*This post is available as a PDF download here.*

## Summary

- The March 2020 equity market sell-off has caused many investors to re-investigate the potential benefits of tail risk hedging programs.
- Academic support for these programs is quite limited, and many research papers conclude that the cost of implementation for naïve put strategies out-weighs the potential payoff benefits.
- However, many of these studies only consider strategies that hold options to expiration. This means that investors can only profit from damage assessed. By rolling put options prior to expiration, investors can profit from damage
- In this research note we demonstrate that holding to expiration is not a required feature of a successful tail hedging program.
- Furthermore, we demonstrate that once that requirement is lifted, the most valuable component of a tail risk hedging program may not actually be the direct link to damage assessed, but rather the ability to profit in a convex manner from the market’s re-pricing of risk.

*“To hedge, or not to hedge, that is the question.”*

Nothing brings tail risk management back to the forefront of investors’ minds like a market crisis. Despite the broad interest, the jury is still out as to the effectiveness of these approaches.

Yet if an investor is subject to a knock-out barrier – i.e. a point of loss that creates permanent impairment – then insuring against that loss is critical. This is often the case for retirees or university endowments, as withdrawal rates increase non-linearly with portfolio drawdowns. In this case, the question is not whether to hedge, but rather about the most cost-effective means of hedging.

Some academics and practitioners have argued that put-based portfolio protection is prohibitively expensive, failing to keep pace with a simple beta-equivalent equity portfolio. They also highlight that naïve put strategies – such as holding 10% out-of-the-money (“OTM”) puts to expiration – are inherently path dependent.

Yet empirical evidence may fail us entirely in this debate. After all, if the true probability and magnitude of tail events is unknowable (as markets have fat tails whose actual distribution is hidden from us), then prior empirical evidence may not adequately inform us about latent risks. After all, by their nature, tail events are rare. Therefore, drawing any informed conclusions from tail event data will be shrouded in a large degree of statistical uncertainty.

Let us start by saying that the goal of this research note is not to prove whether tail risk hedging is or is not cost effective. Rather, our goal is to demonstrate some of the complexities and nuances that make the conversation difficult.

And this piece will only scratch the surface. We’ll be focusing specifically on buying put options on the S&P 500. We will not discuss pro-active monetization strategies (i.e. conversion of our hedge into cash), trade conversion (e.g. converting puts into put spreads), basis risk trades (e.g. buying calls on U.S. Treasuries instead of puts on equities), or exchanging non-linear for linear hedges (e.g. puts for short equity futures).

Given that we are ignoring all these components – all of which are important considerations in any actively managed tail hedging strategy – it does call into question the completeness of this note. While we hope to tackle these topics in later pieces, we highlight their absence specifically to point out that tail risk hedging is a highly nuanced topic.

So, what *do *we hope to achieve?

We aim to demonstrate that the path dependency risk of tail hedging strategies may be overstated and that the true value of deep tail hedges emerges not from the actual insurance of loss but the rapid repricing of risk.

## A Quantitative Aside

Options data is notoriously dirty, and therefore the results of back testing options strategies can be highly suspect. In this note, rather than price our returns based upon historical options data (which may be stale or have prohibitively wide bid/ask spreads), we fit a volatility surface to that data and price our options based upon that surface.

Specifically, each trading day we fit a quadratic curve to log-moneyness and implied total variance for each quoted maturity. This not only allows us to reduce the impact of dirty data, but it allows us to price any strike and maturity combination.

While we limit ourselves only to using listed maturity dates, we do stray from listed strikes. For example, in quoting a 10% out-of-the-money put, rather than using the listed put option that would be closest to that strike, we just assume the option for that strike exists.

This approach means, definitively, that results herein were not actually achievable by any investor*.* However, since we will be making comparisons across different option strategy implementations, we do not believe this is a meaningful impact to our results.

To reduce the impacts of rebalance timing luck, all strategies are implemented with overlapping portfolios. For example, for a strategy that buys 3-month put options and holds them to maturity would be implemented with three overlapping sub-portfolios that each roll on discrete 3-month periods but do so on different months.

Finally, the indices depicted herein are designed such that they match notional coverage of the S&P 500 (e.g. 1 put per share of S&P 500) when implemented as a 100% notional overlay and rebalanced monthly upon option expiration.

