Statistical Distributions of the Volatility Index

VIX related products (ETNs, futures and options) are becoming popular financial instruments, for both hedging and speculation, these days.  The volatility index VIX was developed in the early 90’s. In its early days, it led the derivative markets. Today the dynamics has changed.  Now there is strong evidence that the VIX futures market leads the cash index.

In this post we are going to look at some statistical properties of the spot VIX index. We used data from January 1990 to May 2017. Graph below shows the kernel distribution of spot VIX.

volatility trading strategies: VIX distribution is not normal

Kernel distribution of the spot VIX index

It can be seen that the distribution of spot VIX is not normal, and it possesses a right tail.

We next look at the Q-Q plot of spot VIX. Graph below shows the Q-Q plot. It’s apparent that the distribution of spot VIX is not normal. The right-tail behavior can also be seen clearly. Intuitively, it makes sense since the VIX index often experiences very sharp, upward spikes.

volatility arbitrage: Q-Q plot of spot volatility index

Q-Q plot of spot VIX vs. standard normal

It is interesting to observe that there exists a natural floor around 9% on the left side, i.e. historically speaking, 9% has been a minimum for spot VIX.

We now look at the distribution of VIX returns. Graph below shows the Q-Q plot of VIX returns. We observe that the return distribution is closer to normal than the spot VIX distribution. However, it still exhibits the right tail behavior.

Relative value arbitrage: distribution of VIX returns

Q-Q plot of VIX returns vs. standard normal

It’s interesting to see that in the return space, the VIX distribution has a left tail similar to the equity indices. This is probably due to large decreases in the spot VIX after sharp volatility spikes.

The natural floor of the spot VIX index and its left tail in the return space can lead to construction of good risk/reward trading strategies.

Read More Here: Statistical Distributions of the Volatility Index

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Volatility, Skew, and Smile Trading

Peter Carr recently gave a talk on volatility trading at the Fields institute.


In general, an option’s fair value depends crucially on the volatility of its underlying asset. In a stochastic volatility (SV) setting, an at-the-money straddle can be dynamically traded to profit on average from the difference between its underlying’s instantaneous variance rate and its Black Merton Scholes (BMS) implied variance rate. In SV models, an option’s fair value also depends on the covariation rate between returns and volatility. We show that a pair of out-of-the-money options can be dynamically traded to profit on average from the difference between this instantaneous covariation rate and half the slope of a BMS implied variance curve. Finally, in SV models, an option’s fair value also depends on the variance rate of volatility. We show that an option triple can be dynamically traded to profit on average from the difference between this instantaneous variance rate and a convexity measure of the BMS implied variance curve. Our results yield precise financial interpretations of particular measures of the level, slope, and curvature of a BMS implied variance curve. These interpretations help explain standard quotation conventions found in the over-the counter market for options written on precious metals and on foreign exchange.

In this talk, Carr discussed which options you should trade when

  • You know the realized volatility will exceed 10% and yet the ATM volatility is currently below 10%
  • You know that the correlation of every IV with the underlying will realize positive and yet an OTM call’s IV is currently below an equally OTM put’s IV
  • You know that IVs are themselves volatile and yet 3 IVs currently plot linearly

Click here to watch

Article Source Here: Volatility, Skew, and Smile Trading

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Arbitrage Pricing Theory and Factor Investing

Factor investing is becoming popular these days. It has its roots in Arbitrage Pricing Theory. According to Wikipedia
Arbitrage pricing theory (APT) is a general theory of asset pricing that holds that the expected return of a financial asset can be modeled as a linear function of various macro-economic factors or theoretical market indices, where sensitivity to changes in each factor is represented by a factor-specific beta coefficient. The model-derived rate of return will then be used to price the asset correctly—the asset price should equal the expected end of period price discounted at the rate implied by the model. If the price diverges, arbitrage should bring it back into line.

S. Ross is the Father of Arbitrage Pricing Theory. D. Musto of Wharton recently summarized the key points of Arbitrage Pricing Theory in this podcast

  • … the risk that faces the investor, the risk that ultimately is going to deliver the payoff to his bank account, is going to be the risk of this portfolio. And once you think of it that way, you realize that the correct measure of risk is not a stock’s risk by itself, but instead, the risk that it’s going to add to a diversified portfolio.
  • … stock returns follow what you could call a factor structure … All of the systematic portion of their stock returns is captured by those five factors. Everything that’s not captured by them is idiosyncratic.
  • …the risk of a stock that’s going to matter to investors is its exposure to those factors. And every factor is going to have associated with it what you would call a risk premium
  • …the idiosyncratic component of the stock’s return is not going to give you any additional expected return. It shouldn’t, because to an intelligent investor putting together an optimal portfolio, that’s just going to wash out. It’s the factor-driven part of the return that’s going to matter. And so that’s going to be driving expected returns
  • [factors] could be things like changes in expected inflation. They could be developments to GNP. They could be things having to do with interest rates — and what kind of risk premium you would need to be compensated for exposure to that factor.

