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# Choosing The Right Option: Backtesting Different Hedging Tools For The S&P 500

While the last year saw equities climbing from one all-time high to another, the new year started with a sharp move downwards. As a result, it seems to be the right time to look for a portfolio insurance that can be a reasonable hedge for the S&P 500. But while for a long time SPX put options seemed to be the only way of truly hedging a US equity portfolio, recent years have seen a rise of derivatives that provide insurance against extreme market movements. In my previous articles, I already conducted an in-depth analysis of these hedging tools, which is why I will only provide a brief summary of the key take-aways here:

One insurance vehicle that has increased in popularity is the VIX index, also referred to as the “fear gauche”. Even though VIX derivatives, with the VXX being the most prominent among them, have mainly gained attention through their catastrophic performance, VIX options are a tool that certainly has to be considered. What makes VIX options unique is their volatility surface. While for SPX options (like for most equity options), implied volatility tends to decrease when getting closer to expiry, the opposite is true for VIX options. Here, an increase in implied volatility when getting closer to expiry can be observed.

**Portfolio Optimization**

In order to analyze the different hedging tools, a backtest is conducted that hedges an S&P 500 portfolio with long positions in either VIX call options or SPX put options. The reason for choosing call options on the VIX is that the VIX has a negative correlation to the SPX. In a first step, the impact of different allocation sizes and chosen option deltas on the Sharpe ratio is analyzed with a sensitivity analysis. In a second step, the ideal allocation size and deltas of the options are selected based on the highest Sharpe ratio, in order to receive two optimized portfolios. These portfolios are then compared to each other and the S&P 500.

At first, the SPX option hedged portfolio is analyzed. The header of the table above shows the delta of the option and the first column displays the allocation to options. Here, it can be observed that the highest Sharpe ratio can be achieved with a low allocation and a high delta. It is also worth mentioning that the highest Sharpe ratio is achieved with the lowest possible allocation to the options. Even though the volatility of the portfolio decreases when the option allocation increases, this decrease does not compensate for the massively lower return that comes with a higher option allocation.

In the VIX option hedged portfolio, it can be observed that the Sharpe ratio gets higher, the lower the delta of the option. This observation is true for all allocation sizes, as the highest Sharpe ratio for every allocation size is seen at a delta of 0.25. Across all simulated deltas the same result as with the SPX put options can be seen: A bigger allocation to options might indeed decrease the volatility of the portfolio, but the premium paid for the options does not justify their price.

Taking the sensitivity analysis into consideration, two optimized portfolios can be identified: An SPX Put Options Hedged Portfolio with 1% allocation and 0.75 delta and a VIX Call Options Hedged Portfolio with 1% allocation and 0.25 delta.

**Backtest Results**

The table indicates quite clearly that the VIX Call Options Hedged Portfolio is the most attractive hedged portfolio. Not only does it generate a higher return and Sharpe ratio, but it also has the lowest maximum drawdown and Value at Risk than a portfolio hedged with SPX puts. Moreover, it has the lowest correlation to the S&P 500, indicating that it is also the best hedged portfolio, which is very interesting, given that this portfolio is hedged with a hedging tool that has a different underlying than the portfolio that ought to be hedged. In addition, the comparison with the unhedged portfolio clearly shows that it takes until mid-2014 for the unhedged portfolio to catch up with the VIX Call Options Hedged Portfolio. This demonstrates that on the long term, an unhedged portfolio offers the highest return rate, as it has to compensate the investor for the risk he takes on by fully exposing his portfolio to downside risks. In periods of uncertainty however, the VIX Call Options Hedged Portfolio can generate a quite substantial outer-performance against the unhedged portfolio that sustains for several years.

**Conclusion**

As a result, VIX options seem to offer an attractive hedging opportunity for an S&P 500 portfolio. Their unique volatility surface (which was examined in-depth in a previous article) as well as the mean reverting nature of volatility in general make them excellent hedging tools for an equity portfolio.

**Appendix: Methodology**

For the options portfolio backtest simulation, it is assumed that on every 3rd Wednesday of the month, at close n % of the portfolio will be allocated to options with a 2 months expiry. The reason for choosing the 3rd Wednesday of the month is that this is the expiration day for VIX options. It is not intended however, to hold the options till expiry, which is why this is always the date when the options should be rolled over. As a result, on every 3rd Wednesday of the month, the options in the portfolio which only have 1 month till expiry left are sold and options with 2 months till expiry are bought. In this simulation, it is assumed that every option will be bought exactly at the ask price and sold for the bid price.

On the same day, the whole portfolio is always rebalanced, which means that (1-n) % of the portfolio are allocated to the S&P 500. It is assumed that the S&P 500 is directly investable and does not have a bid/ask spread. The initial allocation (n) is not dynamic and does not change over the whole backtesting period, even though different initial allocation sizes are tested in order to find out the ideal allocation size. Since both the S&P 500 and the option position are not rebalanced on a daily basis, the hedge ratio is not constant over the course of the month. This occurs since an increase in the S&P 500 will automatically increase the value of the S&P 500 position and decrease the value of the options position and vice versa.

What is still left, are the specifications of the options in the portfolio, which are listed below. For the sake of better comparability, it is assumed that both the VIX call options and the SPX put options have the same specifications.

Strike Price Interval: 1 Index Point

Minimum Tick Size: 0.01

Bid/Ask Spread: 10%

Expiration Date: 3rd Wednesday of the Month

Multiplier/Contract Size: 100 USD

The backtest is subject to several limitations: The option prices in this backtest are calculated with the black and scholes formula and are not quotations from an exchange. Even though black and scholes is commonly used for pricing options, it might not necessarily always be true that the option is traded at its (fair) value. Moreover, the volatility surface for the implied volatility of the options is calculated by interpolating values from different deltas. The interpolation used is a linear interpolation. Different interpolation methods might yield different final results. In addition, the actual bid/ask spread might differ from the one in the simulation (just as the minimum tick size) and can have a fairly big impact on the results.