## The Path Dependency of Holding to Expiration

One of the arguments often made against tail hedging is the large degree of path dependency the strategy can exhibit. For example, consider an investor who buys 10% OTM put options each quarter. If the market falls less than 10% each quarter, the options will provide no protection. Therefore, when holding to expiration, we need drawdowns to precisely coincide with our holding period to achieve maximum protection.

But is there something inherently special about holding to expiration? For popular indices and ETFs, there are liquid options markets available, allowing us to buy and sell at any time. What occurs if we roll our options a month or two before expiration?

Below we plot the results of doing precisely this. In the first strategy, we purchase 10% OTM puts and hold them to expiration. In the second strategy, we purchase the same 10% OTM puts, but roll them a month before expiration.

*Source: DiscountOptionsData.com. Calculations by Newfound Research. **Returns are hypothetical and backtested. Returns are gross of all fees including, but not limited to, management fees, transaction fees, and taxes. Returns assume the reinvestment of all distributions.*

We see nearly identical long-term returns and, more importantly, the returns during the 2008 crisis and the recent March turmoil are indistinguishable. And we outright skipped holding each option for 1/3^{rd} of its life!

Our results seem to suggest that the strategies are less path dependent than originally argued.

An alternative explanation, however, may be that during these crises our options end up being so deep in the money that it does not matter whether we roll them early or not. One way to evaluate this hypothesis is to look at the rolling delta profile – how sensitive our option strategy is to changes in the underlying index – over time.

*Source: DiscountOptionsData.com. Calculations by Newfound Research. *

We can see is that during calm market environments, the two strategies exhibit nearly identical delta profiles. However, in 2008, August 2011, Q4 2018, and March 2020 the delta of the strategy that holds to expiration is substantially more negative. For example, in October 2008, the strategy that holds to expiration had a delta of -2.75 whereas the strategy that rolls had a delta of -1.77. This means that for each 1% the S&P 500 declines, we estimate that the strategies would gain +2.75% and +1.77% respectively (ignoring other sensitivities for the moment).

Yet, despite this added sensitivity, the strategy that holds to expiration does not seem to offer meaningfully improved returns during these crisis periods.

*Source: DiscountOptionsData.com. Calculations by Newfound Research. *

Part of the answer to this conundrum is theta, which measures the rate at which options lose their value over time. We can see that during these crises the theta of the strategy that holds to expiration spikes significantly, as with little time left the value of the option will be rapidly pulled towards the final payoff and variables like volatility will no longer have any impact.

What is clear is that delta is only part of the equation. In fact, for tail hedges, it may not even be the most important piece.

## Convexity in Volatility

To provide a bit more insight, we can try to contrive an example whereby we know that ending *in the money* should not have been a primary driver of returns.

Specifically, we will construct two strategies that buy 3-month put options and roll each month. In the first strategy, the put option will just be 10% OTM and in the second strategy it will be 30% OTM. As we expect the option in the second strategy to be significantly cheaper, we set an explicit budget of 60 basis points of our capital each month.^{1}

Below we plot the results of these strategies.

In March 2020, the 10% OTM put strategy returned 13.4% in and the 30% OTM put strategy returned 39.3%. From prior trough (February 19^{th}) to peak (March 23^{rd}), the strategies returned 18.4% and 46.5% respectively.

This is a stark difference considering that the 10% OTM put was definitively in-the-money as of March 20^{th} (when it was rolled) and the 30% OTM strategy was on the cusp. Consider the actual trades placed:

- 10% OTM Strategy: Buy a 3-month 10% OTM put on February 21
^{st}and sell a 2-month 23.3% ITM put on March 20^{th}. When bought, the option had an implied volatility of 20.9% and a price of $45.45^{2}; when sold it had an implied volatility of 39.5% and a price of $1428.21 for a 3042% return. - 30% OTM Strategy: Buy a 3-month 30% OTM put on February 21
^{st}and sell a 2-month 1.4% ITM put on March 20^{th}. When bought, the option had an implied volatility of 35.0% and a price of $5.42; when sold it had an implied volatility of 53.8% and a price of $425.85 for a 7757% return.

