Feb 28, 2011

CBOE to launch Gold VIX Futures and options, piss off CME

CBOE announced today that they are launching derivatives on GVZ - Gold VIX index that is based on implied volatility of GLD ETF. What is surprising that only half of a year ago CBOE licensed to CME the right to develop their own volatility indexes (crude, and gold - GVX based on implied volatility of GC futures options), and CME did already launch futures and options on them. While as far as I know there has been no trading activity in these two products on CME (ok, gold futures traded a 4-lot, options none, oil none for both futures and options) it is really a surprising move for CBOE to move directly into CME space, and also to try to revive a dead product. I would love to see this product gain liquidity, but experience suggests that the product is likely to fail.

Futures contracts specs, options contracts specs - all in parallel to the VIX contracts.

See my previous posts about CME indexes here, and here.

CBOE SKEW Index, Part 3

Read the first part of SKEW post here, part 2 here.

There is a very interesting table on page 8 of SKEW index white paper: Estimated Risk-Adjusted Probabilities of S&P 500 Log Returns Two and Three Standard Deviations below the Mean. I think the author meant risk-neutral, not risk-adjusted, but regardless I reproduce it below:

SKEW -2 Std. Dev -3 Std. Dev.
100 2.30% 0.15%
105 3.65% 0.45%
110 5.00% 0.74%
115 6.35% 1.04%
120 7.70% 1.33%
125 9.05% 1.63%
130 10.40% 1.92%
135 11.75% 2.22%
140 13.10% 2.51%
145 14.45% 2.81%

Since SKEW index is "adjusted" as SKEW Index = 100 - 10 * implied skew, SKEW Index value of 100 is skew of 0, SKEW Index value of 105 is skew of -0.5, etc.

The table is based on Gram-Charlier expansion of normal distribution, for various skew levels with zero excess kurtosis, however the paper that is referenced provides the formula for probability distribution function, and not cumulative density function (see Wikipedia):

PDF(z) = n(z)*[1+skew/6*(z3-3z)+kurtosis/12*(z4 − 6z2 + 3)],

where n(z) is the normal probability distribution function.

Since kurtosis = 0, the expression becomes F(z) = n(z)*[1+skew/6*(z3-3z)]

The formula for CDF is obtained by integration

CDF(z) = N(z) - skew/6*n(z)*(z2-1),

where N(z) is the normal cumulative density function. In excel this formula would read:

NORM.DIST(Z,0,1,TRUE) - SKEW/6*NORM.DIST(Z,0,1,FALSE)*(Z*Z-1),

where Z and SKEW are variables, Z is the Z-score e.g. -2 for negative 2 standard deviations move, and SKEW is implied skew, calculated from SKEW index. The results are not exactly the same, but sufficiently close to the ones in CBOE paper. I suspect the difference is due to weird rounding on CBOE's part. I do realize it may sound arrogant for me to assume CBOE's error rather than mine, however CBOE's numbers don't quite match standard results in the case of normal distribution (where skew is zero): e.g. for 2 stds the number should be 2.27501 to the 5 significant digits, where CBOE rounds it to 2.30, and for 3 stds the number should be 0.13499 to the 5 significant digits, where CBOE rounds it to 0.15, so I confidently stand by my numbers.

However there is a different problem that I would really like to point out - the Gram-Charlier approximation is a rather weak approximation. In some cases it produces rather strange numbers, for example in the case of Z=-1 (down 1 standard deviation) probabilities become independent of the skew parameter, which obviously makes no sense.

SKEW -0.5 Std. Dev -1 Std. Dev. -1.5 Std. Dev.
100 30.85% 15.87% 6.68%
105 28.65% 15.87% 8.03%
110 26.45% 15.87% 9.38%
115 24.25% 15.87% 10.73%
120 22.05% 15.87% 12.08%
125 19.85% 15.87% 13.43%
130 17.65% 15.87% 14.78%
135 15.45% 15.87% 16.12%
140 13.25% 15.87% 17.47%
145 11.05% 15.87% 18.82%

So, if you want to calculate risk-neutral probabilities now you have the same formula CBOE used in the white paper, however know that the formula is only an approximation, and clearly does not work in some cases. I hope in the near future I'll blog about better formula, as well as other uses for SKEW index.

Feb 27, 2011

Week in Volatility

This week saw a lot of action - a strong rise in VIX and subsequent decline. VXX had a draw-up of 20% ~log(34.01/27.73) with some large intraday volatility. Although this draw-up seems significant, I think it is a good time to remember that average draw-up for SPVIXSTR (the index that underlies VXX) for the last 5 years is 49% !, and that most of 2008 the index was making new highs.

For those who want to know about risks of being short VXX I recommend my previous post on the topic, as well as recent analysis on MarketSci blog (link).

Feb 23, 2011

XXV - IVO trade

I've written before about XXV and its price connection to VXX. The formula for historical relationship between two ETFs is

XXV $20 * (1 - (VXX-108.03)/108.03 ) = $40 - VXX * (20/108.03)

That relationship has changed slightly, probably because of accrued fees / transaction costs. As I pointed out, it VXX falls to zero that provides a maximum for theoretical value of the ETF of $40. Having risen 70% since its inception (using base value of $20) in 7 months XXV does not provide much of an upside to investors, and will probably spend the rest of its existence going slowly to $40. Apparently in response to this iPath launched IVO - an identical fund, but with a more recent start date. Now that IVO has been trading for a few weeks, I provide a similar formula for IVO:

IVO ≈ $20 * (1 - (VXX-32.99)/32.99 ) = $40 - VXX * (20/32.99)

The maximum value is still $40, but return upside is greater.

