## COVID-19 Crash: Tail Hedge and the Convexity within IV

Put options are one of the main tools used in hedging against a black-swan event, a market crash with unpredictable cause and severe impact.

The recent COVID-19 market crash where the SPY crashed 33% from 340 to 228 is an example of a black-swan event. Unpredictable with severe impact. Almost all the put options purchased before from the crash profited. Investors that had a tail hedge put option made bank from the convexity (more will be explained later).

Most investors know about convexity within the price movement of the underlying due to gamma. But little know that a similar mechanism lies within IV.

This blog will attempt to explain the convexity within IV.

We will first explore about delta and gamma, which are two measurements used to understand how the price of the option change according to the underlying’s price.

Delta measures the change in option price for every $1 increase of the underlying. Gamma measures the change in delta for every$1 increase of the underlying.

An analogy that might help is seeing delta as speed or velocity and gamma as acceleration. And to build on this analogy, the option’s price is the distance travelled.

Gamma increases as the underlying heads towards the strike price (first image). The increase of gamma is what causes the value of the option to increase convexly as it the price of the underlying increases.

(Second image) The value of the option (in red) does not increase linearly with the underlying. Instead the value of the option has a somewhat ‘exponential-ish’ relation to the price of underlying.

That is to say that not every $1 increase in the underlying is the same. When the option is deep OTM, an increase in the underlying’s price might result in$0.05 increase for the option’s value. But when ATM or ITM, the $1 increase in the underlying’s price might result in$0.50 in the option value.

The ‘curveness’ of the option value in the second image is known as convexity.

This convexity is what makes deep OTM put option appealing as a tail hedge. But what if the investor’s had placed his strike price too far out?

During the crash when SPY dipped 33% from the peak to bottom. What if he had a put option that was 40% OTM? Would he not benefit from convexity, the main selling point of a tail hedge?

There is another mechanism brings out the convexity in an option.

## Convexity within IV

During a market crash, the option’s implied volatility (IV) would spike and the price of the option would increase due to vega.

Vega is a measurement of increase in the option’s value for a 1% increase in IV. Similar to how delta is not constant, vega is also not constant.

Vega increases faster the higher the IV. A 1% increase in IV from 40% to 41% would result in the option’s value increasing more than from 20% to 21%.

Vomma is used to measure the increase in vega for a 1% increase in IV.

And in a black-swan event, the market crash is often quick and severe which causes a big IV spike which even would mean that an investor that invested in a 40% OTM put option would have also made a monstrous return and is very likely to have protected his portfolio.

To make things more intuitive, parallels could be drawn between IV and price of underlying.

Price of underlying           ->           IV

Delta                                   ->           Vega

Gamma                               ->           Vomma

## Analysis of Put Options

Strike PricePrice on 20 Feb 2020Price on 23 Mar 2020Return
$270 (10% OTM)$2.05$75.2736.7x$305 (20% OTM)$0.59$43.2073.2x
$240 (30% OTM)$0.25$24.8999.0x$205 (40% OTM)$0.11$11.99109.0x
Expiration: 15 May 2020
Strike PricePrice on 20 Feb 2020Price on 23 Mar 2020Return
$270 (10% OTM)$6.12$77.2012.6x$305 (20% OTM)$2.54$48.919.3x
$240 (30% OTM)$1.10$32.7129.7x$205 (40% OTM)$0.37$19.5252.8x
Expiration: 18 Sep 2020
Strike PricePrice on 20 Feb 2020Price on 23 Mar 2020Return
$270 (10% OTM)$11.27$79.507.1x$305 (20% OTM)$5.57$52.669.5x
$240 (30% OTM)$2.81$36.8013.1x$205 (40% OTM)$1.15$23.0820.1x
Expiration: 19 Mar 2021

*Data are from TD Ameritrade’s thinkorswim platform

### Strike Price

Across different option expiration dates, the strike price that performed the best was the one that was furthest OTM at 40% down from the market peak. In fact, from the peak to bottom of the crash, this option was never in the money. That is to say, delta and gamma was not the main factor to why its outperformance.

Vega and vomma worked hand-in-hand during an IV spike, bringing the price of the option up. But, here’s the catch, the price movement of the underlying moving towards the strike price also increases vega!