It is also worth noting that since we are spending a fixed budget, we can buy 8.38 contracts of the 30% OTM put for every contract of the 10% OTM put.

So why did the 30% OTM put appreciate so much more? Below we plot the position scaled sensitivities (i.e. dividing by the cost per contract) to changes in the S&P 500 (“delta”), changes in implied volatility (“vega”), and their respective derivatives (“gamma” and “volga”).

*Source: DiscountOptionsData.com. Calculations by Newfound Research. *

We can see that as of February 21st, the sensitivities are nearly identical for delta, gamma, and vega. But note the difference in volga.

What is volga? Volga tells us how much the option’s sensitivity to implied volatility (“vega”) changes as implied volatility itself changes. If we think of vega as a kind of velocity, volga would be acceleration.

A positive vega tells us that the option will gain value as implied volatility goes up. A positive volga tells us that the option will gain value at an accelerating rate as implied volatility goes up. Ultimately, this means the price of the option is convex with respect to changes in implied volatility.

So as implied volatilities climbed during the March turmoil, not only did the option gain value due to its positive vega, but it did so at an accelerating rate thanks to its positive volga.

Arguably *this *is one of the key features we are buying when we buy a deep OTM put.^{3} We do not need the option to end in the money to provide a meaningful tail hedge; rather, the value is derived from large moves in implied volatility as the market re-prices risk.

Indeed, if we perform the same analysis for September and October 2008, we see an almost identical situation.

*Source: DiscountOptionsData.com. Calculations by Newfound Research. *

## Conclusion

In this research note, we aimed to address one of the critiques against tail risk hedging: namely that it is highly path dependent. For naively implemented strategies that hold options to expiration, this may be the case. However, we have demonstrated in this piece that holding to expiration is not a necessary condition of a tail hedging program.

In a contrived example, we explore the return profile of a strategy that rolls 10% OTM put options and a strategy that rolls 30% OTM put options. We find that the latter offered significantly better returns in March 2020 despite the fact the options sold were barely in the money.

We argue that the primary driver of value in the 30% OTM put is the price convexity it offers with respect to implied volatility. While the 10% OTM put has positive sensitivity to changes in implied volatility, that sensitivity does not change meaningfully as implied volatility changes. On the other hand, the 30% OTM put has both positive vega *and *volga, which means that vega will increase with implied volatility. This convexity makes the option particularly sensitive to large re-pricings of market risk.

It is common to think of put options as insurance contracts. However, with insurance contracts we receive a payout based upon damage assessed. The key difference with options is that we have the ability to monetize them based upon potential damage *perceived. *When we remove the expectation of holding options into expiration (and therefore only monetizing damage assessed), we potentially unlock the ability to profit from more than just changes in underlying price.

## Option-Based Trend Following

By Nathan Faber

On June 23, 2020

In Risk Management, Trend, Weekly Commentary

This post is available as a PDF download here.## Summary

The non-linear payoff of trend following strategies has many similarities to options strategies, and by way of analogy, we can often gain insight into which market environments will favor trend following and why.

In our previous research piece, Straddles and Trend Following, we looked at purchasing straddles – that is, a call option and a put option – with a strike price tied to the anchor price of the trend following model. For example, if the trend following model invested in equities when the return over the past 12 months was positive, for a security that was at $100 12-months ago and is at $120 today, we would purchase a call and a put option with a strike price of $100. In this case, the call would be 20% in-the-money (ITM) and the put would be out-of-the-money (OTM).

In essence, this strategy acted like an insurance policy where the payout was tied to a reversion in the trend signal, and the premium paid when the trend signal was strong was small.

This concept of insurance is an important discussion topic in trend following strategies. The risk we must manage in these types of strategies, either directly through insurance or some other indirect means like diversification, is whipsaw.

In this commentary, we will construct an options strategy that is similar to a trend following strategy. The option strategy will pay a premium

up-frontto avoid whipsaw. By comparing this strategy to trend following that bears the full risk of whipsaw, we can set a better practical bound for how much investors should expect to pay or earn for bearing this risk.## Methodology and Data

For this analysis, we will use the S&P 500 index for equity returns, the 1-year LIBOR rate as the risk-free rate, and options data on the S&P 500 (SPX options).

