only days with a sufficient number of available maturities. A maturity is regarded as available if it has a cross-section with at least two OTM puts and two OTM call options. Therefore, we require that there are at least four available maturities that cover the next two months and either the third or fourth month (or both) following the current month. Moreover, we do not take into consideration

8_{We do not estimate implied moments for maturities longer than 120 days since the availability}

of long maturity options is not high enough to provide accurate estimates of long-maturity implied moments.

days that do not have available at least one maturity shorter than or equal to 30 days and at least one maturity longer than or equal to 120 days.9

In order to create costant maturity implied moments, we follow the interpola- tion technique of Kostakis et al. (2011) and Neumann and Skiadopoulos (2013). In particular for each cross-section of options, we interpolate across implied volatilites in the delta space to obtain a grid of 1000 data points with deltas ranging from 0.01 to 0.99. Inside the available delta range we interpolate using a cubic smoothing spline with smoothing parameter 0.99 while outside the available delta range, we extrapolate using the respective boundary values. The interpolation across the time dimension for a given day proceeds as follows: First, from all the available inter- polated implied volatility curves of a given day we keep the data points with delta values of 0.1, 0.2,...,0.9. Using a cubic smoothing spline, we then interpolate across the time dimension for the given constant maturities. Second, we create constant maturity implied volatility curves by fitting a cubic spline to the available nine im- plied volatilities. Third, the delta grid of the constant maturity implied volatility curve is converted to strike prices and the respective implied volatilities are trans- formed to option prices. Finally, equations (2.37) and (2.38) are discretized and estimated using the trapezoidal approximation.

Once we have the estimates of constant maturity implied moments for 30-, 60-, 90- and 120-days ahead, we use equations (4.7) and (4.8) to create vectors of forward 1-month moments. In particular we create:

fv0 ≡ h F V₀(1) F V₀(2) F V₀(3) F V₀(4)i 0 ≡ [F V0;0,30 F V0;30,60 F V0;60,90 F V0;90,120] 0 , (4.11) fs0 ≡ h F S₀(1) F S₀(2) F S₀(3) F S₀(4) i0 ≡ [F S0;0,30 F S0;30,60 F S0;60,90 F S0;90,120] 0 . (4.12)

9_{In fact, our sample is restricted to the 1996:01-2012:12 period because of the relatively limited}

Then using equation (4.10) we create a vector of forward 1-month skewness coeffi- cients: fsc0 ≡ h F SC₀(1) F SC₀(2) F SC₀(3) F SC₀(4)i 0 ≡ [F SC0;0,30 F SC0;30,60 F SC0;60,90 F SC0;90,120] 0 . (4.13)

Table 4.1 reports the descriptive statistics for the estimated forward variances and skewness coefficients. All forward variances exhibit very similar statistics and their autocorrelations range from 0.775 to 0.834. In contrast, forward skewness co- efficients become more negative and volatile as the horizon increases. Moreover, forward skewness coefficients are much less persistent with autocorrelation coeffi- cients ranging from 0.300 to 0.548. Table 4.2 provides the correlation coefficients for the forward moments. Forward variances are all positively and highly correlated with correlation coefficients ranging from 85% to 94%. The respective correlations between forward skewness coefficients range from 37% to 63%. It is apparent that while each forward variance has an idiosyncratic component depending on the month it refers to, all of them share a strong common component. On the contrary, the idiosyncratic information embedded in each forward skewness coefficient is more pronounced. This can be also confirmed by looking at Figure 4.1 which plots the forward moments across time. Forward variances tend to move in lockstep, taking their highest values during the recent financial crisis. Forward skewness coefficients exhibit similar patterns but the idiosyncratic variation of each variable is evident. The correlations between forward variances and skewness coefficients are low and consistently negative apart from the case of F V(1) _{and F SC}(1) _{whose correlation is}

positive but very close to zero (2%).