When we implement tail hedges, we have traditionally used one year as the hedge horizon. To calculate the premium for these hedges, we start by pricing the explicit one-year hedge and then explore ways of reducing the premium by using indirect hedges. Some of these indirect hedges might consist of cheaper “option-like” positions from other markets that take advantage of potential tail correlations. In addition, we actively manage the hedges in an effort to reduce the realized cost of hedging over this investment horizon.
Many tail risk clients have expressed a desire to move toward more frequent rebalancing of hedges. One popular frequency is quarterly. There are numerous reasons for this: First, implementing more frequent hedges “averages” out the premium spent and the attachment point when the hedge kicks in (for example, a 15% attachment point will hedge the portfolio against losses greater than 15%). Second, it avoids a “cliff” effect, i.e., the preset hedge strike becomes more distant as it nears expiration. Finally, more frequent rebalancing might allow a recalibration of the portfolio exposures within the one-year period and capture other rebalancing activity.
Of course, we cannot make a blanket statement about the efficacy or superiority of any one hedge algorithm. Even without conducting a lot of work analyzing these algorithms, one can anticipate that trade-offs will arise simply because of the variations in volatility term structures (i.e., the variation of implied volatility during the time to maturity). Since option prices are roughly proportionate to implied volatility as well as the square root of time to expiration, we can expect a more frequent hedge to do better if the volatility curve is steep and upward-sloping (shorter-term implied volatility lower than longer-term implied volatility), or when the volatility term structure is flat. However, if the volatility curve is inverted, as usually happens after a market shock, hedging more frequently costs more than it does when the curve is steep or flat. This observation also supports the argument for maintaining some flexibility when adjusting the frequency of rebalancing in the aftermath of market events. Transaction costs also should play an important role in any investment decision and some of the benefits of rebalancing can quickly disappear if the transactions costs are large.
The trade-offs associated with more frequent hedging
As a hypothetical example of the key considerations involved in tail hedging, let’s look at a one-year hedge on the S&P 500 Index with a strike that is 25% out of the money. This is a common “benchmark” direct hedge because when hedging a 60/40 portfolio of stocks and bonds with a 15% attachment point, the effective strike for the direct S&P 500 put option is calculated as 15%/0.6, which is 25%. Let’s also assume that the one-year volatility for this 25% out-of-the-money strike is 30%. According to the Black Scholes method of pricing this option, the one-time “bullet” cost would be 250 basis points (bps), or 2.5%.
Now let’s take this one-year hedge and consider the trade-offs we might expect to make if we increased the frequency of the hedge (cost, sensitivities, etc.). For theoretical clarity, we will ignore the transaction costs of unwinding existing hedges and replacing them with new hedges, although these cost considerations are critical in running an actual hedge program. We also keep the hedge horizon fixed at the initial one-year, i.e., we will not replace the expiring shorter-dated hedges. This assumption allows us to focus on the sensitivities of the original portfolio of hedges.
Suppose first that the volatility of the S&P 500 options market is flat at 30% for the whole year. Then we can think of replacing the annual option with a string of daily options expiring in one day, two days, three days etc. all the way to 365 days, each of which would have 1/365th of the notional value of the option associated with it. Clearly no one would want to have 365 daily options for practical reasons, but as with all problems, taking the limit sheds important light on the nature of this problem: Since the time value of options with shorter expirations is lower than those of longer-dated ones, the total cost of the daily options is only 94 bps (see Figure 1 and Figure 2) – which is obtained by pricing each option and dividing by 365. Also, the shorter-dated options will expire with the passage of time and will need replacement, while the longer-dated options remain alive. If we consider the more sensible approach of quarterly rebalancing (where 250/4 of the premium is used each quarter), the total cost is only 127 bps, or roughly half of the “bullet” one-year hedge of 250 bps mentioned earlier. Figure 1 also shows that the pricing depends on the shape of the volatility curve.
For this exercise, we created three idealized curves: The “steep” curve starts at 15% volatility and goes to 30% volatility linearly, which is usually associated with a calm equity market (note all volatilities here refer to the out-of-the-money put options). A flat volatility curve stays at 30% all the way, and an inverted curve starts at 50% volatility for the shortest option (a one-day option) and ends up at the 30% volatility for the one-year option with linear interpolation, i.e., a straight line between the two coordinate points.
Figures 1 and 2 show the inverted volatility curve requires almost twice as much premium for daily rebalancing as the flat volatility curve and almost three times as much as the steep volatility curve. For quarterly rebalancing, the inverted volatility curve creates a 60% higher cost than the flat volatility curve and the steep volatility curve creates a discount of 25%. In terms of higher frequency rebalancing, a flat volatility curve makes the total cost of quarterly hedging 127 bps compared with 95 bps for the steep volatility curve and 202 bps for the inverted volatility curve. In all three cases, the more frequent hedging is cheaper than locking in the one-year hedge.
But there is no such thing as a free lunch: As Figure 3 shows, the more frequent hedging has a lower delta than the longer-term hedge. Since the delta represents the sensitivity of the hedge to the movements in the underlying security, this analysis shows that the sensitivity of a hedge portfolio goes down as costs are lowered.
Intuitively, it is clear to see why this happens: First, more frequent hedging means hedges expire more frequently, so the daily hedge will have 1/365th of the notional amount expire each day. Second, the shorter hedges have lower delta. Since the cumulative probability of a large market move is much lower in a shorter time period, the hedge is theoretically less likely to move into an “in-the-money” position. Of course, one might argue that if event risk is higher in the short term, using cheaper short-term hedges might be the better approach. This short-term approach is a reflection of an “active” view on the level of the volatility term structure – and an approach one might want to take around the time of a major policy decision, election or any similar event with potential unpriced jump risk.
Figure 4 illustrates the ratio of the potency, or deltas, of the various frequencies of hedges relative to the benchmark one-year hedge. Obviously, the quarterly delta ratio of 0.86 for the inverted volatility curve is higher than the quarterly delta ratio of 0.47 when the volatility curve is steep and upward-sloping.
Bringing it all together
To bring all these discussions about trade-offs together, Figure 5 shows the cost for the same unit of delta (the delta-adjusted cost) under the three volatility curves and various hedging frequencies. In other words, if we want the same delta regardless of the frequency of hedging, what would the total equivalent cost be? The conclusion is that rebalancing more frequently than once a year offers some benefit under all volatility curves (ignoring transactions costs) – but the benefits are greatest when the volatility curve is flat or steep.
Also, under all scenarios, new extension hedges have to be added to replace hedges when they expire. Since the normal volatility term structure is expected to appear more than 75% of the time (inversions are unusual and are typically associated with large downside market events), we generally expect to improve hedging efficiencies by moving to a reasonable (quarterly) hedging program.
This also highlights two other major points that we think any tail hedging client should consider: One, hedging has to be a systematic, repeated, asset allocation decision to obtain best long-term benefits, and two, the hedge program has to be active and consider pricing levels so efficient rebalancing can be implemented.