To bridge the gap between practice and abstraction, we will utilize a volatility surface calibrated to real option data to price options. We will constrain our SPX options to $5 increments and interpolate total implied variance to get prices for options that were either illiquid or not included in the data set.

For the most part, we will stick to options that expire on the third Friday of each month and will mention when we deviate from that assumption.

The long/short trend equity strategy looks at total returns of equities over 12 months. If this return is positive, the strategy invests in equities for the following month. If the return is negative, the strategy shorts equities for the following month and earns the risk-free rate on the cash. The strategy is rebalanced monthly on the options expiration dates.

For the option-based trend strategy, on each rebalance date, we will purchase a 1-month call if the trend signal is positive or a put if the trend signal is negative. We will purchase all options at-the-money (ATM) and hold them to expiration. The strategy is fully cash-collateralized. Any premium is paid on the options roll date, interest is earned on the remaining account balance, and the option payout is realized on the next roll date.

Why are we now using ATM options when previous research used ITM and OTM options, potentially deeply ITM or OTM?

Here we are looking to isolate the cost of whipsaw in the premium paid for the option while earning a payout that is close to that of the underlying in the event that our trend signal is correct. If we utilized OTM options, then our premium would be lower but we would realize smaller gains if the underlying followed the trend. ITM options would have downside exposure before the protection kicked in.

We are also not using straddles since we do not want to pay extra premium for the chance to profit off a whipsaw. The underlying assumption here is that there is value in the trend following signal. Either strategy is able to capitalize on that (i.e. it’s the control variable); the strategies primarily differ in their treatment of whipsaw costs.

## The High Cost of ATM Options

The built-in whipsaw protection in the options does not come cheap. The chart below shows the –L/S trend following strategy–, the –option-based trend strategy–, and the ratio of the two (dotted).

Source: DiscountOptionsData.com. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees including, but not limited to, management fees, transaction fees, and taxes. Returns assume the reinvestment of all distributions.During normal market environments and even in prolonged equity-market drawdown periods like 2008, trend following outperformed the option-based strategy. Earning the full return on the underlying equity is generally beneficial.

However, something that is “generally beneficial” can be erased very quickly. In March 2020, the trend following strategy reverted back to the level of the option-based strategy. If you had only looked at cumulative returns over those 15 years, you would not be able to tell much difference between the two.

The following chart highlights these tail effects.

Source: DiscountOptionsData.com. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees including, but not limited to, management fees, transaction fees, and taxes. Returns assume the reinvestment of all distributions.In most months, the option-based strategy forfeits its ~1.5% premium for the ATM option. The 75

^{th}percentile cutoff is 2.2% and the 90^{th}percentile cutoff is 2.9%. These premiums have occasionally spiked to 6-7%.While these premiums are not always forfeited without some offsetting gain, they are always paid relative to the trend following strategy.

A 3% whipsaw event in trend should definitely not be a surprise based on the typical up-front cost of the option strategy.

Source: DiscountOptionsData.com. Calculations by Newfound Research.## But What About a 30% Whipsaw?

Now that’s a good question.

Up until March 2020, for the 15 years prior, the largest whipsaws relative to the options strategy were 12-13%. This is the epitome of tail risk, and it can be disheartening to think that now that we have seen 30% underperformance, we should probably expect more at some point in the (hopefully very distant) future.

However, a richer sample set can shed some light on this very poor performance.

Let’s relax our assumption that we roll the options and rebalance the trend strategies on the third Friday of the month and instead allow rebalances and rolls on any day in the month. Since we are dealing with one-month options, this is not beyond implementation since there are typically options listed that expire on Monday, Wednesday, and Friday.

The chart below shows all of these option strategies and how large of an effect that roll / rebalance timing luck can have.

Source: DiscountOptionsData.com. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees including, but not limited to, management fees, transaction fees, and taxes. Returns assume the reinvestment of all distributions.With timing luck in both the options strategies and trend following, there can be large effects when the luck cuts opposite ways.

The worst returns between rebalances of trend following relative to each options strategy highlight how bad the realized path in March 2020 truly was.

In many of the trend following and option strategies pairs, the worst underperformance of trend following over any monthlong period was around 10%.