It is clear that arbitrage is possible among all three ETFs. For example the following relationship holds between IVO and XXV

IVO ≈ $40 - (40-XXV)*108.03/32.99

Good luck, traders!

Feb 21, 2011

VIX Tuesday Open

It's Presidents Day in the US, a national and exchange holiday, and markets are closed. But in Asia and Europe markets were down by over 1% with volatility indexes up across the board:
VSTOXX +2.45
VDAX +1.99
VSMI +1.32
Asian volatility indexes (the ones that I know of) are up as well, and at the time of writing Asia opened lower, and S&P E-minis are down 1.17%. I guesstimate that VIX will open tomorrow at about 18, and probably will be volatile throughout the day.

Feb 19, 2011

Week in Volatility

From the volatility view it was a decidedly mixed week - VIX up, VSTOXX down.


While the volatility did its own thing I was watching the skew on VIX and VXX, and while I cannot point to any quantitative measures I have a strong feeling that something unusual was taking place this week. ATM vols on the VIX were up, but that is the usual pattern with VIX vols, and nothing out of the ordinary.

Feb 16, 2011

VIX Expiration

Today VIX expired at 16.49, up 0.11 since last expiration. My prediction made on Jan 20, 2011 was for 18.42 at the time when futures were trading at 18.80. While the forecast is slightly closer than the futures, it is considerably off this month.

At the time the forecast was made the index, futures and my forecast were around the same level of about 18-19. While lower forecast certainly made sense my model did not anticipate the unrest in Egypt, or susequent volatility decline. Since about Feb 4th VIX remained unchanged at around 16, and futures slowly declined to that level.

My forecast for March VIX is 17.44 vs 18.05 (yesterday's settlement) and VSTOXX at 21.39 vs 22.35 in the futures market. Question for the readers: is there an official settlement for VSTOXX, like VRO for the VIX?


Feb 13, 2011

Week in Volatility

Another positive week for the market, and negative week for volatility. VIX and VSTOXX long term contracts continued to fall, declining by about a vol point. VIX Feb futures having only two full trading days to go settled at 16.60, a vol point above the index.

Feb 12, 2011

CBOE SKEW Index, Part 2

Read the first part of SKEW post here, part 3 here.

In the press release CBOE described SKEW index as "a benchmark measure of the perceived risk of extreme negative moves – often referred to as "tail risk" or a "black swan" event ". I guess they really needed a catchy moniker similar to "fear index" for VIX. However the main measure of an outlier is in volatility, not skew. SKEW index elimenates skew dependence on volatility, and there is almost no correlation between SKEW and VIX. What SKEW measures is the degree of assymetry and risk aversion that is not already reflected in VIX. By the very construction SKEW index is not redundant to the VIX.

If we apply SKEW in combination with VIX to market analysis (for example using it as technical indicator) we can see some interesting patterns. I separate SPX returns into 4 categories and calculate annualized returns (I assume everyone knows the importance of using geometric average returns for evaluation of long-term strategies)

Regime Arithmetic average Geometric average Sharpe ratio
Low VIX Low SKEW 1.9% 1.3% 0.18
High VIX Low SKEW 12.3% 9.5% 0.49
Low VIX High SKEW 11.3% 11.3% 1.07
High VIX High SKEW 5.2% 2.7% 0.23

Right now we're in the Low VIX High SKEW regime, which can be considered a bullish signal. This is just the basic way of using SKEW index, more complicated indicators can be developed. For example Bollinger bands indicators seem to work reasonable well. In the next skew post I'll examine what SKEW means for option traders.

Feb 8, 2011

CBOE SKEW Index, Part 1

Part 2 of the SKEW post is available here, part 3 here.

Finally! CBOE had SKEW page up for years, and yesterday they announced that they will start publishing the data. Note that SKEW index is not 100 * Implied skew (like VIX), but rather 100-10*Implied skew.

White paper, FAQ, and time series in Excel are now available from CBOE website.

EDIT: The dataset is missing 9/20/2000 to 9/29/2000 and a few other random days for some reason.
EDIT: Here is a combined dataset of SPX, VIX, and SKEW indexes. Missing days are left as blanks. Data from CBOE and Yahoo! Finance.

Feb 6, 2011

Week in Volatility

Equity indexes were up this week with S&P up 2.7% and Stoxx up 1.6%. VIX index sank more than four points from 20 to just below 16. Long term futures declined by about 1.5, with the longest expiration now trading at 23.30. VStoxx fall was more modest, with long-term volatility still above 26%.

While uncertainly in Egypt may have been an excuse for volatility rise last week, we see that market discounted it as a non-event. Even now with more recent news about gas pipeline explosion, would probably not be a catalyst for higher volatility. Long-term futures tell us unambiguously that we're still in the low-volatility regime.

One of the reasons for such fall in volatility can be falling correlations. After the crisis in 2008 we've seen a spike in correlation among all asset classes, what became colloquially known as risk on / risk off. Lately that pattern started to wane. For example KCJ implied correlation index closed near a 6-month low (chart) Looking at historical charts of  ICJ, JCJ, KCJ it looks like implied correlation will likely not to go much lower, and that is probably going to have a negative effect on VIX skew.

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