Updated: Jun 9

One of the hallmarks of the COVID-19 crisis has been extreme uncertainty about how the pandemic will play out: will there be a V- or U-shaped recovery, will there be a second wave, and if so, when, how the equity markets will sell off if there’s a second wave, and more. In the early days of the crisis there was even less clarity about immediate economic impact.

Options markets give an interesting window into the market’s expectations of future scenarios. This post gives a few examples of those insights, and highlights how important it is to have a flexible risk system that can respond to rapidly-changing market conditions like these.

**W-Shaped Implied Volatilities**

Normally implied volatility by strike has a fairly simple shape: U-shaped to give smile, and tilted to give a skew. Here is an example from February 13, 2020 closing S&P 500 (SPX) options data, just before the COVID-19 crisis, for February 26, 2020 expiration, two weeks later. There are two lines here: the orange is market implied volatilities strike by strike, and blue is fitted implied volatility based on a cubic spline to smooth out noise with individual strikes; in normal markets these are very close.

In the COVID-19 crisis the options markets showed more complex shapes, like W-shapes, in addition to a much higher overall level of volatility. Here are data for SPX on the April 3, 2020 close – near the height of the pandemic in New York – for the April 17, 2020 expiration (also about two weeks to expiry):

In this case the implied volatility by strike data is noisier due to less liquidity, so the spline fit is more important for smoothing and getting stable pricing and risk analytics. But the negative convexity vs strike is clear in the 2,300-2,500 strike region.

This W shape is relatively rare in equity index options markets, but does occur sometimes in options markets for individual equities before earnings announcements – and for similar reasons: the market is predicting that the underlying price will jump up or down but probably not stay where it is.

**The Market Expects One of Two States**

This W shape is an example of the market making a statement about future realization of the SPX index price. We can see this more clearly if we look at the “risk neutral probability density function”, which shows us what the options market predicts about the probability of the SPX price being at different levels on the option expiration:

The height of the function is related to the probability that the SPX index level ends at that point. The probability density function is telling us an interesting story: that options traders expect one of two qualitatively different outcomes for the SPX over the subsequent two weeks until expiration, one where the market sells off to around 2,400, and another where it rallies to around 2,600, but that it’s less likely to stay around where it currently is.

This kind of probability density function does not arise in standard derivatives pricing models, like geometric Brownian motion (which underlies the Black-Scholes model), or standard extensions like local volatility and stochastic volatility. Instead it suggests something more like a model where the equity price can suddenly jump into one of two possible states and then subsequently diffuse.

Other markets outside equities show similar behavior; if we look at the WTI crude oil futures option market on the same April 3, 2020 date, for the June 2020 option expiration, we see a similar W shape for implied volatilities and a two-peaked shape for the probability densities, if not as dramatic as for SPX:

**Second Wave COVID-19 Risk**

The examples above were from the height of the crisis, where there was significant short-term uncertainty in market outcome. Later in the crisis the market reflected the improved information about the short term evolution of the crisis while noting the downside risk from a second wave.

Here are implied volatilities and probability densities from the Apr 27, 2020 close, for May 13, 2020 expiration – again, roughly two weeks expiration:

There is still some mild W shape to the implied volatilities, and some hint of two closely-overlapping peaks in the probability densities, but implied volatilities look much closer to pre-crisis volatilities, except for the overall higher level of volatility.

When we turn to a more distant expiration, however, we see some of that structure start to reappear in the options data. Here are implied volatilities and probability densities for the March 31, 2021 expiration – about eleven months to expiry – on that same April27, 2020 close:

There is a significant amount of probability density for much lower index prices, around 1,500, that reflects the possibility that a second wave of COVID-19 causes a new round of market selloffs.

**You Need a Flexible Risk System**

Non-standard market conditions like those described above often cause real problems for portfolio managers when the markets violate assumptions baked into their risk systems.

As an example: a standard algorithm for implied volatility interpolation is “SVI”, which stands for “stochastic volatility inspired”. It is a five-parameter function, and you tune the five parameters to best fit the market-implied volatilities. Many risk systems use SVI for volatility interpolation because it usually gives a nice match to the overall shape of implied volatility and has a simple closed form that makes implied volatility interpolation calculations very fast.

However, SVI cannot fit all shapes – in particular, it is too rigid to match W-shaped implied volatilities. Here are best-fit volatilities for that SPX April 3, 2020 close, for April 17, 2020 expiration:

The blue line shows the best-fit SVI implied volatilities, calibrated most closely to near-the-money options – the SVI best fit deviates significantly from the market (and the spline fit) for low strike options especially, and does not show any material W shape.

These problems with SVI interpolation highlight how important it is to have a flexible risk system. For the analysis described above, we used the Beacon platform, which lets you toggle between different volatility interpolation schemes, and quickly construct new ones when market conditions warrant. **If you have some feedback or want to learn more about Beacon, let me know!**

**About the Author**

*Before co-founding Beacon in 2014, Mark Higgins spent eight years at JP Morgan, launching and delivering the Athena project, co-heading the quantitative research group for the investment bank and running the electronic market-making business for currency options. Prior to JP Morgan he spent eight years at Goldman Sachs as a desk strategist on the foreign exchange and interest rate market-making desks.*

*Beacon Platform, Inc.*

*Mark Higgins, Chief Operating Officer,*

*+1 917 721 8150, mark.higgins@beacon.io*

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