There is a new page on my blog called Forecasts Tracker. It is located at the bottom of the right-hand menu. I created this page for the sake of full disclosure to the readers of this blog. Every month around the time of VIX expiration I make a forecast for the following month. Now you can see all my forecast in one place with dates and compare them to market forecasts.
One of the things that I do not like about some other blogs is that they make nebulous, equivocal forecasts, often using very wide ranges, or without specifying a time-frame. I would like to distinguish myself by making point forecasts and specifying the exact dates. You can review all the forecasts I made, judge their accuracy for yourself, or compare them with your own.
Matlab Computational Finance
It is not a secret that I use Matlab almost daily, and it is my main tool for preparing analytics you see on my blog. Which is why I'm so stoked to learn about a virtual conference on Matlab applications to finance to take place on June 9th. Some of the highlights among presentations:Building Models for High-Frequency Algorithmic Trading Strategies Using MATLAB
This presentation provides an overview of how MATLAB is used at Deutsche Bank for research, development, calibration, and monitoring of quantitative models that inform high-frequency trading strategies used in the latest suite of Autobahn Equity algorithms. [I don't know much about DB's algos, as I use Knight for executions at work, but some of the DB's research on algos that I came across was super]
MATLAB: A Corporate Development Tool at Banc Sabadell
In this presentation, we show how Banc Sabadell uses MathWorks products in the day‐to‐day operations of the trading desk. We focus on Banc Sabadell's deployment environment and how MATLAB CompilerTM enables us to reach, and cater to the specific needs of, more than a thousand users. In particular, we examine the notion of "parallel deployment" with MATLAB, and how we have sped up some of our performance-critical functions by running them in parallel. [Matlab compiler was greatly improved in the v 2011a, and I would like to find out more about how other trading desk use matlab tools]
Finally, there is a presentation from Attilo Meucci that does not seem to do anything with Matlab, but rather about different steps from model estimation to execution, to post-trade analysis. His paper on this topic is available from SSRN.
Using New Volatility Products
Reminder, there will be a meetup this Thursday at 7 pm, ISE buiding at 60 Broad St., 26th floor New York, NY
From the event's description:
“It is no longer a VIX centric universe. Large banks are pushing vol products that are trying to capture the attention of options traders. Those products include VXX, VXZ, XXV, Volatility SKEW index, and others. How can we use them?”
From the event's description:
“It is no longer a VIX centric universe. Large banks are pushing vol products that are trying to capture the attention of options traders. Those products include VXX, VXZ, XXV, Volatility SKEW index, and others. How can we use them?”
VSTOXX Derivatives Growth
Eurex introduced VSTOXX futures and options (in their new incarnation) last June. Since it has been almost a year, I decided to look at the growth of volume and open interest on VSTOXX derivatives. Left axis for open interest, right axis for volume.
VSTOXX open interest has interesting V-shaped pattern - I think what happened is that a trader initiated a big trade soon after the contracts were listed. Initial data that I have points to a number of statistical / relative arbitrage opportunities, which have since disappeared. Otherwise volume and open interest in both futures and options exhibit consistent and robust growth.
Meanwhile iPath listed two ETFs VSXX(on XETRA and LSE), VSXY(XETRA), VSYG(LSE) - VSTOXX futures tracking ETF denominated in GBP and EUR. VSXX tracks short term futures index (similar to VXX) while VSXY/VSYG track the mid term futures index (similar to VXZ).
EDIT: VSTOXX derivatives were launched in March 2011, however my data set only started in July 2011.
VSTOXX open interest has interesting V-shaped pattern - I think what happened is that a trader initiated a big trade soon after the contracts were listed. Initial data that I have points to a number of statistical / relative arbitrage opportunities, which have since disappeared. Otherwise volume and open interest in both futures and options exhibit consistent and robust growth.
Meanwhile iPath listed two ETFs VSXX(on XETRA and LSE), VSXY(XETRA), VSYG(LSE) - VSTOXX futures tracking ETF denominated in GBP and EUR. VSXX tracks short term futures index (similar to VXX) while VSXY/VSYG track the mid term futures index (similar to VXZ).
EDIT: VSTOXX derivatives were launched in March 2011, however my data set only started in July 2011.
VOLT ETF / Voltage ETF
[EDIT: Updated info on VOLT here]
The fund is developed by Nomura and launched as Source ETF. The problem is - there's just not much info on the ETF that I can find, particularly no technical details whatsoever. Here is the official website for the ETF, and factsheet .The fund is based on NMEDVMU3 index, for which there is no information. From the two sources above I would like to quote the following:
"The Reference Index adjusts its exposure to the S&P 500 VIX Mid-Term Futures Index between 0% and 100% on a daily basis by reference to the previous 30-day volatility of the S&P 500 VIX Mid-Term Futures Index. This adjustment process aims to reduce the roll costs associated with holding a constant long volatility position (i.e. being 100% exposed to the S&P 500 VIX Mid-Term Futures Index at all times) while still seeking to capture the major spikes in volatility by adjusting its allocation to the S&P 500 VIX Mid-Term Futures Index on a daily basis based on the previous 30-day volatility of the S&P 500 VIX Mid-Term Futures Index."
