*Step 2: Benchmark Selection*

The second step is to create a “naive” or systematic hedging benchmark index (the benchmark) using a plain-vanilla options strategy to gauge the relative performance of the tail-hedging strategy. In our example, our benchmark is designed as follows:

- Strategy: Every month, on each listed expiry date, we execute a rolling strategy whereupon we purchase new S&P and EuroStoxx 50 90 percent strike put options with a two-month expiry. At any time, we would therefore have four options in the portfolio with maturities equal to front-month and back-month expiries. All options are let to run until they expire.
- The notional of the purchased options is equal to one-fourth of the mark-to-market value of the benchmark on that same day in order to match exposures.
- Performance calculation: The benchmark is calculated in USD. Payoffs or premiums are paid in and out of a synthetic USD cash account earning the federal funds rate.

The simulated history of the benchmark is shown in Figure 4. We also show the cumulative P&L of the S&P 500 Index and the cumulative P&L of the S&P with a one-to-one overlay of the benchmark as a hedge.

Figure 4 demonstrates the conundrum faced by many systematic plain-vanilla hedging strategies:

When a tail event does materialize, such a strategy can successfully cushion the initial blow of a tail event. In our example, for $100 invested in the portfolio in April 2008, the hedging strategy would have saved the investor up to $20 by November 2008. However, if a tail event does not materialize, it also shows how the long-term running cost (the carry) of the strategy may gradually eat up the accrued hedging benefits.

This then illustrates the disadvantage of a static tail-hedge strategy: By systematically investing in the same notional, it tends to be under-invested in the period leading up to the shock, causing the investor to be under-hedged, and it tends to be over-invested immediately after the tail event when the price of options is high and the risks have dissipated, resulting in higher performance drag.

*Step 3: Choice Of A Tail-Hedging Strategy*

Rather than the benchmark and its short-delta/long-volatility bias, we prefer a cleaner tail-hedging strategy that aims to isolate the volatility component typical of tail events.

The underlying fundamental strategy of the Equity Tail Hedge S&P Index can be broken down into five steps:

- The algorithm completes a monthly sale of vanilla ratio-put spreads on the underlying equity index consisting of:

- short a number of three-month 95 percent puts
- long a number of three-month 80 percent puts

- The quantity of puts is chosen such that each leg generates a profit/loss of 1% of the strategy notional (i.e. 1 percent vega exposure) per point change in the underlying index volatility. The position thus naturally adapts to the prevailing level of equity volatility. Specifically, during times of low volatility, when options’ vega is low, the quantity of options needed to generate 1 volatility point increases, resulting in higher exposure to a tail event before it has happened. Likewise, when a tail event has occurred and equity volatility and options’ vega are high, the quantity of options needed to generate 1 volatility point is lower, and the strategy naturally deleverages itself at each reset. The ratio of 95 percent puts to 80 percent puts has a historical average of 1-to-3.15.
- The position is delta-hedged. (Once the directional component of the position is removed via delta-hedging, what remains is pure exposure to volatility at each strike, thus resulting in a long skew exposure.)
- The puts are unwound a day before expiration to avoid expiration-day effects and are rolled on a monthly basis.
- Any cash balance accrues at the relevant rate.