Another significant contribution of S. Ross is the binomial option pricing model.

This is a very elegant way to price the whole range of derivative securities out there. So Steve, building on the work of [Fisher] Black and [Myron] Scholes, showed how you could take what they did and think about it as a binomial framework that would help you price a wide range of securities, and show you how you go about replicating the payoff of any derivative security you might be interested in, with this binomial trading technique.


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Is Value at Risk a Good Risk Measure?

Value at Risk (VaR) is an important risk measure that large financial institutions use for managing the risks and allocating capital. Wikipedia defines VaR as follows:

Value at Risk (VaR) is a measure of the risk of investments. It estimates how much a set of investments might lose, given normal market conditions, in a set time period such as a day. VaR is typically used by firms and regulators in the financial industry to gauge the amount of assets needed to cover possible losses.

For a given portfolio, time horizon, and probability p, the p VaR can be defined informally as the maximum possible loss during the time if we exclude worse outcomes whose probability is less than p. This assumes mark-to-market pricing, and no trading in the portfolio.

Value at Risk

Bloomberg recently reported that the combined VaR of the six largest US banks has decreased from $1 billion in 2009 to $279 million. Does this mean that we have much less risk now than before?

Not so if we adjust for the decreasing trend in volatility

What if we strip it out to get a sense of whether risk-taking has really declined, independent of the broader market? The result won’t be perfect, 3 but it should give us a rough idea.

It indicates that, relative to the broader market, the banks’ trading operations are only about 25 percent less risky than they were in 2009 — and have actually become a bit riskier over the past year. Read more

In a similar context, Peter Guy pointed out that, generally speaking, risk models are vulnerable because they were developed and tested in a market environment that can change in the future

… risk models are vulnerable because a decade of zero interest rates have never occurred before in financial and economic history. No one possesses accurate historical data to predict the future. And quantitative models and algorithms heavily depend on historical data for forecasting risk.

“There are no models that are able to accurately capture the effect of rising interest rates. You need to reach back to the period before quantitative easing began,” Read more

So what are the solutions?

One solution is to develop stress scenarios, then use them to calculate probability-weighted, forward-looking risk measures. Additionally, we can implement other VaR variants that better account for the tail risks.



Originally Published Here: Is Value at Risk a Good Risk Measure?

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Are Short Out-of-the-Money Put Options Risky? Part 2: Dynamic Case

This post is the continuation of the previous one on the riskiness of OTM vs. ATM short put options and the effect of leverage on the risk measures. In this installment we’re going to perform similar studies with the only exception that from inception until maturity the short options are dynamically hedged. The simulation methodology and parameters are the same as in the previous study.

As a reference, results for the static case are replicated here:

ATM  (K=100)   OTM (K=90)
Margin Return Variance VaR Return Variance VaR
100% 0.0171 0.0075 0.1940 0.0118 0.0031 0.1303
50% 0.0370 0.0292 0.3844 0.0206 0.0133 0.2783
15% 0.1317 0.3155 1.2589 0.0679 0.1502 0.9339


Table below summarizes the results for the dynamically hedged case

ATM  (K=100)   OTM (K=90)
Margin Return Variance VaR Return Variance VaR
100% -0.0100 1.9171E-05 0.0073 -0.0059 1.4510E-05 0.0062
50% -0.0199 7.6201E-05 0.0145 -0.0118 5.8016E-05 0.0121
15% -0.0660 8.7943E-04 0.0480 -0.0400 6.5201E-04 0.0424


From the Table above, we observe that:

  • Similar to the static case, delta-hedged OTM put options are less risky than the ATM counterparts. However, the reduction in risk is less significant. This is probably due to the fact that delta hedging itself already reduces the risks considerably (see below).
  • Leverage also increases risks.

It is important to note that given the same notional amount, a delta-hedged position is less risky than a static position. For example, the VaR of a static, cash-secured (m=100%) short put position is 0.194, while the VaR of the corresponding dynamically-hedged position is only 0.0073. This explains why proprietary trading firms and hedge funds often engage in the practice of dynamic hedging.

Finally, we note that while Value at Risk takes into account the tail risks to some degree, it’s probably not the best measure of tail risks. Using other risk measures that better incorporate the tail risks can alter the results and lead to different conclusions.