1 Calculate 30-day historical volatility of SPVXMTR index
2 Create robust score for volatility based on median absolute deviation from the median, using all historical data
3 Allocate min(max(score+2,0)/9,1) % to SPVXMTR index.
As you can see below my version of VOLT does not look exactly like the official version.
So at this point there is not much more that I can add; if you have any info on the ETF please forward it to me. Also note that the fund is not available in the US but is available to European investors.
Trading VIX Derivatives
EDIT: see my book review here.
Just came across this: Trading VIX Derivatives: Trading and Hedging Strategies Using VIX Futures, Options, and Exchange Traded Notes by Russell Rhoads is set to come out in September. This is afaik the first book that is dedicated to VIX derivatives. The table of contents is available here.
Looks like the first half of the book covers the basics like the historical background on implied volatility, the VIX, other volatility indexes, volatility strategy indexes, VIX derivatives, VIX-based ETFs and and their specs.
The second part of the book seems to have all the good stuff: hedging equity portfolios with VIX derivatives, speculating with VIX derivatives, and 3 chapters on different calendar spreads (a topic that I'm particularly interested in), and 2 chapters on verticals. Overall the book looks like an good read, and I will definitely review it again when I get a copy.
Ukrainian Exchange Launches Options Trading
In the news that probably will not have much practical interest to most readers, but for the sake of complete volatility coverage ... Ukrainian Exchange listed options on UX Index - the benchmark stocks index of the Ukraine. I could not find volume information but open interest is high - higher than GVZ, for example. Implied volatility for June is about 35-40, and more or less closely follows the RTSVX - the Russian Volatility Index.
Press release here, exchange's website here.
Press release here, exchange's website here.
VIX Expiration
VIX setteled this morning surprisingly high, at 18.02, up 3.16 since previous expiration. This value was higher than 16.51 I forecasted a month ago. VSTOXX closed at 22.36, also higher than my forecast of 20.60. It seems that short-term options premiums have collapsed too much, and market corrected up to the current levels. My forecasts for the next VIX expiration, June 15th, 2011 are
This is the first expiration for GVZ at 18.79. The volume remains very light - May futures had total volume of 31 and open interest of 24 contracts. Open interest in May options is 35 contracts. I will not be forecasting GVZ this time.
- VIX at 18.83 vs 18.70 in futures
- VSTOXX at 23.63 vs 23.75 in futures
This is the first expiration for GVZ at 18.79. The volume remains very light - May futures had total volume of 31 and open interest of 24 contracts. Open interest in May options is 35 contracts. I will not be forecasting GVZ this time.
Week in Volatility
In the equities market it was another relatively quite week with major indexes almost unchanged. Volume in GVZ options this week was only 4 contracts. That does not look good for the product. As I mentioned last week with volatility this low I remain mostly on the sidelines, having drastically reduced all my positions. I don't trade silver, but I think that there interesting opportunities in volatility for SLV, AGQ, and DBS.
Commodities Volatility
Everyone is talking about recent spike in commodities volatility, some even going as far as to call it commodities crash. I calculated 30 day interpolated ATM IV for different commodity ETF, and it seems to me that it is only SLV is that is the outlier, USO vol spike is coincidental, and other commodities just have elevated volatility but not out of the norm. See for yourself.
SLV - silver
USO - oil
GLD - gold
UNG - natural gas
DBA - agricultural commodities basket holding coffee, cattle, cocoa, and sugar futures
JJC - copper
CORN - you should be able to guess this one
See more commodity volatility indexes here.
SLV - silver
USO - oil
GLD - gold
UNG - natural gas
DBA - agricultural commodities basket holding coffee, cattle, cocoa, and sugar futures
JJC - copper
CORN - you should be able to guess this one
See more commodity volatility indexes here.
A Useful Risk Index
Recently CBOE introduced SKEW index which many sources have covered and explained extensively. Despite being dubbed Black Swan index and Tail index, values of SKEW index defy simple interpretation. In what follows I hope to construct a simple and intuitive measure of risk that incorporates the same information as the SKEW.
The idea for measure is not mine; credit for inspiration belongs to Mark Sebastian from Optionspit who few weeks ago proposed an index that he called CBOP. CBOP was (originally) defined as CBOP = VIX * SKEW / 100. The index is intended to be a more complete measure of downside risk of the SPX index. By construction CBOP is increasing with both VIX and SKEW. If implied skew from the options is zero – SKEW index equals 100, and CBOP equals to the VIX, signifying no “additional” risk on the downside.
