A lot of things about Bitcoin defy conventional logic. For example, valuation of bitcoin can be treacherous and common sense ranges are a tad too wide for traditional economists or investors

^{1}. "Exceptionally ambiguous", so to speak. Then we have confusions as captured in the tweet below.Bitcoin investor: "I want to get rich really quickly. Oh and by the way, I think Bitcoin will replace fiat currencies." If you really wanted Bitcoin to replace fiat currencies, you would spend it on goods & services, not try to profit from it in fiat currency.— (((FrancesCoppola))) (@Frances_Coppola) December 12, 2017

The attitude of the crypto-currency enthusiasts is best understood in terms of Keynesian beauty contest. It is not important to have a specific valuation in mind to HODL Bitcoin. All you need is your estimates of what others might value it at. It is not important to start using Bitcoin to pay for your morning coffee to realize the dream of replacing fiat currencies. All you need is to believe that enough of other people believe in doing so. The entire thing is a second order guessing game.

Theoretically, this set-up makes for an interesting characteristics of Bitcoin price. Since there is hardly any well-accepted valuation, a sudden rally or sell-off will not see the typical stabilizing intervention we see from value investors in other asset classes. Also since most (if not entire) valuation depends on the collective average of investors' expectation, a rally will see further buying and vice versa. This means the time series of price will show intrinsic characteristics of a momentum strategy.

The chart below captures this feature. The chart on the left shows the empirical distribution of MA(1) coefficients for daily returns for Bitcoin (black) and S&P 500 (red). The MA(1) coefficients can be interpreted as the response to a random shock in price. A series showing momentum nature will tend to magnify these moves, i.e. will have higher response coefficient. As shown in the chart, the Bitcoin response is highly skewed towards the right compared to S&P. The chart on the right similarly shows the response to a previous trends (AR(1) coefficients here). Here again the Bitcoin distribution is skewed towards the right. It even crosses the 1.0 mark, the upper limit of stationarity - turning in to what is called explosive process occasionally

The second characteristics of Bitcoin is its fat tails. The chart on the left below shows standardized returns across different asset classes (Bitcoin - thick red, equities - black, FX - blues, gold - gold, commodities - purple, rates - green, inflation - brown, VIX - orange, credits - grey). The peaked value and fat tails for Bitcoin far exceeds other asset classes (closest is the option adjusted spreads of AA credit). The right hand chart shows an easy to understand metric for fatness of tails. This displays the ratios of mean absolute deviation to standard deviation. Since standard deviation is a square root of square measure, it is more sensitive to large tail values than mean absolute deviation. The ratio of them shows the fatness of tails. A lower ratio means fatter tails. As the distribution approaches normal distribution (no excess kurtosis), this ratio approaches 0.8. Here we see by far, the ratio for Bitcoin is the smallest. It also shows strong positive skew.

The above two observation makes Bitcoin quite a bit of unique asset class. Nevertheless, we tried to explore which asset class comes close to it, in terms of time series characteristics. Here is the outcome - where Bitcoin stands among different asset classes based on a few measures:

There are multiple ways of clustering time series. One is "shape-based" - e.g. measuring a geometric distance like Euclidean distance. The top left chart shows the asset class hierarchy based on this measure. Bitcoins in this respect behaves similar to gold - very distinct from the carry assets block (VIX and credits), FX, rates, equities and commodities block. On "structural" measure (bottom left), like based on time series correlation, Bitcoin teams up with the commodities block. Finally, based on complexity or information based models (top right and bottom right), Bitcoin reflects the distribution and fat tails characteristics discussed above and behaves more like AA spread or VIX. On the whole, Bitcoin does not appear to be either digital gold, or a currency at all. It is more like a commodity showing fat tails similar to credit spreads and volatility.

In fact this fatness of tails, along with the momentum characteristics discussed above makes Bitcoin less like an asset class, and more like a derivative. Momentum strategies are comparable to options. Most momentum trading strategies display two characteristics - many small losing trades and a few big winners. The big winners give rise to fat tails (with positive skew). And those small losing trades can be compared to carry cost of an option (theta cost or time value). This interpretation can be readily extended to Bitcoin as well. The good thing about Bitcoin is, given the upwards trends till date, the so-called carry cost has been positive!