Returning to the premise that the options strategies are analogous to trend following, we see the same effects of timing luck that we have explored in previous research: effects that make comparing variants of the same strategy or similar strategies more nuanced. Whether an option strategy is used for research, benchmarking, or active investing, the implications of this timing luck should be taken into account.

But even without taking a multi-model approach at this point to the options strategy, can we move toward a deeper understanding of when it may be an effective way to offset some of the risk of whipsaw?

## I’d Gladly Pay You Tuesday for a Whipsaw Risk Today

With the two extremes of paying for whipsaw up front with options and being fully exposed to whipsaw through trend following, perhaps there is a way to tailor this whipsaw risk profile. If the risk of whipsaw is elevated but the cost of paying for the insurance is cheap, then the options strategy may be favorable. On the other hand, if option premiums are high, trend following may more efficiently capture the market returns.

The price of the options (or their implied volatilities) is a natural place to start investigating this topic since it encapsulates the premium for whipsaw insurance. The problem is that it may not be a reliable signal if there is no barrier to efficiency in the options market, either behavioral or structural.

Comparing the ATM option implied volatilities with the trend signal (12-month trailing returns), we see a negative correlation, which indicates that the options-based strategy will have a higher hurdle rate of return in strongly downtrending market environments.

Source: DiscountOptionsData.com. Calculations by Newfound Research.But this is only one piece of the puzzle.

Do these implied volatilities relate to the forward 1-month returns for the S&P 500?

Based on the above scatterplot: not really. However, since we are merely sticking implied volatility in the middle of the trend following signal and the forward return, and we believe that trend following works over the long run, then we must believe there is some relationship between implied volatility and forward returns.

While this monthly trend following signal is directionally correct over the next month 60% of the time, historically, that says nothing about the magnitude of the returns based on the signal.

Without looking too much into the data to avoid overfitting a model, we will set a simple cutoff of 20% implied volatility. If options cost more than that, we will utilize trend following. If they cost less, we will invest in the options strategy.

We will also compare it to a 50/50 blend of the two.

The switching strategy (gray line) worked well until around 2013 when the option prices were cheap, but the risk of whipsaw was not realized. It did make it through 2015, 2016 and 4Q 2018 better than trend following.

When viewed in a broader context of a portfolio, since these are alternative strategies, it does not take a huge allocation to make a difference. These strategies manage equity risk, so we can pair them with an allocation to the S&P 500 (SPY) and see how the aggregate statistics are affected over the period from 2005 to April 2020.

The chart below plots the efficient frontiers of allocations to 100% SPY at the point of convergence on the right of the graph) to 40% SPY on the left of the graph with the remainder allocated to the risk- management strategy.

The Sharpe ratio is maximized at a 35% allocation to the switching strategy, a 25% allocation to the option-based strategy, and 10% for the trend following strategy.

## Conclusion

In this research note, we explored the link between trend following and options strategies using 1-month ATM put and call options, depending on the sign of the trend.

The cost of ATM options Is generally 1.5% of the portfolio value, but the fact that this cost can spike upwards of 9% should justify larger whipsaws in trend following strategies. Very large whipsaws, like in March 2020, not only show that the cost can be seemingly unbounded but also that there is significant exposure to timing luck based upon the option roll dates.

Then, we moved on to investigating a simple way to allocate between the two strategies based upon the cost of the options, When the options were cheap, we used that strategy, and when they were expensive, we invested in the trend following strategy. A modest allocation is enough to make a different in the realized efficient frontier.

Deciding to pay the up-front payment of the whipsaw insurance premium, bear the full risk a whipsaw, or land somewhere in between is largely up to investor preferences. It is risky to have a large downside potential, but the added benefit of no premiums can be enough to offset the risk.

An implied volatility threshold was a rather crude signal for assessing the risk of whipsaw and the price of insuring against it. Further research into one or multiple signals and a robust process for aggregating them into an investment decision is needed to make more definitive statements on when trend following is better than options or vice versa. The extent that whipsaw can be mitigated while still maintaining the potential to earn diversified returns is likely limited, but the optimal blend of trend following and options can be a beneficial guideline for investors to weather both sudden and prolonged drawdowns.