The Estimation Bias

I present the construction procedures of static and dynamic trading strategies. The differences in returns reveal the inference bias on the magnitude of correlation risk premium.

3.4. The Impact of Hedging Errors 87

Static Dispersion Portfolio

A static trading is to buy-and-hold the portfolio without any rebalance during the life of the trade. On each Tuesday immediately following the expiration date of the month, I short index options and buy the individual options and stocks. The investment weights are obtained from equation 3.5 and 3.6, respectively. I hold the portfolio until the maturity date of the options in the next month (usually the Saturday after the third Friday) and start a new trade on the following Tuesday.

Repeating the strategy at each month, I obtain the non-overlapping monthly returns of the static dispersion trade.

I first implement the trade with individual options written on all index constituents.

Table 3.1 reports the summary statistics. The sample average return is about 13%

per month with annualized Sharpe ratio of 0.68. The estimation results compare well with the literature. Driessen, Maenhout, and Vilkov (2009) report the average correlation risk premium of 10.37% with annualized Sharpe ratio of 0.73 over the sample period from January 1996 to December 2003. The difference in magnitude between my and Driessen, Maenhout, and Vilkov (2009)’s estimations is due to two reasons. One is that Driessen, Maenhout, and Vilkov (2009) use S&P 100 options and I use S&P 500 options, and the other reason is the difference in sample period.

As it is usually too expensive to trade all individual options, a more realistic design is to trade a subset of options. I select the 28 options with largest market capitaliza-tion of their underlying assets and implement the dispersion trade with the basket of options ⁴. The sample average is about 10% per month with annualized Sharpe ratio of 0.52. The correlation between returns from the dispersion trade with the entire list of index members and returns from the trade with a subset of assets is 0.99, indicating that there is no materially different by using the subset of options.

4The method to construct a dispersion trade with subset of options has been used in other studies. For example, Buraschi, Trojani, and Vedolin (2010)

3.4. The Impact of Hedging Errors 88

Therefore, I construct the dispersion trade with the 28 options in the following anal-ysis. The strategy outperforms the trading with market index and index straddles in terms of higher average returns and higher Sharpe ratios, as shown in the last two columns in Table 3.1.

Delta-Neutral Dispersion Portfolio

I then construct the delta-neutral dispersion trade, which aim to eliminate the ex-posure to stock market risk. As shown in Bertsimas, Kogan, and Lo (2000), Bakshi and Kapadia (2003a), and Branger and Schlag (2008), the delta hedging errors have an asymptotic distribution that is symmetric with zero mean for relatively low re-balancing frequencies. Especially, a delta hedging at daily frequency is sufficient to eliminate the hedging errors. I therefore implement the delta-neutral dispersion trade by rebalancing the investment 𝑧_𝑖 on each individual stocks at daily frequency, where 𝑧_𝑖 is computed using the average of bid-ask prices of options and stocks. The results are reported in Table 3.2. The sample average is about 4% per month, which is dramatically lower than the 10% average return of static dispersion portfolio. The results suggest that hedging errors on delta exposures accounts about 60% of the total returns of static dispersion portfolio and causes severe upward bias on esti-mated correlation risk premium. Further, the correlation between returns of static and neutral portfolios is only 0.39, indicating the remarkable impact of delta-hedging errors on the dynamics of correlation risk.

Delta-Vega-Neutral Dispersion Portfolio

Finally, I construct the delta-vega-neutral dispersion trade, by correcting for the time-varying vega exposures in the above daily delta-hedged dispersion portfolios.

Due to liquidity of individual options, daily rebalance on all the individual options are not feasible. Therefore, the vega exposures are rebalanced at weekly frequency.

The empirical results in Table 3.2 show that the sample average of the delta-vega-neutral dispersion portfolio is about 6%, which is about 40% lower than the static

3.4. The Impact of Hedging Errors 89

portfolio and indicating a remarkable influence due to vega exposures. Notably, the correlation between the returns of the static and the delta-vega-neutral portfolios is only about 0.01. Hedging errors on vega exposure introduces upward bias on correlation risk premium and lead to contaminated inference about the time series dynamics.

Beyond the differences in sample means between static and dynamic portfolios, another important discrepancy goes to the higher moments. In general, the static portfolio has much higher standard deviations, higher (negative) skewness and kur-tosis than the statistics of dynamic portfolios. This result suggests that the static portfolio is more volatile and exposes to higher left tail risk.

To further illustrate the impact of discrete hedging errors, I plot the delta and vega exposures of the static portfolios and the dynamic-hedged portfolios in Figure 3.2. The exposures are computed at 1-week prior to maturity of each trade over time. From the plot, we can see that the exposures in static portfolios are far away from zeros and vary over time, while the delta-vega-neutral portfolio has nearly any exposures to stock market risk and individual variance risk. This plot provide visi-ble evidence that static hedging fails to immune the portfolio from market risk and individual volatility risk after the initiation of the strategy.