But, if that is all, it would still meant that although the 30% OTM put option would benefit more from vega as it would have the higher vega with the option being slightly ITM. (Put options expiring on 15 May 2020 is taken as example)

There are 2 reasons for this:

1. The spike in IV is smaller at 16% (from 32% to 48%) compared to the 22% (from 41% to 66%) for the 40% OTM put option’s . However, the difference in increase in IV is not drastically different, 22% vs 16%. However, remember vomma. The increment of IV at a higher IV would have a bigger impact on vega.
2. Difference in premium paid. The premium paid for the 40% OTM put option is less than half the price of the 30% OTM put option.

### Expiration Date

Now comes the question of buying long-dated or short-dated put option.

One of the important factor when buying deep OTM put options as a tail hedge, most of the time it would do nothing but lose value due to theta decay (losing value as it has less time to be ITM).

So, it is important to choose option with lower theta decay. And contrary to what some investors would think, theta decay does not speed up towards its expiration for OTM options. In fact, theta decay slows down when heading towards expiration.

Especially for tail-hedging where the options would be ‘bleeding’ the overall portfolio’s return, it is important to reduce the theta decay. Therefore, short-dated option (2-4 months) is better than a long-dated option (6months to a year) as it has less theta decay.

Furthermore, as explained before, options that have cheaper premium is more convex in nature.

And so, as a tail hedge it is better to go for short-dated deep OTM put option to reduce theta decay and increase convexity.

*Note that the option should be rolled forward monthly to keep it within the 2-4 months DTE.

## % Allocation to Tail Hedging

Although the event of the COVID-19 market crash is fresh on our minds, it is important not to over-allocate to the tail hedge in hopes of making it rich. Remember that the main purpose of a tail hedge is to protect your assets.

The allocation of the tail hedge should be sized accordingly to the equity position (and equity-correlated assets).

In the case of the COVID-19 crash, a 0.5% allocation to the tail hedge assuming a return of 70x would protect the entire portfolio of 99.5% stocks and 0.5% tail hedge. And most portfolio consist of bonds and other assets that are negatively correlated to stocks. In which case, the portfolio would even make a positive return during the COVID-19 crash.

A relatively conservative allocation to tail hedge would be around 0.5%. But then again, it should be weighted against the investors’ equity exposure.

Some other factors to consider are:

1. How much downside risk on equity are you willing to take on?
2. How much return are you willing to sacrifice annually?

## Exit Plan

Before entering a position, it is important to have a vague exit plan so that when the market crashes, investors would not be at a lost as to what to do.

What would the point of a tail hedge that when its purpose is fulfilled, investors are instead at a lost of what to do.

Below are some scenarios that the return of the tail hedge could be used for:

1. After a 10x return, re-balance the tail hedge back to its original allocation (maybe 0.5%)
2. Buy the dip in the equity (Anticipation of market recovery or FED propping up the market)
3. Keep the return in cash or deflationary assets like bonds (Long periods of low economic growth or even outright deflation like in Japan)
1. Store of value assets like gold or bitcoin that are limited by scarcity (Growing concerns of hyperinflation due to growing fiscal stimulus and growing M2 money supply)

Nobody are able to predict the market and what would be the ideal action to take. It is all about managing the portfolio so that it aligns with the investors’ market outlook.

My personal opinion is to never assume that you would be right and go ‘irresponsibly long’. After all, even if your thesis is right, if the timing is wrong then you are wrong.

Even in the popular movie The Big Shortif Michael Burry had been wrong for another few years, he could very well be driven to bankruptcy.

## Conclusion

Convexity of a deep OTM put option comes from both gamma and vomma.

There is no need to be afraid of buying put options with strike price 30% or more out of the money. Even does not end up ITM, the return due to IV spike could still protect investors’ overall portfolio.

The expiration date could be around 2-4 months out, so that it would provide the convexity which is an important criteria for a tail hedge.

During a steep market crash similar to COVID-19 situation, there are various ways to use the return from the tail hedge. Buying the dip, bonds, gold and bitcoin are all plausible depending on the investors’ market outlook.