Why so Serious ? Smile with Crypto Options!
This article is the second in a series co-authored by Gerardo Lemus from Quanto and Kumar Suppiah of Project Jarvis. Project Jarvis is a pre-incorporation stealth mode digital asset trading startup supported by Quanto.
Gerardo does the quantitative heavy lifting while Kumar tries to break it down into layman’s terms with a dash of dry humor. Our co-authored articles are published by each individual author and you may find it reproduced here.
It is a universally acknowledged truth that any serious article worth its salt should eventually contain a cat meme (How Cats took over the Internet), and as a proof that even quantitative finance or crypto assets are not immune to this rule just take a look at the following link between Cheshire Cats and Black Swans:
We made it a point to publish our analysis after the expiry so that our article would not be construed as investment advice in any context. One can never be too careful with the hype surrounding cryptoassets.
In a previous blog (Black Crypto Swans) we clearly uncovered so many Black Swans in the recent past of a cryptoasset that any illusion that they can be modeled as log-normal random walks should have been completely shattered: Fat tails were prevalent.
The fat tails that we found in the historical data explain the ‘smile’: the price of the options (think insurance) of the tails is much higher than that of an option in the middle to cover for drops of prices (investors seem to be very worried of the prices falling even further). In other words, the world in general or crypto option traders at least, are worried that a sharp drop in price is more likely at expiration rather than a movement in price clustered around the current price.
The beauty of options is that we can use them to estimate the (risk-neutral) implied probabilities, which can tell us the greed and fear of crypto investors.
A quick note here: risk-neutral is mathematical speak for ‘cost of insurance’ versus ‘probability of happening’. Think of it this way: the price of fire insurance on your house has an implied probability of a total loss (let’s say 0.1%) which is expectedly different from the real-life probability of it actually happening. It is called ‘risk-neutral’ because the theory to obtain the price assumes there is a hedging strategy to replicate the option (hence, there is no ‘risk’ being taken).
Note: there is a method to determine the ‘probability of happening’ but it requires math and many option prices well beyond what this blog can do (and prices available in the market) — you can read more here: Ross recovery theorem.
With that in mind, we can compute implied range probabilities (roughly the probability of a return being within the range) — and which looks a lot like the histograms we obtained in the Black Swan blog. Notice the ‘fat tails’ and the implied probability of a large negative return. (The link above is the ‘back of the envelope practitioners’ formula — for the rigorous mathematical version read this paper, section 4.0.1)
(Note: as usual, you can find the code to replicate the work below — but it is worthwhile to mention that we run this specific result for the 29 March 2019 option expiry when the time to expiration was 12 days)
In the Black Swan blog, we mentioned that cumulative distribution functions could help us determine the probability of falling below a certain return. We can also get them from the options pricing page:
Do you remember in the past blog that we said ‘past performance is not an indicator of future results’? Well,
- Risk-neutral implied probabilities are not an indicator of a future result but …
- … options can be used to hedge the undesired event. Are you worried about a larger than 20% crypto fall? buy insurance and sleep at night. (The guy who sold you the insurance, on the other hand, might need some help sleeping.)
Just as the fire insurance, the price of the insurance implies a probability which does not really forecast the ‘real life’ probability of the event happening.
Time Scaling Hack
Option date intervals tend to fall days or weeks away from ‘now’; how can they help us if what we care are the following 5 minutes?
A known property of Brownian motion is that the volatility scales with the square root of time which allows us to scale up risk by multiplying (or dividing) the volatility of a known period of time by the square root.
Above we had a ‘fat tail’ event of a maximum loss of 45% that could happen in 12 days. To get the maximum loss for the next minute, we could divide it by the square root of the total number of trading minutes in 12 days:
45% / sqrt( 12 days x 24 hours x 60 minutes) = 0.34%
In our black swan blog, we found a day when the maximum loss was well beyond 0.34%: it was close to 2.5%
The square root hack underestimates risk: I call it hack in the ‘a tool for rough cutting’ sense. There are tons of reasons why — read “On time-scaling of risk
and the square–root–of–time rule” for an academic perspective, but remember that we already found that black swans disproved the Brownian motion model. There are no standard models readily available for fat-tailed processes, or time-varying volatility.
What to do then? Always be on the search for black swans, and question established formulas and use them with a ‘pinch of salt’ if there are no alternatives.
As we get closer to the expiry date the square root approximation works better. Take a look at this example taken only 2 days before the 29 expires. The blue line represents the realized fat tails cumulative probability distribution (historical realized volatility) done on a minute by minute return for the past month, the red line the equivalent ‘normal’ result (which awfully underestimates the tails) …
… but the green dots represent the values using all the available data (option implied probability) — with only two days left the probability of a loss (a negative return) distribution is well represented, but the optimistic probability of a gain is unbelievable: some people think a gain of more than 17.5% in one minute is possible!
(additional works need to be done to assure the green dots follow the properties of a cdf — monotonically increasing and always between 0 and 1 — the dots that violate this rule are artifacts from the bid/ask spread)
Use implied probabilities for risk … but with a pinch of salt
We at Project Jarvis with the help of our friends at Quanto use such quantitative tools to design our risk management framework but we also recognize, these tools are no magic silver bullets and there is no substitute for well-designed money management discipline. Remember, remember the crash of November (See what we did there with our pun…)
If you’re an aspiring quantitative cryptoasset trader, please do your research on the limitations of any techniques ported over from traditional asset classes and be careful with how much you risk. Otherwise, you’re probably better off sticking to your day job, HODLing and leaving the management of your cryptoassets to professionals as the cryptoasset management industry matures.
Gist: