The devil’s defense of Bitcoin volatility

What a day:

Bitcoin is now 44 percent off its intraday high of $266.

Maybe it’s in the nature of Bitcoin to have volatile prices (and Megan comments here).

But that’s part of the point, isn’t it?  (Have we ever posted on the two envelopes problem?  I think so but I can’t find it through search.)  Imagine you hold a currency which, over the next period will either double or halve in value.  The expected return of such a Bitcoin is in fact (0.5 x .5) + (0.5 x 2) = 1.25.

What a good deal that is!  Holding a single Bitcoin — a very volatile Bitcoin that is — seems like a lot of fun.  It’s unlikely that simple risk aversion will  take away the expected gain there.

Does this not mean that exchange rate variability is desirable per se, a kind of automatic utility machine?  The party holding the other currency reaps a comparable gain from the ex ante volatility.

Fischer Black was obsessed with this problem for a few years, though I don’t think he ever quite nailed it.  The mathematics behind Jensen’s Inequality are relevant here, but again that’s not the same as an explanation of the puzzle.  My preferred path is to start with the Sumnerian “never reason from a price change” insight, but in any case this is a good brain teaser for your evening.

I thank John W. for a relevant pointer.  And Jerry Brito says that Bitcoin’s volatility doesn’t matter, but he isn’t quite devilish enough for my taste, at least not on this one.


Bitcoin volatility may have magnified Dornbusch overshooting properties.

Ok, I'll bite.

Over n repeated gambles, your expected return will be 1.25^n.

BUT - your wealth will not approach 1.25^n * initial wealth.

Why not? Expected values are the asymptotic targets for things that are adding up over time, not things that are multiplying over time like wealth. One needs to calculate geometric (log) returns rather than arithmetic returns to get a good picture of where things are going asymptotically.

Try simulating it on your computer to see why.

The idea here is that you hold a *single* bitcoin which makes risk aversion arguments irrelevant.

If you hold a diversified portfolio, what is important when considering a small investment in bitcoins is the expected returns and how those returns are associated with your personal wealth.

The later is likely small; however, the former is likely negative and so bitcoins aren't obviously a good investment.

Perhaps a Kelly Criterion sort of approach? Invest in the 2 envelopes at first, and then gradually reduce the amount invested as the envelopes pay off.

This problem was solved 50 years ago. It's called the Kelly Criterion. Any repeated game with positive expected value can produce positive growth in wealth, and to maximize the long term rate of wealth growth you use edge over odds.

What that means is the envelope game is indeed a fantastic investment. The simple strategy that Kelly tells us that maximizes our long-term wealth is to invest half our wealth every period in the game, and leave half our wealth sitting on the table.

For example if you start with $100, you invest $50. If you win you now have $150, in which case you invest $50 in the next round. If you lose you now have $75 in which case you invest $37.5. Over time this strategy is expected to increase our wealth by 12.5% per period.

The game is a geometric random walk. The walk has an expected position for any number of iterations n but although that position is not an asymptote, that is not the claim being made here. (The walk has asymptotic *bounds* given by the law of the iterated logarithm, but again that is not directly relevant.)

No, the devilish aspect of the game is that it has assumed the conclusion: the problem statement has a specious patina of neutrality because it implicitly implies that a zero bias in the walk implies a zero return assumption and that one therefore ought to be able to modify volatility without affecting expected returns. But mathematically, that is not true, which is all that the arithmetic in this example is demonstrating.

In theoretical terms, the limiting case of this game is geometric Brownian motion, and Ito's lemma tells us that an instantaneous return of r must be modified by - s^2 / 2 to solve for the corresponding discretized drift. In other words, assuming equiprobable price changes of equal geometric proportions is mathematically equivalent to assuming a positive return that increases with volatility. So if you assume that bitcoin returns increase in volatility, then they increase in volatility. Woo hoo!

In the practical terms of Black's problem, if one finds that one rate, say BRL/USD, is more volatile than another, say CAD/USD, then one is likely to find that the expected forward value of the first is less than that of the second. The expected return cannot be treated independently from the volatility.

Bitcoin's current volatility stems from the limited USD/EUD/JPY trading facilities. The primary (Mt.Gox- 80%) trading site was essentially closed today by a DDOS attack. They stopped accepting most new trades, those orders that were filled were all existing limit orders, which resulted in some very weird trading.

I"m so confused by bitcoin supporters pointing to rapid price shifts (upwards of course) as support for the currency. I don't expect this to be a rational argument, but why would you want to store value in something like that (obviously could be okay as a speculative investment of sorts). Is the expectation that prices will remain somewhat stable in bitcoins? It seems at best destined to be a medium of exchange, but never a reliable medium of account. XBox Live points have a better shot at being reliable to use as a currency.

Bitcoin needs to go through a bubble phase in order to become legitimate because it is attempting to become money. It is like a lottery ticket, the value is either zero or a large amount, but this large amount is not dependent on random numbers selected by the state lottery commission, rather on how many people clap. If enough people clap, Bitcoin is money. But when they start clapping, the price goes vertical. We'll see how Bitcoin behaves now as this upward move is broken. If you look at a chart, Bitcoin was very stable post-2011 bubble. If it behaves the same way again, it will tumble to somewhere between $40 and $120 and stay there for quite some time (almost 2 years in the previous case, but only 1 data point) before another bubble phase launches Bitcoins to somewhere in the $1000s.

There will never be stability as we know it today because there's no central bank to offer stability. Bitcoins wouldl more closely resemble a digital gold standard. There will be volatility because there is no stabilizer, no government or central bank. However, if you subscribe to the theory that stability breeds instability, then the Bitcoin economy (however large it may be) will suffer repeated corrections and panics that prevent the buildup of unsustainable trends.

"digital gold standard": yup. And yet we have goldbugs and Teabaggers who want to get rid of the Fed and other central banks to leave the macroeconomies of the world on the tender mercies of capital markets.

Touche if bitcoin-like stuff were the only alternative to central-bank managed fiat money. Otherwise, not. In fact: not.

I once asked Victor Niederhoffer what he thought of this problem and he responded that market volatility is arithmetic not logarithmic.


This is the only rational way to approach shorting an irrational price. Still hard to get the timing right.

By the way, I'm not sure I quite see the connection to the two-envelope problem. That problem is generated by the fact that one cannot put an uniform prior over all real numbers (that would not be a proper probability distribution). Once you put, say, a Gaussian distribution on the amount X in the envelope, the paradox goes away.

Yes, there are variants of the two-envelope problem in which you put crazy heavy-tailed distributions on the amount in the envelopes, which bring back the paradox. But ultimately those cases are being driven by expected utility being infinite in certain cases - this is possible with heavy tailed distributions that lack first and second moments. That is a more general problem with expected utility theory (unless you put certain bounds on the set of utility functions you are allowed to consider).

Megan McArdle seems to be echoing the Moldbug argument against Bitcoin:

Why does Megan McArdle get linked to here so frequently? A) Friend of the family?, B) genuflecting towards Wichita and the Koch bros?, or C) does MR have an outreach program for sheltered workshops?

Did you actually read the links you posted? They are basically a really, really long summary of her life that they apparently think is embarrassing or shameful in some way but is not at all. "Megan McArdle is concerned with consumption inequality and not income inequality! The horror!!!"

Pal, if you don't think she's embarrassing, her POV shameful and her hack writing pedestrian, you need to take a long look in the mirror.

I don't understand why. Isn't she a popular mainstream blogger? I don't read her regularly but I never found her to be embarrassing in any way.

She's a girl who isn't liberal, so all attacks are legitimate.

I've read her and she does seem a tad fluffy. Even for my dose of non-liberal bloggers I'd rather read others.

Wow. What happened dude? No second date or something?

"Not Safe For Work Corp" is the latest (quickly failing) project of scumbag journalists Mark Ames and Yasha Levine. The links they're providing are a shamless plug of their own crappy, manipulative and half-truth filled work.

Basically these guys used to run a paper called "The Exile" back in Yeltsin's day in Russia. They made themselves semi-famous over in a "newspaper" that was notorious for non-existent fact-checking and presenting one-sided editorials as "investigative journalism." They had some mildly entertaining stories about some sophomoric, poorly-written, low-rent Hunter S. Thompson like personal escapades. Back in those days Matt Taibbi (another second-rate would-be muck-racker) used to work there too. He fit in well because both him and Ames are drug addicts. Ames is a speed-freak and Taibbi is a heroin junkie. (By the way if you think this is personal accusation this is all stated in the Exile's own publications at the time).

The Exile's early success soon cratered into bankruptcy and they had to flee Russia to escape bankruptcy and prosecution for fraud. Taibbi saw the writing on the wall and used his daddy's connections to get him a gig at Rolling Stone where he pretends to be a grown-up writer.

