Convexity of Vega (Volga) Part 1

How your vega exposure changes as implied volatility changes, backed with academic papers

Hello!

Welcome to the start of a new two-part series. Two and a half months ago, I explained the concept of skew and vanna, and the anomalous findings in academic papers (Part 1 and Part 2). Today, we are going to investigate Volga (not the river in Europe) and kurtosis.

Volga (also known as Vomma) is a second order option Greek (like gamma and vanna) and is very important on institutional trading desks but not understood by the average retail trader.

Let’s get into it.

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The Fall of LTCM

In 1998, Long-Term Capital Management (LTCM) collapsed because its risk models were built on the assumption that market returns follow a normal (Gaussian) distribution. The fund was led by Myron Scholes and Robert Merton, two authors of the Black-Scholes model (so investors trusted that they knew what they were doing), and focused on relative value arbitrage.

Their strategy involved identifying tiny price discrepancies between related securities (such as buying “off-the-run” Treasuries and shorting “on-the-run” Treasuries) and betting that these spreads would eventually converge. Because these spreads were so small, the fund utilized extreme leverage, often reaching 30:1, to increase returns. Their models suggested that the probability of these spreads widening significantly and simultaneously was statistically zero.

Despite warnings from economists like Eugene Fama, who argued that real-world markets are prone to extreme outliers, LTCM maintained its heavy positioning. The first signs of trouble appeared in early 1998 as the fund’s capital began to shrink.

The situation worsened when Salomon Brothers shuttered its own arbitrage desk. Since Salomon held many of the same positions as LTCM, their liquidation drove prices against LTCM’s portfolio. Because other hedge funds and bank desks were crowded into these same trades, a feedback loop began.

Market participants began to anticipate LTCM’s forced liquidation, leading partner Victor Haghani to remark that “it was as if there was someone out there with our exact portfolio,... only it was three times as large as ours, and they were liquidating all at once.”

The final straw for LTCM was when Russia defaulted on its debt in August 1998. Instead of the expected convergence, the market experienced a global flight to safety. Asset classes that were supposed to be uncorrelated all diverged at the same time due to market contagion. LTCM’s models characterized this move as a “10-sigma” event, an occurrence so rare it should happen once in a million years.

In reality, the model had failed to account for kurtosis (we’ll talk about this later). The fund lost $4.6 billion in four months because it assumed the “tails” of the distribution were thin when they were actually fat. Ultimately, the Federal Reserve had to coordinate a $3.6 billion bailout by major creditors to prevent a systemic collapse of the financial markets.

We now know that these tail events occur much more frequently than a normal distribution would suggest. This is the basis for our discussion today.

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Kurtosis

One of the major assumptions of the Black-Scholes Model is that the underlying stock’s returns are normally distributed (and therefore the prices of the stock are lognormally distributed). We know this is not true, and asset classes also have specific third moment (skew) and fourth moment (kurtosis) characteristics.

It’s fitting how Black and Scholes’ erroneous assumption in the option pricing model led to the downfall of their hedge fund.

While we have already talked about skew in a prior post, we have not discussed kurtosis, or fat tails. We know that “high-sigma” events occur much more often than they should. Earlier this year, we heard that silver and gold had a 6-sigma move, which should happen 3.4 times or fewer per million opportunities. The 1987 Black Monday Crash was a 22-sigma (!!) event, which should only occur about 1 in every one googol (a 1 with 100 zeros behind it) occurrences (which is a number so small it essentially exceeds the number of atoms in the visible universe).

Clearly, large, outlier moves happen more than a normal distribution suggests in financial markets. This is the reason that we model distributions with fat tails.

If a distribution has fat tails, it has positive kurtosis, and is considered leptokurtic. A normal distribution is called a mesokurtic distribution. A depiction of these distributions is below.

As you can see in the leptokurtic distribution, the center is much higher and pointier than the normal distribution, but the “shoulders” are thinner. Crucially, the ends of the curve (the tails) of the leptokurtic distribution are above the normal distribution.

If the market actually followed a normal distribution, the Black-Scholes model would be “correct,” and implied volatility (IV) would be a flat line across all strike prices. If the market expected a 15% volatility for a stock, a 10% out-of-the-money (OTM) Put would cost exactly 15% IV, just like an At-the-Money (ATM) option.

However, institutional traders aren’t naive (at least anymore). They know that the “tails” are fat. To account for the fact that a 5-sigma crash happens much more often than it should, they manually bid up the price of OTM options. This creates the volatility smile.

The curvature of this smile is essentially the market’s way of “fixing” the Black-Scholes model. The steeper the smile, the more the market is pricing in high kurtosis.

This brings us to the option greek Volga, and the ways that the street trades Volga. Additionally, I will cover papers that show the relationship between implied tail risk (from options pricing) and the future returns of that underlying stock.