Salviati: Simplicio, do you think the Bitcoin markets are efficient?

Simplicio: If you'd asked me two years ago, I would have said yes. I know hindsight is 20/20, but even at the time, I think the fact that relatively few people were trading it would have risen to prominence in my analysis.

Salviati: And what about today?

Simplicio: Today, it seems like there's no shortage of trading volume. The hedge funds of the world have heard of Bitcoin, and had their quants do their fancy analyses on it, and they actively trade it.

Salviati: Well, I'm certainly not a quant, but I think I've spotted a systematic market inefficiency. Would you like to hear it?

Simplicio: Nah, I'm good.

Salviati: Did you hear what I said? I think I've spotted an exploitable pattern of price movements in a $10 Billion market. If I'm right, it could make us a lot of money.

Simplicio: Sure, but you won't convince me that whatever pattern you're thinking of is a "reliable" one.

Salviati: Come on, you don't even know what my argument is.

Simplicio: But I know how your argument is going to be structured. First you're going to identify some property of Bitcoin prices in past data. Then you'll explain some causal model you have which supposedly accounts for why prices have had that property in the past. Then you'll say that your model will continue to account for that same property in future Bitcoin prices.

Salviati: Yeah, so? What's wrong with that?

Simplicio: The problem is that you are not a trained quant, and therefore, your brain is not capable of bringing a worthwhile property of Bitcoin prices to your attention.

Salviati: Dude, I just want to let you know because this happens often and no one else is ever going to say anything: you're being a dick.

Simplicio: Look, quants are good at their job. To a first approximation, quants are like perfect Bayesian reasoners who maintain a probability distribution over the "reliability" of

*every single property*of Bitcoin prices that you and I are capable of formulating. So this argument you're going to make to me, a quant has already made to another quant, and the other quant has incorporated it into his hedge fund's trading algorithms.Salviati: Fine, but so what if quants have already figured out my argument for themselves? We can make money on it too.

Simplicio: No, we can't. I told you I'm pretty confident that the market is efficient, i.e. anti-inductive, meaning the quants of the world haven't left behind any reliable patterns that an armchair investor like you can detect and profit from.

Salviati: Would you just shut up and let me say my argument?

Simplicio: Whatever, knock yourself out.

Salviati: Ok, here goes. Everyone knows Bitcoin prices are volatile, right?

Simplicio: Yeah, highly volatile. But at any given moment, you don't know if the volatility is going to move the price up or down next. From your state of knowledge, it looks like a random walk. If today's Bitcoin price is $1000, then tomorrow's price is as likely to be $900 as it is to be $1100.

Salviati: I agree that the Random Walk Hypothesis provides a good model of prices in efficient markets, and that the size of a each step in a random walk provides a good model of price volatility in efficient markets.

Simplicio: See, I told you you wouldn't convince me.

Salviati: Ah, but my empirical observation of Bitcoin prices is inconsistent with the Random Walk hypothesis. So the only thing I'm led to conclude is that the Bitcoin market is not efficient.

Simplicio: What do you mean "inconsistent"?

Salviati: I mean Bitcoin's past prices don't look much like a random walk. They look more like a random walk

*on a log scale*. If today's price is $1000, then tomorrow's price is equally likely to be $900 or $1111. So if I buy $1000 of Bitcoin today, I expect to have 0.5($900) + 0.5($1111) = $1005.50 tomorrow.Simplicio: How do you know that? Did you write a script to loop through Bitcoin's daily closing price on Mt. Gox and simulate the behavior of a Bayesian reasoner with a variable-step-size random-walk prior and a second Bayesian reasoner with a variable-step-size log-random-walk prior, and thus calculate a much higher Bayesian Score for the log-random-walk model?

Salviati: Yeah, I did.

Simplicio: That's very virtuous of you.

[This is a fictional dialogue. The truth is, I was too lazy to do that. Can someone please do that? I would much appreciate it. --Liron.]

Salviati: So, have I convinced you that the market is anti-inductive now?

Simplicio: Well, you've empirically demonstrated that the log Random Walk Hypothesis was a good model for predicting Bitcoin prices in the past. But that's just a historical pattern. My original point was that you're not qualified to evaluate which historical patterns are *reliable* patterns. The Bitcoin markets are full of pattern-annihilating forces, and you're not qualified to evaluate which past-data-fitting models are eligible for future-data-fitting.

Salviati: Ok, I'm not saying you have to believe that the future accuracy of log-Random-Walk will probably be higher than the future accuracy of linear Random Walk. I'm just saying you should perform a Bayesian update in the direction of that conclusion.

Simplicio: Ok, but the only reason the update has nonzero strength is because I assigned an a-priori chance of 10% to the set of possible worlds wherein Bitcoin markets were inefficient, and that set of possible worlds gives a higher probability that a model like your log-Random-Walk model would fit the price data well. So I update my beliefs to promote the hypothesis that Bitcoin is inefficient, and in particular that it is inefficient in a log-Random-Walk way.