At the end of the Exile's days, they managed to ensnare one brilliant writer: Gary Brecher, a.k.a. The War Nerd. They've managed to ride his coattails to not having to quit journalism and become dishwashers. Nowadays their site, NotSafeForWork Corp is completely buoyed by the traffic generated by Brecher. Without him they'd all be washed-up nobodies. But they still like to do little hit and run character assassination pieces like the one on McCardle, because they're still pretending that somehow despite all these decades of failure (Ames is pushing 50 at this point), that they actually have any talent or chance of success.

I first read the links posted by NotSafeforWorkCorp via Naked Capitalism. I read them and they caused my respect for Yves Smith to crash through the floor. Anyone who thinks that the project shame attack on Megan McArdle is worthy journalism has no integrity. And I use to read Gary Brecher on occasion (who I suspect is one and the same as Ames).

John Dolan admits to being the War Nerd. It's possible that it's a shared alias, though.

Thanks for the info. I had not heard that.

Nice summary. I had no clue who they were but my mental impression after reading that site was similar.

A fun and relevant to econ extension of your brainteaser is that risk neutral firms actually like volatility -- all else equal, firms like prices to move around a lot!


My manager would go crazy with the invoices.

Way of shorting bitcoins:

I cannot vouch for the legitimacy of this website.

What if it is not a bubble, after all?

You can tell when to sell in a bubble, when people start speculating that it's not really a bubble.

Dow 50,000!!!

Sweet! You've completed the fourth corner of the "bitcoin no-win"!

Bitcoins are cheap -> LOL what a piddling, pathetic currency, no one cares
Bitcoins are expensive -> LOL deflationary currency, everyone knows that doesn't work
Bitcoins are volatile --> LOL volatile currencies are too risky to actually deal with
Bitcoins stabilize in price -> LOL volatility is actually good

I still hold some bitcoins, as I was very early to jump on the bandwagon. The very fact that I still have them serves to state the problem with these coins, it is not easy to spent them.

Anyway, the current volatility makes me think that this great idea has been hijacked, and I am really interested in who is making a killing now.

I'm trying to get rid of mine while the getting is good; what's the least hassle way to convert to USD at around 5% transaction costs?

I will not give my bank account information to Mt Gox.

Volatility is great if you want a vehicle for speculation. And it's funny that bitcoin has no actual intrinsic value, so now we're speculating on imaginary bits in a virtual world. Even tulip bulbs you can plant.

But volatility is terrible if what you are trying to build is a viable alternate currency. If you actually want people to start buying and selling things in bit coin they have to have some reasonable way of pricing goods in bitcoins. If the value of a bit coin doubles in a day, you don't want to be charging twice as much as your competitors. And you don't want the value to drop by half in the time it takes for the transaction to clear and you to do something with the bitcoins.

And we're not even talking about lending money demoninated in bitcoins yet. What does volatility do to interest rates? What does it do to the risk associated with taking out a loan, or making one?

Sure you could buy bitcoins to escape the devaluation of the US dollar. But what is more likely to happen first: A) a complete collapse in the value of bitcoins, or B) a complete collapse in the value of dollars. Bitcoin volatility makes bitcoins NOT a safe place to park money.

The claim is that simple risk aversion doesn't explain aversion to these games, but that is not true. Anyone with log utility is indifferent between the two gambles, and anyone with a coefficient of relative risk aversion greater than 1 will reject the gamble. (Assuming you have to put in all your wealth; everyone with CRRA utility wants at least a slice of this gamble, at least, but that's a general fact about gambles with positive expected value.)

Consider investing all your wealth in the gamble, with log utility:

EU(gamble) = 0.5*log(0.5W) + 0.5*(log(2W)) = log W,

which is of course the utiltiy from the sure thing.

Tyler, I think you have discovered Ito's lemma. The key insight of Ito's lemma is that if X_t follows Brownian motion without drift, then unless the second derivative of f is 0, f(X_t) is no longer driftless. The "diffusion" term in X_t contributes to a "drift" term in f(X_t). In particular, if the volatility of X_t is sigma_x, then f(X_t) now has drift f''(X_t)/2*sigma_x^2.

When you say "there is an equal probability that Bitcoin will go up or down by a factor of 2", you are describing a random walk in logs. So if f is the exponential function, Ito's lemma says that applying f to this log random walk, to get back the actual price, gives us a process with drift. (We are going from "arithmetic brownian motion" in the log, to "geometric brownian motion with drift" in the proces.) This, of course, will be a source of unlimited profit to anyone who is risk-neutral, in which case it cannot be expected to hold in equilibrium.

Nnother day volunteering at Mt. Gox. everyone keeps asking me if they can fuck the bitcoin. Buddy they won't even let me fuck it

Comments for this post are closed