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.

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 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.

## Tail Hedging

By Corey Hoffstein

On June 8, 2020

In Risk Management, Weekly Commentary

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

“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

dowe 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 moneyshould 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.

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 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:^{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.^{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

thisis 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

andvolga, 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.