# Volatility: One Theorem, One Lemma, Zero Paradoxes

A common theme I’ve mentioned — in pieces on mortgages, industry demographics, energy, and bubbles generally — is the idea that the perception of lower risk creates the reality of higher risk. This theme is hardly original to me; Hyman Minsky wrote about it, and anybody can intuit it from thinking about how crowded trades work.

It’s worth exploring in more detail. If you start from the historical perspective, it looks like a law, i.e. an observation that’s always true but doesn’t necessarily have a more fundamental theory behind it. But you can derive this pattern purely from theory.

Let’s start with the classic ostensibly low-volatility bet: the Martingale. A simple martingale is a strategy where you’re playing a game in which you either double your money or lose your stake. You start out betting $1 at a time. If you lose money, though, you double your bet; you continue doubling after each loss.

A simple simulation (you can do this in Excel) will show you that the martingale strategy is guaranteed to work as long as you have an infinite bankroll. Otherwise, it’s eventually guaranteed to fail. If you go to the casino with a thousand dollars in your pocket and play a game with even odds, there’s a 1/1024 chance that you’ll lose ten times in a row, doubling your bet each time and eventually losing all of your money.

As it turns out, any strategy that involves insuring against volatility is fundamentally equivalent to a martingale *in the best-case scenario*. At best, if you’ve accurately estimated the distribution of returns, you will either make a little or lose a lot. Since “a lot” is unbounded, over a long enough time period you’ll eventually lose your stake.

To your counterparty, this means that any kind of insurance against volatility has the following payoff structure:

- Most of the time, the insurance provider pays out, reducing your volatility.
- Some of the time, the insurer can’t pay out, so you feel all the volatility past that point — and since they’re in default, some of the insurance you thought you’d get does not actually arrive.

So any attempt to reduce risk will necessarily shrink the center of the bell curve while fattening the worst tail.

This makes sense, because all else being equal, finance can’t create new wealth, just change the distribution. Anything that reduces the impact of real-world changes in output in one segment of the bell curve needs to have an equal and opposite effect at another point, because the quantity of goods is unchanged.

# The Not-So-Best-Case Scenario

That’s the best case scenario of the Vol Theorem. There’s a painful lemma, though: all else is not equal, because insurance lowers *perceived *risk, and that encourages people to take more risk. So the presence of volatility-mitigating institutions — whether they’re market-makers selling put options, companies selling insurance, or politicians and central banks pushing counter-cyclical policies — actually *increases* underlying volatility.

One surprising case study of this is central bank transparency. Back in the bad old days, central banks didn’t necessarily broadcast their intentions to the market. When the Fed raised rates 50 basis points, it was not strictly clear whether they’d decided that rates were 50 basis points too low or whether this was the first of ten planned hikes. This meant that speculation about their intentions (and parsing of Alan Greenspan’s pronouncements) made rates more volatile.

In the last two decades, the Fed has gotten much more transparent. For example, the Fed now offers a dot plot showing the distribution of expected rate changes at future dates. At one level, this reduces volatility: for any given rate change, you can put an upper bound on how far the most aggressive member of the Open Market Committee wants to push things.

However, this introduces a new risk. Now that volatility is lower, people *know* volatility is lower. They price vol-sensitive instruments accordingly. Which means that in the event that the dot plot becomes misleading (for example, if the Fed is forced to raise or cut rates well outside the bounds of the previous dot plot), underlying volatility will return to its pre-dot plot norm, but realized volatility will be higher as the short-vol side of options trades painfully unwinds.

This applies to other policies as well: many countercyclical policies are designed to reduce the magnitude of economic fluctuations by taxing windfall gains and providing money to the unemployed or financially underwater. In the short term, this works, but in the long term it creates an incentive for people who expect windfalls to have them in less taxable jurisdictions (or to invest their fortune in tax-efficient R&D instead of inefficient direct consumption). And it increases the scope of the government’s promises at a time when its finances are strained due to lower tax receipts.

In most recessions, there’s a flight to quality, i.e. investors buy the debt of stable countries like the US, Japan, and Germany. But the more severe a recession gets, the more uncertain the definition of “quality” — in a severe recession like 2008, the flight to quality is really just a bet on prolonged deflation, which risks destroying output levels and thus constrains the government’s ability to provide social insurance.

# Not a Paradox

Financial writers like to look at a case like this and call it a paradox: you wanted lower volatility, but you got higher volatility that’s harder to model! But like a lot of things popularly known as paradoxes (Simpson’s, for example) aren’t paradoxes. They’re just difficult truths.

It’s necessarily true that reducing how much volatility someone feels is not going to reduce the amount of volatility there is, just that it’ll reduce their sensitivity to it. It’s as much a paradox as the notion that having your cake precludes eating it, too.

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