Originally Published Here: Are Short Out-of-the-Money Put Options Risky? Part 2: Dynamic Case

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Lessons From an Option Seller

Victor Niederhoffer is a famous option seller. According to Wikipedia:

Niederhoffer studied statistics and economics at Harvard University (B.A. 1964) and the University of Chicago (Ph.D. 1969). He was a finance professor at the University of California, Berkeley (1967–1972). In 1965, while still at college, he co-founded with Frank Cross a company called Niederhoffer, Cross and Zeckhauser, Inc., an investment bank which sold privately held firms to public companies. This firm is now called Niederhoffer Henkel, and was run by Lee Henkel (who died May 30, 2008), the former general counsel to the IRS. Niederhoffer pioneered a mass marketing approach in investment banking and did a large volume of small deals at this firm. He also bought many privately held firms with Dan Grossman, his partner during this period.

As a college professor in the 1960s and 1970s, Niederhoffer wrote academic articles about market inefficiencies, which led to the founding in 1980 of a trading firm, NCZ Commodities, Inc. (aka Niederhoffer Investments, Inc.). The success of this firm attracted the attention of George Soros. Niederhoffer became a partner of Soros and managed all of the fixed income and foreign exchange from 1982 to 1990.[15] Soros said in The Alchemy of Finance that Niederhoffer was the only one of his managers who retired voluntarily from trading for him while still ahead. Soros held Niederhoffer in such high esteem that he sent his son to work for him to learn how to trade. Read more

His trading strategy provided high returns for more than 20 years. However, he suffered a huge loss in 1997.

…reward comes with risk, and Niederhoffer embraced risk in ways that would eventually become costly. He got caught leaning the wrong way when the Asian financial crisis hit in 1997, all but completely wiping his fund out. But he slowly rebuilt, and once again amassed another fortune, only to see this capital pool destroyed by the financial crisis of 2007-09.

Bloomberg recently interviewed Niederhoffer.

Niederhoffer is a brilliant and fascinating character, a study of rich contrasts. He is a nationally ranked squash champion, and former Berkeley professor of finance and statistics. He is an undeniably talented trader, except for that small issue of occasionally blowing up and getting wiped out.

I am not sure that he fully accepts responsibility for his various disasters. His trading record is akin to setting the track record on the straightaways, only to crash into the wall on the curves. Still, he teaches an important lesson for any trader. As revealed in his first book, “The Education of a Speculator,” the risk-embracing style that created his first fortune comes with some caveats. Read more

Click here to listen to the interview.

Is shorting volatility a dangerous game?


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What Do Creators of the VIX Think of Volatility?

The volatility index was created more than 30 years ago. Since then it has become a favorite tool for both speculation and risk management.  There is now strong evidence that VIX futures and related exchange-traded products are changing the market dynamics. Specifically, in the early days of the VIX, the cash market led the futures. But since 2012, VIX futures leads cash 75% of the time.

VIX contango

VIX Contango as at Sep 15, 2017. Source:

Business Insider recently interviewed two of the creators of the volatility index, Robert Whaley and Dan Galai. Here are the key takeaways from R. Whaley interview,

  • Where VIX is the volatility over the next 30 days, VIX futures is the expectation of the volatility 30 days from now. Those two series don’t behave like one another, in fact quite differently.
  • If you look at VXX and go to 13F filings, you’ll see it’s largely an instrument used by retail customers, not institutions. If you go over to the ownership of XIV, it’s largely institution.
  • Retail customers do not look at the prospectus because those things are 300 pages long. Institutional investors do read them and they know exactly what’s going on.
  • His investment strategy is to own long term VXX put options and it has worked out well for him. Read more

Regarding bullet point #1, we have repeatedly said that VIX futures are (risk-neutral) expectation values of forward volatilities, and not spot VIX. Furthermore, since they are expection values in the risk-neutral world, they do not represent the future expected value of the spot VIX in the physical measure.

Here are the key takeaways from Dan Galai interview

  • It’s hard to say the market is biased one way or another. The VIX is actually reflecting market expectations in the sense that people pay for the options, it’s traded, and anybody can be on any side of the fence. It’s doing its job of reflecting the market consensus going forward.
  • The market always has its own dynamics, and the effect is marginal. I don’t think they change the market. Volatility is low, and it’s been low. If the market was expected to change abruptly, we’d see it in options prices.
  • When interest rates start moving up, so will volatility.
  • There’s no doubt that passive investment and ETFs changed the nature of correlations in the marketplace among different securities. Once you create baskets, you create artificially high correlations between the members of those baskets. Whether it changes the environment in the long term, he’s not sure. We don’t have enough history to make any strong conclusions about it.
  • You can find what’s happening in the US on a global basis. You have the same phenomenon of low volatility. Maybe the US is the dominant force, but it’s happening everywhere. Read more

Originally Published Here: What Do Creators of the VIX Think of Volatility?

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