The problem with CBOP that it suffers from arbitrary construction – for example an alternative way to construct an index with the same qualities could have been VIX + SKEW – 100, or infinitely many other ways. The second problem is that it is only comparable to to its own time series, and does not have the same mathematical “meaning” as the VIX.
The solve these two issues I propose to expand on the idea of the VIX with so-called downside VIX, or put VIX, and upside VIX, or call VIX. VIX is defined as standard deviation of returns, or square root of variance, or √(E[R2]). Similarly call VIX and put VIX defined as √(E[R2|R>0]) and √(E[R2|R<0]) or standard deviation condition on return being positive or returns being negative. Just like VIX index corresponds to the value of variance swap, put VIX and call VIX correspond to values of weighted variance swaps, namely down variance swap and up variance swap.
These “partial” risk measures are not original in any way, for example Sortino ratio uses downside volatility in denominator, which is realized analogue to PUT VIX. Again, nothing above is original research, I’m just proposing to use a simple intuitive and existing concept instead of a new one.
Calculating these new indexes is trivial. VIX is calculated as weighted sum of OTM calls and puts, so by definition it contains call VIX (using only calls) and put VIX (using only puts). Please note that VIX contains an adjustment term that converts lowest call (that is ITM) to OTM put. I believe that calculating these indexes in real-time and also providing historical back-calculations should be really easy for CBOE. And as I mentioned before these indexes contain the same information as SKEW index.
But for illustration purposes I created a numerical approximation for these indexes. VIX and SKEW indexes are back-calculated from 1990 and provide information on the second and third moments. Non-zero skew imposes a lower bound on kurtosis, and these can be parametrized with normal inverse gaussian distribution. Then I calculate up-variance and down variance with numerical integration, so its all very straight-forward.
The image above shows the time-series of put VIX (highest line in light blue), VIX - middle, and call VIX (lowest line in dark blue) . The effect of skew is obvious, for example during 2008 VIX topped at 80, which put VIX showed higher number at over 90. Below is the YTD plot of indexes. Raw data is available here.
As I mentioned before, the numbers are intuitive - for example the latest set of numbers is:
call VIX = 14.66, VIX = 18.40, and put VIX 21.50. We can infer that market expects upside volatility of about 15%, but downside volatility of 22%.
The idea for measure is not mine; credit for inspiration belongs to Mark Sebastian from Optionspit who few weeks ago proposed an index that he called CBOP. CBOP was (originally) defined as CBOP = VIX * SKEW / 100. The index is intended to be a more complete measure of downside risk of the SPX index. By construction CBOP is increasing with both VIX and SKEW. If implied skew from the options is zero – SKEW index equals 100, and CBOP equals to the VIX, signifying no “additional” risk on the downside.
The problem with CBOP that it suffers from arbitrary construction – for example an alternative way to construct an index with the same qualities could have been VIX + SKEW – 100, or infinitely many other ways. The second problem is that it is only comparable to to its own time series, and does not have the same mathematical “meaning” as the VIX.
The solve these two issues I propose to expand on the idea of the VIX with so-called downside VIX, or put VIX, and upside VIX, or call VIX. VIX is defined as standard deviation of returns, or square root of variance, or √(E[R2]). Similarly call VIX and put VIX defined as √(E[R2|R>0]) and √(E[R2|R<0]) or standard deviation condition on return being positive or returns being negative. Just like VIX index corresponds to the value of variance swap, put VIX and call VIX correspond to values of weighted variance swaps, namely down variance swap and up variance swap.
These “partial” risk measures are not original in any way, for example Sortino ratio uses downside volatility in denominator, which is realized analogue to PUT VIX. Again, nothing above is original research, I’m just proposing to use a simple intuitive and existing concept instead of a new one.
Calculating these new indexes is trivial. VIX is calculated as weighted sum of OTM calls and puts, so by definition it contains call VIX (using only calls) and put VIX (using only puts). Please note that VIX contains an adjustment term that converts lowest call (that is ITM) to OTM put. I believe that calculating these indexes in real-time and also providing historical back-calculations should be really easy for CBOE. And as I mentioned before these indexes contain the same information as SKEW index.
But for illustration purposes I created a numerical approximation for these indexes. VIX and SKEW indexes are back-calculated from 1990 and provide information on the second and third moments. Non-zero skew imposes a lower bound on kurtosis, and these can be parametrized with normal inverse gaussian distribution. Then I calculate up-variance and down variance with numerical integration, so its all very straight-forward.
The image above shows the time-series of put VIX (highest line in light blue), VIX - middle, and call VIX (lowest line in dark blue) . The effect of skew is obvious, for example during 2008 VIX topped at 80, which put VIX showed higher number at over 90. Below is the YTD plot of indexes. Raw data is available here.