Salviati: Thanks. And hey, guess what: I think I've traced the source of the log-Random-Walk regularity.

Simplicio: I'm surprised you waited this long to mention that.

Salviati: I figured that if I mentioned it earlier, you'd snap back about how efficient markets sever the causal connection between would-be price-regularity-causing dynamics, and actual prices.

Simplicio: Fair enough.

Salviati: Anyway, the reason Bitcoin prices follow a log-Random-Walk is because they reflect the long-term Expected Value of Bitcoin's actual utility.

Simplicio: Bitcoin has no real utility.

Salviati: It does. It's liquid in novel, qualitatively different ways. It's kind of anonymous. It's a more stable unit of account than the official currencies of some countries.

Simplicio: Come on, how much utility is all that really worth in expectation?

Salviati: I don't know. The Bitcoin economy could be anywhere from hundreds of millions of dollars, to trillions of dollars. Our belief about the long-term future value of a single BTC is spread out across a range whose 90% confidence interval is something like [$10, $100,000] for 1BTC.

Simplicio: Are you saying it's spread out over the interval [$10, $100,000] in a uniform distribution?

Salviati: Nope, it's closer to a bell curve centered at $1000 on a log scale. It gives equal probability of ~10% both to the $10-100 range and to the $10,000-100,000 range.

Simplicio: How do you know that everyone's beliefs are shaped like that?

Salviati: Because everyone has a causal model in their head with a node for "order of magnitude of Bitcoin's value", and that node varies in the characteristically linear fashion of a Bayes net.

Simplicio: I don't feel confident in that explanation.

Salviati: Then take whatever explanation you give yourself to explain the effectiveness of Fermi estimates. Those output a bell curve on a log scale too, and seems like estimating Bitcoin's future value should have a lot of methodology in common with doing back-of-the-envelope calculations about the blast radius of a nuclear bomb.

Simplicio: Alright.

Salviati: So the causality of Bitcoin prices roughly looks like this:

[Beliefs about order of magnitude of Bitcoin's future value] --> [Beliefs about Bitcoin's future price] --> [Trading decisions]

Simplicio: Okay, I see how the first node can fluctuate a lot in reaction to daily news events, and that would have a disproportionately high effect on the last node. But how can an efficient market avoid that kind of log-scale fluctuation? Efficient markets always reflect a consensus estimate of an asset's price, and it's rational to arrive at an estimate that fluctuates on a log scale!

Salviati: Actually, I think a truly efficient market shouldn't just skip around across orders of magnitudes, just because expectations of future prices do. I think truly efficient markets show some degree of "drag", which should be invisible in typical cases like publicly-traded stocks, but become noticeable in cases of order-of-magnitude value-uncertainty like Bitcoin.

Simplicio: So you think you're the only one smart enough to notice that it's worth trading Bitcoin so as to create drag on Bitcoin's log-scale random walk?

Salviati: Yeah, I think maybe I am.

**Salviati is claiming that his empirical observations show a lack of drag on Bitcoin price shifts, which would be actionable evidence of inefficiency. Discuss.**

If you were a quant, you would know that random walks on a log scale (geometric Brownian motion) are what people normally use for asset prices. It's what's beneath Black-Scholes, for example. An additive random walk can go negative, which prices can't, but a log random walk is always positive.

(Also note that the fact that the EV is higher tomorrow than today isn't that meaningful, because of time discounting- if the EV tomorrow is the same as the EV today in nominal terms, you should sell and buy something that's expected to go up. How does the expected future growth rate compare to other opportunities?)

The book Fortune's Formula describes a simple investing scheme invented by Claude Shannon, referred to as "Shannon's Demon", that's specifically designed to make money in markets described by log random walks. I found a blog post describing the scheme here. (Some previous discussion.) I'd expect this kind of volatility harvesting scheme to work better for Bitcoins than for other assets because Bitcoins are more volatile.

However, I'm not convinced that the market for Bitcoins is efficient... for example, there are going to be 84 million Litecoins to Bitcoins' 21 million, but typical investors don't know that, so 4 Litecoins for $100 feels like more of a steal than 1 Bitcoin for $100 (even Silicon Valley software engineers commonly forget to account for this basic division operation). There was talk on /r/bitcoin about how once the price got to the $1000 range, people seemed reluctant to invest since it seemed so expensive and how things should be reframed as "mBTC". And I'd expect that quant firms are reluctant to trade bitcoins due to factors like institutional regulation and it not being serious-seeming enough for themselves or their investors.

Do we really ? My own view is quite the opposite - a kinda reverse bell curve, with two possible outcomes :

Bitcoin dies, either because the crypto behind it is broken (due to mathematical progress or Moore's law) or because it gets replaced by other, "second generation" cryptocurrencies, or because states successfully fight it, or any other reason - and then it'll have a very low value, maybe even less than $1 for a BTC.