Beyond the economists and valuation experts communities, I suspect most traders understand this optionality of Bitcoin by instinct. You invest an amount you are happy to lose, with a possibility of large upside - this is the optionality of Bitcoin without all these technical details. This says nothing about Bitcoin being a bubble or not. But a long position in Bitcoin in a positive trend (or a short position in a negative trend) is going to act like a long option position. A proper valuation of the Bitcoin, therefore, must include this embedded optionality.

Theoretically, this set-up makes for an interesting characteristics of Bitcoin price. Since there is hardly any well-accepted valuation, a sudden rally or sell-off will not see the typical stabilizing intervention we see from value investors in other asset classes. Also since most (if not entire) valuation depends on the collective average of investors' expectation, a rally will see further buying and vice versa. This means the time series of price will show intrinsic characteristics of a momentum strategy.

The chart below captures this feature. The chart on the left shows the empirical distribution of MA(1) coefficients for daily returns for Bitcoin (black) and S&P 500 (red). The MA(1) coefficients can be interpreted as the response to a random shock in price. A series showing momentum nature will tend to magnify these moves, i.e. will have higher response coefficient. As shown in the chart, the Bitcoin response is highly skewed towards the right compared to S&P. The chart on the right similarly shows the response to a previous trends (AR(1) coefficients here). Here again the Bitcoin distribution is skewed towards the right. It even crosses the 1.0 mark, the upper limit of stationarity - turning in to what is called explosive process occasionally

^{2}!The second characteristics of Bitcoin is its fat tails. The chart on the left below shows standardized returns across different asset classes (Bitcoin - thick red, equities - black, FX - blues, gold - gold, commodities - purple, rates - green, inflation - brown, VIX - orange, credits - grey). The peaked value and fat tails for Bitcoin far exceeds other asset classes (closest is the option adjusted spreads of AA credit). The right hand chart shows an easy to understand metric for fatness of tails. This displays the ratios of mean absolute deviation to standard deviation. Since standard deviation is a square root of square measure, it is more sensitive to large tail values than mean absolute deviation. The ratio of them shows the fatness of tails. A lower ratio means fatter tails. As the distribution approaches normal distribution (no excess kurtosis), this ratio approaches 0.8. Here we see by far, the ratio for Bitcoin is the smallest. It also shows strong positive skew.

The above two observation makes Bitcoin quite a bit of unique asset class. Nevertheless, we tried to explore which asset class comes close to it, in terms of time series characteristics. Here is the outcome - where Bitcoin stands among different asset classes based on a few measures:

There are multiple ways of clustering time series. One is "shape-based" - e.g. measuring a geometric distance like Euclidean distance. The top left chart shows the asset class hierarchy based on this measure. Bitcoins in this respect behaves similar to gold - very distinct from the carry assets block (VIX and credits), FX, rates, equities and commodities block. On "structural" measure (bottom left), like based on time series correlation, Bitcoin teams up with the commodities block. Finally, based on complexity or information based models (top right and bottom right), Bitcoin reflects the distribution and fat tails characteristics discussed above and behaves more like AA spread or VIX. On the whole, Bitcoin does not appear to be either digital gold, or a currency at all. It is more like a commodity showing fat tails similar to credit spreads and volatility.

In fact this fatness of tails, along with the momentum characteristics discussed above makes Bitcoin less like an asset class, and more like a derivative. Momentum strategies are comparable to options. Most momentum trading strategies display two characteristics - many small losing trades and a few big winners. The big winners give rise to fat tails (with positive skew). And those small losing trades can be compared to carry cost of an option (theta cost or time value). This interpretation can be readily extended to Bitcoin as well. The good thing about Bitcoin is, given the upwards trends till date, the so-called carry cost has been positive!

Beyond the economists and valuation experts communities, I suspect most traders understand this optionality of Bitcoin by instinct. You invest an amount you are happy to lose, with a possibility of large upside - this is the optionality of Bitcoin without all these technical details. This says nothing about Bitcoin being a bubble or not. But a long position in Bitcoin in a positive trend (or a short position in a negative trend) is going to act like a long option position. A proper valuation of the Bitcoin, therefore, must include this embedded optionality.

1. For example here is a rather ridiculous attempt. And a rather commonsensical overview.

2. Notice how the S&P AR(1) coefficients distribution gets truncated at 1.0, as it should be for stationary processes