As I mentioned before, the numbers are intuitive - for example the latest set of numbers is:
call VIX = 14.66, VIX = 18.40, and put VIX 21.50. We can infer that market expects upside volatility of about 15%, but downside volatility of 22%.
Using New Volatility Products
EDIT: This event has been moved to May 27th 26th!
There is a meetup event in New York tomorrow focused on volatility ETFs. I have never attended these events, but this one looks really interesting and I'm going. 7 pm, ISE buiding at 60 Broad St., 26th floor New York, NY
From the event's description:
“It is no longer a VIX centric universe. Large banks are pushing vol products that are trying to capture the attention of options traders. Those products include VXX, VXZ, XXV, Volatility SKEW index, and others. How can we use them?”
There is a meetup event in New York tomorrow focused on volatility ETFs. I have never attended these events, but this one looks really interesting and I'm going. 7 pm, ISE buiding at 60 Broad St., 26th floor New York, NY
From the event's description:
“It is no longer a VIX centric universe. Large banks are pushing vol products that are trying to capture the attention of options traders. Those products include VXX, VXZ, XXV, Volatility SKEW index, and others. How can we use them?”
Mixture of Normal, Formulas for Skew and Kurtosis
Mixture of normal distributions is a simple and intuitive way to create distributions with skew and kurtosis by mixing or adding two or more normal distributions. These are well known formulas, but for some reason every time I need to use them I have trouble locating them. So here they are for everyone's reference.
Define cdf and pdf as:
=\sum_{i=1}^{n}{p_i}\Phi%20(\frac{x-\mu_i}{\sigma_i})%20\\%20f(x)=\sum_{i=1}^{n}{p_i}\phi%20(x;\mu_i,\sigma_i^2)%20\\%20\phi%20(x;\mu_i,\sigma_i^2)=\frac{1}{\sqrt{2\pi}\sigma_i}e^{-\frac{(x-\mu_i)^2}{2\sigma_i^2}}%20\\%200\leq%20p_i%20\leq%201,%20\sum_{i=1}^n{p_i}=1 "\dpi{150} \bg_black \dpi{150} \bg_black \\ F(x)=\sum_{i=1}^{n}{p_i}\Phi (\frac{x-\mu_i}{\sigma_i}) \\ f(x)=\sum_{i=1}^{n}{p_i}\phi (x;\mu_i,\sigma_i^2) \\ \phi (x;\mu_i,\sigma_i^2)=\frac{1}{\sqrt{2\pi}\sigma_i}e^{-\frac{(x-\mu_i)^2}{2\sigma_i^2}} \\ 0\leq p_i \leq 1, \sum_{i=1}^n{p_i}=1")
Then moments of x are:
-\mu^2%20\\%20skew%20=%20\frac{1}{\sigma^3}\sum_{i=1}^{n}{p_i}(\mu_i-\mu)[3\sigma_i^2+(\mu_i-\mu)^2]%20\\%20kurtosis%20=%20\frac{1}{\sigma^4}\sum_{i=1}^{n}{p_i}[3\sigma_i^4+6(\mu_i-\mu)^2\sigma_i^2+(\mu_i-\mu)^4] "\dpi{150} \bg_black \\ \mu=\sum_{i=1}^{n}{p_i}\mu_i \\ \sigma^2 = \sum_{i=1}^{n}{p_i}(\sigma_i^2+\mu_i^2)-\mu^2 \\ skew = \frac{1}{\sigma^3}\sum_{i=1}^{n}{p_i}(\mu_i-\mu)[3\sigma_i^2+(\mu_i-\mu)^2] \\ kurtosis = \frac{1}{\sigma^4}\sum_{i=1}^{n}{p_i}[3\sigma_i^4+6(\mu_i-\mu)^2\sigma_i^2+(\mu_i-\mu)^4]")
Source: Modeling and Generating Daily Changes in Market Variables Using A Multivariate Mixture of Normal Distributions. My special thanks to Dr. Jin Wang for his correction to the skew formula.
Define cdf and pdf as:
Then moments of x are:
Source: Modeling and Generating Daily Changes in Market Variables Using A Multivariate Mixture of Normal Distributions. My special thanks to Dr. Jin Wang for his correction to the skew formula.
Week in Volatility
This was a very boring week for equity indexes, with volatility practically unchanged for VIX and VSTOXX. My trading account is practically flat now because of all the volatility that I bought back. However gold rose 3% for the week, pushing GVZ from 15.70 last week to 20.25 on Friday, its highest closing value this year. Futures rose over 2 points, and implied volatility in GVZ contracts rose as well. As guys in Options Pit explained here, unlike VIX that has negative correlation with SPX levels, GVZ having both call and put skew could rise with the GLD, just like we saw this week. GV futures traded 27 contracts this week, and options traded 38.
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