Bitcoin survives, and then, because it's inherently deflationary (fixed monetary mass for an always growing amount of real world wealth) there is no limit to how high the value of single BTC can grow.

But maybe it depends what exactly "long-term" is ?

Simplicio: There are zillions of exotic markets out there. I think you fixate on Bitcoins because they are fun and shiny. Why not instead try to outguess local real estate markets? You are much more likely to be successful.

Is there a way to outguess real estate markets that doesn't involve buying relatively illiquid properties for thousands of dollars?

BTW, I would be interested in seeing a list of exotic markets if you've got one handy.

This bitcoin conversation has run for almost a week now, and given the site I'd expect the level of reasoning to be quite high, yet when I hit "^Ftax" or "^Fgovern" or "^Fpolitic" almost nothing shows up, which causes me a measure of confusion, because these (much more than "magnitudes") are key nodes in my causal reasoning about the future value of bitcoins.

From my perspective, the plausible socio-political implications of bitcoin are large enough, and different enough from what I see commonly discussed, that it cau... (read more)

I'd take bets against Bitcoin resulting in any significant restructuring of government. Remember, Warren Buffett pays lower tax rates than his secretary. Criminals around the world are already quite successful at money laundering. And yet society has not collapsed. This won't collapse it either.

So a word of warning if you are thinking of playing the long-game here. The source of your observation is that bitcoin is (or has been) apparently undervalued.

But an other explanation is that its a bubble. While spot trading in bitcoin is quite liquid, the derivatives markets in bitcoin are fairly untested, relatively new, and fairly low volume. Right now, if I believe bitcoin is expected to climb, I can easily pile in, but shorting bitcoin takes more effort.

As the new bitcoin derivatives markets grow, its will give counterparties with negative opin... (read more)

The first line and Simplicio's response seem to be incongruous. Was the question meant to be, "Do you think the Bitcoin markets are

inefficient?"Not surprising - the risk free rate (http://en.wikipedia.org/wiki/Risk-free_rate) is exponential, and any efficient asset has to do at least as well in expectation. So expected exponential growth in asset value is exactly the behaviour you'd expect.

Or put more prosaically: if I invest money at x% interest in a bank, I have exponential growth. Therefore any investment that would tempt me away from a bank account, must offer at least exponential growth.

Several people already pointed out that what is most frequently used as a model for prices is precisely the log random walk. This should have been obvious since no asset can have a negative price, and it is known that the long-term probability of the one-dimensional random walk reaching any specified point (such as zero) is 1 (same for 2D, not so in 3+ D) - this part of how casinos make money.

It is very easy to spot why the random walk model doesn't make sense for prices, just take the sentence that says "If today's Bitcoin price is $1000, then tomorr... (read more)

The fact that there are a stock has a random walk behavior in the log-scale does not imply that its expected future price is larger than its current price. Imagine that when a bitcoin is worth 1000$ today it has equal chance tomorrow of going up 100$ or going down 100$. Then if it goes to 900$, it has an equal chance of going up or down 90$ the next day, and similarly, if the price becomes 1100$, it has an equal chance of going up or down 110$. Then the log-price would have variations on the log scale, but at each step the expected value stays the same.

Please place more trust in the competence of mainstream institutions next time.

Salviati's claim at the end reminds me of Einstein asserting that mass has more inertia than you think, but you'll only notice that when its velocity is really high.

Can you elaborate on why you think this is true?

I mean this in the least hostile way possible -- this was an awful post. It was just a complicated way of saying "historically speaking, bitcoin has gone up". Of course it has! We already know that! And for obvious reasons, prices increase according to log scales. But it's also a well known rule of markets that "past trends does not predict future performance".

Of course, I am personally supportive and bullish on bitcoin (as people in IRC can attest). All I'm saying is that your argument is an unnecessarily complex way of arguing that bitcoin is likely to increase in the future because it has increased in price in the past.

Financial markets typically exhibit leptokurtosis, meaning that rare large declines influence the expected value more than a lognormal distribution predicts. A few years of data are often inadequate to measure that.

I have posted this to /r/bitcoin, so as to allow the market to learn of this proposed inefficiency, and thus remove it, should it in fact exist.

Not sure what the lesson is, or the question. An improvised model for explaining the seemingly exponential growth is the belief that there is inherent risk of being forgotten as a currency but that this risk falls exponentially with time or price itself. In this model we have efficient markets as in all knowledge is encoded, random walk and exponential growth.

I am sleepy so have mercy with my reasoning.

Given that it sounds like nobody has actually run the numbers, is there a reason to suspect that bitcoin prices are best modeled by a

uniformrandom walk on the log scale? If it is modeled instead by a weighted random walk with an appropriate negative drift term the expected value of a BTC tomorrow could be exactly equal to today's expected value (or more exactly today's value adjusted for inflation).Salviati is claiming that his empirical observations show a lack of drag om price shifts, which would be actionable evidence of inefficiency.